Complex Functional Transforms and Their Applications

concave domains in CPn. Funktsional. Anal. i Prilozhen. 1978;**12**(4):6-23

*Advances in Complex Analysis and Applications*

[17] Gel'fand IM, Graev MI, Shapiro IIP. Generalized Functions. New York, USA:

residues", Transactions of the American Mathematical Society. 1982. pp. 308-316

[19] Derbyshire S. An illustration to Cauchy's integral formula in complex analysis. Available from: http://en. wikipedia.org/wiki/File:ComplexResid

[20] Bulnes F. Cohomology of moduli spaces in differential operators

classification to the field theory (II). In: Proceedings of FSDONA-11 (Function Spaces, Differential Operators and Non-linear Analysis, 2011); Tabarz Thur. Vol. 1, No. 12; 2011. pp. 1-22

Academic Press; 1952

uesExample.png

**98**

[18] S. Huggett and Singer, "Cohomology of contours and

**Chapter 7**

**Abstract**

**1. Introduction**

**101**

Wavelet Analysis

*Peng Du, Haibao Hu and Xiao Huang*

defined at last to prove the validity of our extraction procedure.

significance to undercover the physics and to realize flow control.

Extracting Coherent Structures in

To analyze the properties of the coherent structures in near-wall turbulence, an extraction method based on wavelet transform (WT) and a verification procedure based on correlation analysis are proposed in this work. The flow field of the turbulent boundary layer is measured using the hot-film anemometer in a gravitational low-speed water tunnel. The obtained velocity profile and turbulence intensity are validated with traditional boundary layer theory. The fluctuating velocities at three testing positions are analyzed. Using the power spectrum density (PSD) and WT, coherent and incoherent parts of the near-wall turbulence are extracted and analyzed. The probability density functions (PDFs) of the extracted signals indicate that the incoherent structures of turbulence obey the Gaussian distribution, while the coherent structures deviate from it. The PDFs of coherent structures and original turbulence signals are similar, which means that coherent structures make the most contributions to the turbulence entrainment. A correlation parameter is

**Keywords:** coherent structure, wavelet transform, correlation analysis, turbulence

Turbulence is a commonly seen but very complicated phenomenon in nature. Numerous tests have proven that turbulence is not a pure random process but contains different scales of fluctuations called coherent structures [1–3]. These structures significantly contribute to fluid entrainment and mass, momentum, and heat transfer [4, 5]. Therefore, investigating the coherent structures is of great

Among the techniques of turbulence analysis, wavelet transform has been proven feasible and power to detect and extract the coherent structures in turbulence [6–9]. Early works are based on continuous wavelet transform (CWT). Liandrant [10] and Jiang [11, 12] proposed the maximum energy principle, which considered the signal at the maximum energy scale as the burst events in turbulence. Kim [13] identified the coherent structure around a vibrating cantilever based on CWT. However, a drawback of CWT is that it is unable to reconstruct the signal if the mother wavelet is not orthogonal [14–16]. To solve this problem, Longo [17] used the multiresolution analysis technique based on the discrete wavelet transform (DWT) and extracted the structures in turbulence. DWT has evident

Near-Wall Turbulence Based on

#### **Chapter 7**

## Extracting Coherent Structures in Near-Wall Turbulence Based on Wavelet Analysis

*Peng Du, Haibao Hu and Xiao Huang*

#### **Abstract**

To analyze the properties of the coherent structures in near-wall turbulence, an extraction method based on wavelet transform (WT) and a verification procedure based on correlation analysis are proposed in this work. The flow field of the turbulent boundary layer is measured using the hot-film anemometer in a gravitational low-speed water tunnel. The obtained velocity profile and turbulence intensity are validated with traditional boundary layer theory. The fluctuating velocities at three testing positions are analyzed. Using the power spectrum density (PSD) and WT, coherent and incoherent parts of the near-wall turbulence are extracted and analyzed. The probability density functions (PDFs) of the extracted signals indicate that the incoherent structures of turbulence obey the Gaussian distribution, while the coherent structures deviate from it. The PDFs of coherent structures and original turbulence signals are similar, which means that coherent structures make the most contributions to the turbulence entrainment. A correlation parameter is defined at last to prove the validity of our extraction procedure.

**Keywords:** coherent structure, wavelet transform, correlation analysis, turbulence

#### **1. Introduction**

Turbulence is a commonly seen but very complicated phenomenon in nature. Numerous tests have proven that turbulence is not a pure random process but contains different scales of fluctuations called coherent structures [1–3]. These structures significantly contribute to fluid entrainment and mass, momentum, and heat transfer [4, 5]. Therefore, investigating the coherent structures is of great significance to undercover the physics and to realize flow control.

Among the techniques of turbulence analysis, wavelet transform has been proven feasible and power to detect and extract the coherent structures in turbulence [6–9]. Early works are based on continuous wavelet transform (CWT). Liandrant [10] and Jiang [11, 12] proposed the maximum energy principle, which considered the signal at the maximum energy scale as the burst events in turbulence. Kim [13] identified the coherent structure around a vibrating cantilever based on CWT. However, a drawback of CWT is that it is unable to reconstruct the signal if the mother wavelet is not orthogonal [14–16]. To solve this problem, Longo [17] used the multiresolution analysis technique based on the discrete wavelet transform (DWT) and extracted the structures in turbulence. DWT has evident

#### *Advances in Complex Analysis and Applications*

advantages compared with CWT since it is invertible and multi-scaled scales can be analyzed. Kadoch [18] combined DWT and direct numerical simulation (DNS), whose results proved that coherent structures preserve the vortical structures with only about 4% of the wavelet coefficients but retain 99.9% of the turbulence energy. flow to measure the flow field. A coordinate frame was used to move the probe in the vertical direction, with a precision of 0.01 *mm*. During the experiment, the

By using the experimental setups in **Figure 2**, the flow velocity of the turbulence boundary layer was measured at a series of positions in the vertical direction. The mean velocity profile and the turbulence intensity distribution at the water speed 0.4 *m=s* can are analyzed in **Figures 3** and **4**, which agree with the turbulence boundary

sampling frequency and sampling time were set to 50 *kHz* and 10.24 *s*.

*Extracting Coherent Structures in Near-Wall Turbulence Based on Wavelet Analysis*

**2.2 Verification of turbulent boundary layer flow**

*DOI: http://dx.doi.org/10.5772/intechopen.92015*

**Figure 3.**

**Figure 4.**

**103**

*The profile of mean velocity at the water speed 0.4 m=s.*

*Turbulence intensity at the water speed 0.4 m=s.*

In this work, measurement of the turbulent boundary layer is carried out using hot-film anemometer in a gravitational low-speed water tunnel. A procedure based on the WT and correlation analysis is proposed to extract and verify the coherent and incoherent structure in turbulence.

#### **2. Experimental tests and analysis**

#### **2.1 Experimental apparatus**

A gravitational low-speed water tunnel was constructed for the experiment. The gravity generated by the water level difference drives the water flow in the tunnel, and the flow can be tested in the experimental section (**Figure 1**). A maximum water speed of 2.0 *m=s* can be reached, and the turbulence intensity is less than 2%. The IFA300 hot-wire anemometer was used to measure the turbulence boundary layer flow at a series of positions in the vertical direction (**Figure 2**). Detailed setups in the experimental section can be observed in **Figure 2**. A probe penetrates into the

#### **Figure 1.**

*Sketch (left) and photo (right) of the gravitational low-speed water tunnel. (1) Water tank; (2) stabilization section; (3) contraction section; (4) experimental section; (5) electromagnetism flowmeter; (6) water level observation section; (7) water pump; (8) water storage basin; (9) switch valve.*

#### **Figure 2.**

*Sketch of experimental setups. (1) PC; (2) hot-film anemometer; (3) coordinate frame; (4) experimental plate; (5) hot-film probe.*

*Extracting Coherent Structures in Near-Wall Turbulence Based on Wavelet Analysis DOI: http://dx.doi.org/10.5772/intechopen.92015*

flow to measure the flow field. A coordinate frame was used to move the probe in the vertical direction, with a precision of 0.01 *mm*. During the experiment, the sampling frequency and sampling time were set to 50 *kHz* and 10.24 *s*.

#### **2.2 Verification of turbulent boundary layer flow**

advantages compared with CWT since it is invertible and multi-scaled scales can be analyzed. Kadoch [18] combined DWT and direct numerical simulation (DNS), whose results proved that coherent structures preserve the vortical structures with only about 4% of the wavelet coefficients but retain 99.9% of the turbulence energy. In this work, measurement of the turbulent boundary layer is carried out using hot-film anemometer in a gravitational low-speed water tunnel. A procedure based on the WT and correlation analysis is proposed to extract and verify the coherent

A gravitational low-speed water tunnel was constructed for the experiment. The gravity generated by the water level difference drives the water flow in the tunnel, and the flow can be tested in the experimental section (**Figure 1**). A maximum water speed of 2.0 *m=s* can be reached, and the turbulence intensity is less than 2%. The IFA300 hot-wire anemometer was used to measure the turbulence boundary layer flow at a series of positions in the vertical direction (**Figure 2**). Detailed setups in the experimental section can be observed in **Figure 2**. A probe penetrates into the

*Sketch (left) and photo (right) of the gravitational low-speed water tunnel. (1) Water tank; (2) stabilization section; (3) contraction section; (4) experimental section; (5) electromagnetism flowmeter; (6) water level*

*Sketch of experimental setups. (1) PC; (2) hot-film anemometer; (3) coordinate frame; (4) experimental*

*observation section; (7) water pump; (8) water storage basin; (9) switch valve.*

and incoherent structure in turbulence.

*Advances in Complex Analysis and Applications*

**2. Experimental tests and analysis**

**2.1 Experimental apparatus**

**Figure 1.**

**Figure 2.**

**102**

*plate; (5) hot-film probe.*

By using the experimental setups in **Figure 2**, the flow velocity of the turbulence boundary layer was measured at a series of positions in the vertical direction. The mean velocity profile and the turbulence intensity distribution at the water speed 0.4 *m=s* can are analyzed in **Figures 3** and **4**, which agree with the turbulence boundary

**Figure 3.** *The profile of mean velocity at the water speed 0.4 m=s.*

**Figure 4.** *Turbulence intensity at the water speed 0.4 m=s.*

layer theory. This means that the flow field in the experimental section is fully developed. Our setups and techniques are ready for turbulent boundary layer tests.

#### **3. Theoretical background of wavelet transform**

WT is a mapping of a time function, in a one-dimensional case, to the two dimensional time-scale joint representation. The temporal aspect of the signal can be preserved. The wavelet transform provides multiresolution analysis with dilated windows. The high-frequency part of the signal is analyzed using narrow windows, and the low-frequency part is done using wide windows. WT decomposes the signal into different frequency components and then studies each component with a resolution matched to its scale. It has advantages over traditional Fourier methods in analyzing physics where the signal contains discontinuities and sharp spikes.

WT of a signal *s t*ð Þ is defined as the integral transform of *<sup>ψ</sup><sup>a</sup>*,*<sup>b</sup>* <sup>¼</sup> <sup>1</sup>ffiffi *<sup>a</sup>* <sup>p</sup> *<sup>ψ</sup> <sup>t</sup>*�*<sup>b</sup> a* � �, which can be expressed as:

$$\begin{split} \mathcal{W}\_f(a,b) &= \int\_{-\infty}^{\infty} s(t) \varphi\_{a,b} dt \\ &= \int\_{-\infty}^{\infty} s(t) \frac{1}{\sqrt{a}} \mathcal{W}\left(\frac{t-b}{a}\right) dt \end{split} \tag{1} $$

*s t*ðÞ¼ <sup>X</sup><sup>∞</sup>

<sup>¼</sup> <sup>X</sup><sup>∞</sup> *n*¼�∞

<sup>¼</sup> <sup>X</sup><sup>∞</sup> *n*¼�∞

into two subparts:

**Figure 5.**

**105**

*Sketch of the multiresolution analysis.*

statistically independent.

<sup>¼</sup> *Am*<sup>0</sup> <sup>þ</sup>X*<sup>m</sup>*

*m*¼1 *Dm*

is the details of the signal *s t*ð Þ [15, 20].

where ~*s* is the coherent part and *s*

lence signal when *m* ¼ 0 (i.e., *a* ¼ 1), *f* ¼ *f* <sup>0</sup>.

*m*¼*m*0þ1

X∞ *n*¼�∞

*DOI: http://dx.doi.org/10.5772/intechopen.92015*

<sup>&</sup>lt;*s*, *<sup>ψ</sup> <sup>m</sup>*,*<sup>n</sup>* <sup>&</sup>gt;*<sup>ψ</sup> <sup>m</sup>*,*n*ðÞþ*<sup>t</sup>* <sup>X</sup>*m*<sup>0</sup>

*Extracting Coherent Structures in Near-Wall Turbulence Based on Wavelet Analysis*

*m*¼�∞

*m*¼�∞

X∞ *n*¼�∞

where *ψ <sup>m</sup>*,*<sup>n</sup>*ð Þ*t* is the wavelet function. *φ<sup>m</sup>*,*<sup>n</sup>*ð Þ*t* can be viewed as a low-pass filter, while *ψ <sup>m</sup>*,*<sup>n</sup>*ð Þ*t* as a band-pass filter. The first part of the above equation is the lowfrequency approximation of the signal *s t*ð Þ at the scale 2�*<sup>m</sup>*0; the high-frequency part

For turbulence, the fluctuating velocity of turbulence can be normally divided

*s* ¼ ~*s* þ *s*

By adopting the multiresolution analysis (**Figure 5**), the turbulence signal *s t*ð Þ

reconstructed in a selected frequency domain. Other redundant signals can then be eliminated. Therefore, the frequency range determination and localization of the coherent structures are critical in this process. The frequency can be determined as:

> *<sup>f</sup>* <sup>¼</sup> *<sup>f</sup> <sup>c</sup> fi a*

where *fs* and *f <sup>c</sup>* are the sampling frequency and the central frequency of a particular wavelet basis. a is the scale, denoted as 2*<sup>m</sup>* (*m* is a particular level of decomposition) in OWT. It represents the original frequency range of the turbu-

can be divided into different frequencies. Coherent structures can thus be

<sup>&</sup>lt;*s*, *<sup>φ</sup>m*,*<sup>n</sup>* <sup>&</sup>gt;*φm*0,*n*ðÞþ*<sup>t</sup>* <sup>X</sup>*m*<sup>0</sup>

*am*<sup>0</sup> ½ � *<sup>n</sup> <sup>φ</sup>m*0,*n*ðÞþ*<sup>t</sup>* <sup>X</sup>*m*<sup>0</sup>

*m*¼�∞

*dm*½ � *n ψ <sup>m</sup>*,*n*ð Þ*t*

X∞ *n*¼�∞

X∞ *n*¼�∞

<*s*, *ψ <sup>m</sup>*,*<sup>n</sup>* >*ψ <sup>m</sup>*,*n*ð Þ*t*

<*s*, *ψ <sup>m</sup>*,*<sup>n</sup>* >*ψ <sup>m</sup>*,*n*ð Þ*t*

<sup>0</sup> (6)

<sup>0</sup> is the incoherent part. The signals ~*s* and *s*

(5)

<sup>0</sup> are

(7)

where *φ<sup>m</sup>*,*<sup>n</sup>*ð Þ*t* is the scaling function, which is defined as *φ<sup>m</sup>*,*<sup>n</sup>*ðÞ¼ *t* 2�*m=*<sup>2</sup> *<sup>φ</sup>* <sup>2</sup>�*<sup>m</sup>* ð Þ *<sup>t</sup>* � *<sup>n</sup>* , and where *<sup>a</sup>* and *<sup>b</sup>* are the scale and position [19].

Scale a and position b should be discretized for applications. Usually we choose *<sup>a</sup>* <sup>¼</sup> <sup>2</sup>*<sup>m</sup>*ð Þ *<sup>m</sup>* <sup>∈</sup> *<sup>Z</sup>*, *<sup>a</sup>*<sup>0</sup> <sup>&</sup>gt; <sup>1</sup> , *<sup>b</sup>* <sup>¼</sup> *<sup>n</sup>* � <sup>2</sup>*<sup>m</sup>*ð Þ *<sup>b</sup>*<sup>0</sup> <sup>&</sup>gt; 0, *<sup>n</sup>* <sup>∈</sup>*<sup>Z</sup>* . When *<sup>ψ</sup>*ð Þ*<sup>t</sup>* obeys the orthogonal condition Ð *ψ <sup>m</sup>*,*<sup>n</sup>*ð Þ*t ψ <sup>m</sup>*<sup>0</sup> ,*<sup>n</sup>*0ð Þ*t dt* ¼ *δ<sup>m</sup>*,*m*0*δ<sup>n</sup>*,*n*0, the functions of the orthogonal basis can be written as:

$$\begin{split} \boldsymbol{\Psi}\_{m,n}(t) &= \frac{1}{\sqrt{2^m}} \boldsymbol{\Psi} \left( \frac{t - n \cdot 2^m}{2^n} \right) \\ &= 2^{-m/2} \boldsymbol{\Psi} (2^{-m} t - n) \end{split} \tag{2}$$

The corresponding DWT can be expressed as:

$$\begin{split} \langle s, \psi\_{m,n} > &= 2^{-m/2} \Big|\_{ -\infty }^{ \infty } s(t) \psi\_{m,n}(t) dt \\ &= 2^{-m/2} \Big|\_{ -\infty }^{ \infty } s(t) \psi(2^{-m}t - n) dt \end{split} \tag{3}$$

The orthogonality of *ψ <sup>m</sup>*,*<sup>n</sup>*ð Þ*t* eliminates the relevance between the points in wavelet space because of redundancy. The analyzing result of WT can thus reflect the characteristics of the original signal. Based on OWT, the signal *s t*ð Þ can be written as:

$$\kappa(t) = \sum\_{-\infty}^{\infty} \sum\_{-\infty}^{\infty} < \varsigma , \psi\_{m,n} > \psi\_{m,n}(t) \tag{4}$$

By choosing the scale *m*<sup>0</sup> as the critical value, the signal *s t*ð Þ can be divided into the approximate and detailed parts:

*Extracting Coherent Structures in Near-Wall Turbulence Based on Wavelet Analysis DOI: http://dx.doi.org/10.5772/intechopen.92015*

$$\begin{aligned} \varphi(t) &= \sum\_{m=m\_0+1}^{\infty} \sum\_{n=-\infty}^{\infty} < s, \boldsymbol{\mathcal{w}}\_{m,n} > \boldsymbol{\mathcal{w}}\_{m,n}(t) + \sum\_{m=-\infty}^{m\_0} \sum\_{n=-\infty}^{\infty} < s, \boldsymbol{\mathcal{w}}\_{m,n} > \boldsymbol{\mathcal{w}}\_{m,n}(t) \\ &= \sum\_{n=-\infty}^{\infty} < s, \boldsymbol{\mathcal{q}}\_{m,n} > \boldsymbol{\mathcal{q}}\_{m\_0,n}(t) + \sum\_{m=-\infty}^{m\_0} \sum\_{n=-\infty}^{\infty} < s, \boldsymbol{\mathcal{w}}\_{m,n} > \boldsymbol{\mathcal{w}}\_{m,n}(t) \\ &= \sum\_{n=-\infty}^{\infty} \boldsymbol{d}\_{m\_0}[n] \boldsymbol{\varrho}\_{m,n}(t) + \sum\_{m=-\infty}^{m\_0} \sum\_{n=-\infty}^{\infty} \boldsymbol{d}\_m[n] \boldsymbol{\upmu}\_{m,n}(t) \\ &= \boldsymbol{A}\_{m\_0} + \sum\_{m=1}^{m} \boldsymbol{D}\_m \end{aligned} \tag{5}$$

where *ψ <sup>m</sup>*,*<sup>n</sup>*ð Þ*t* is the wavelet function. *φ<sup>m</sup>*,*<sup>n</sup>*ð Þ*t* can be viewed as a low-pass filter, while *ψ <sup>m</sup>*,*<sup>n</sup>*ð Þ*t* as a band-pass filter. The first part of the above equation is the lowfrequency approximation of the signal *s t*ð Þ at the scale 2�*<sup>m</sup>*0; the high-frequency part is the details of the signal *s t*ð Þ [15, 20].

For turbulence, the fluctuating velocity of turbulence can be normally divided into two subparts:

$$
\mathfrak{s} = \tilde{\mathfrak{s}} + \mathfrak{s}'\tag{6}
$$

where ~*s* is the coherent part and *s* <sup>0</sup> is the incoherent part. The signals ~*s* and *s* <sup>0</sup> are statistically independent.

By adopting the multiresolution analysis (**Figure 5**), the turbulence signal *s t*ð Þ can be divided into different frequencies. Coherent structures can thus be reconstructed in a selected frequency domain. Other redundant signals can then be eliminated. Therefore, the frequency range determination and localization of the coherent structures are critical in this process. The frequency can be determined as:

$$f = \frac{f\_c f\_i}{a} \tag{7}$$

where *fs* and *f <sup>c</sup>* are the sampling frequency and the central frequency of a particular wavelet basis. a is the scale, denoted as 2*<sup>m</sup>* (*m* is a particular level of decomposition) in OWT. It represents the original frequency range of the turbulence signal when *m* ¼ 0 (i.e., *a* ¼ 1), *f* ¼ *f* <sup>0</sup>.

**Figure 5.** *Sketch of the multiresolution analysis.*

layer theory. This means that the flow field in the experimental section is fully developed. Our setups and techniques are ready for turbulent boundary layer tests.

WT is a mapping of a time function, in a one-dimensional case, to the two dimensional time-scale joint representation. The temporal aspect of the signal can be preserved. The wavelet transform provides multiresolution analysis with dilated windows. The high-frequency part of the signal is analyzed using narrow windows, and the low-frequency part is done using wide windows. WT decomposes the signal into different frequency components and then studies each component with a resolution matched to its scale. It has advantages over traditional Fourier methods in analyzing physics where the signal contains discontinuities and sharp spikes.

WT of a signal *s t*ð Þ is defined as the integral transform of *<sup>ψ</sup><sup>a</sup>*,*<sup>b</sup>* <sup>¼</sup> <sup>1</sup>ffiffi

ð<sup>∞</sup> �∞

¼ ð<sup>∞</sup> �∞

where *φ<sup>m</sup>*,*<sup>n</sup>*ð Þ*t* is the scaling function, which is defined as *φ<sup>m</sup>*,*<sup>n</sup>*ðÞ¼ *t*

*<sup>φ</sup>* <sup>2</sup>�*<sup>m</sup>* ð Þ *<sup>t</sup>* � *<sup>n</sup>* , and where *<sup>a</sup>* and *<sup>b</sup>* are the scale and position [19].

1 ffiffiffiffiffi <sup>2</sup>*<sup>m</sup>* <sup>p</sup> *<sup>ψ</sup>*

<sup>¼</sup> <sup>2</sup>�*m=*<sup>2</sup>

<sup>¼</sup> <sup>2</sup>�*m=*<sup>2</sup>

ð<sup>∞</sup> �∞

ð<sup>∞</sup> �∞

The orthogonality of *ψ <sup>m</sup>*,*<sup>n</sup>*ð Þ*t* eliminates the relevance between the points in wavelet space because of redundancy. The analyzing result of WT can thus reflect the characteristics of the original signal. Based on OWT, the signal *s t*ð Þ can be

By choosing the scale *m*<sup>0</sup> as the critical value, the signal *s t*ð Þ can be divided into

*ψ <sup>m</sup>*,*<sup>n</sup>*ðÞ¼ *t*

<sup>&</sup>lt;*s*, *<sup>ψ</sup> <sup>m</sup>*,*<sup>n</sup>* <sup>&</sup>gt; <sup>¼</sup> <sup>2</sup>�*m=*<sup>2</sup>

*s t*ðÞ¼ <sup>X</sup><sup>∞</sup>

�∞

X∞ �∞

The corresponding DWT can be expressed as:

*s t*ð Þ*ψ<sup>a</sup>*,*bdt*

*t* � *b a* � �

,*<sup>n</sup>*0ð Þ*t dt* ¼ *δ<sup>m</sup>*,*m*0*δ<sup>n</sup>*,*n*0, the functions of the orthogonal basis can

*<sup>t</sup>* � *<sup>n</sup>* � <sup>2</sup>*<sup>m</sup>* 2*m* � �

*<sup>ψ</sup>* <sup>2</sup>�*<sup>m</sup>* ð Þ *<sup>t</sup>* � *<sup>n</sup>*

*s t*ð Þ*ψ <sup>m</sup>*,*<sup>n</sup>*ð Þ*t dt*

*s t*ð Þ*<sup>ψ</sup>* <sup>2</sup>�*<sup>m</sup>* ð Þ *<sup>t</sup>* � *<sup>n</sup> dt*

<*s*, *ψ <sup>m</sup>*,*<sup>n</sup>* >*ψ <sup>m</sup>*,*<sup>n</sup>*ð Þ*t* (4)

*dt*

*s t*ð Þ <sup>1</sup> ffiffiffi *<sup>a</sup>* <sup>p</sup> *<sup>ψ</sup>*

Scale a and position b should be discretized for applications. Usually we choose *<sup>a</sup>* <sup>¼</sup> <sup>2</sup>*<sup>m</sup>*ð Þ *<sup>m</sup>* <sup>∈</sup> *<sup>Z</sup>*, *<sup>a</sup>*<sup>0</sup> <sup>&</sup>gt; <sup>1</sup> , *<sup>b</sup>* <sup>¼</sup> *<sup>n</sup>* � <sup>2</sup>*<sup>m</sup>*ð Þ *<sup>b</sup>*<sup>0</sup> <sup>&</sup>gt; 0, *<sup>n</sup>* <sup>∈</sup>*<sup>Z</sup>* . When *<sup>ψ</sup>*ð Þ*<sup>t</sup>* obeys the orthogonal

*W <sup>f</sup>*ð Þ¼ *a*, *b*

*<sup>a</sup>* <sup>p</sup> *<sup>ψ</sup> <sup>t</sup>*�*<sup>b</sup> a* � �, which

(1)

(2)

(3)

**3. Theoretical background of wavelet transform**

*Advances in Complex Analysis and Applications*

can be expressed as:

2�*m=*<sup>2</sup>

condition Ð

written as:

**104**

be written as:

*ψ <sup>m</sup>*,*<sup>n</sup>*ð Þ*t ψ <sup>m</sup>*<sup>0</sup>

the approximate and detailed parts:

#### **4. Extraction and verification of turbulent structures**

To extract the coherent structures in turbulence, the signals at the central area of turbulence should be selected. According to previous studies [20–22], the formation of the coherent structures in turbulence is formed in the area of 0 <*y*<sup>þ</sup> <30, and the self-sustaining of the coherent structures is in the area of 20<*y*<sup>þ</sup> <60. As a result, three testing positions with the *y*<sup>þ</sup> 20.8, 33.5, and 42.6 were selected, whose fluctuating velocity signals are shown in **Figure 6**.

#### **4.1 Preliminary evaluation of coherent structures**

For preliminary evaluations of the coherent structures, CWT is first utilized for the analysis. CWT is a mathematical mapping similar to the Fourier transform [23, 24]. It is linear, invertible, and orthogonal. However, the Fourier transform uses basis functions, including the sines and cosines, which extend to infinity in time, while wavelet basis functions drop towards zero outside a finite domain (compact support). This allows for an effective localization in both time and frequency. CWT uses inner products to measure the similarity between the turbulence signal and the wavelet function, which defines a mapping between the two. CWT compares the turbulence signal to shifted and compressed/stretched versions of the wavelet function. Compressing/stretching is also referred to as dilation or scaling and corresponds to the physical notion of scale. By continuously varying the values of the scale parameter, *a*, and the position parameter, *b*, one can obtain the CWT coefficients at last.

**4.2 Extraction of coherent structures**

*DOI: http://dx.doi.org/10.5772/intechopen.92015*

**Figure 7.**

coherent structures in turbulence [9, 25, 26].

structures, which is calculated by:

**4.3 Verification of extracted signals**

ute the most to turbulence entrainment.

**107**

To obtain the frequency range of coherent structures in turbulence, power spectrum densities of the three selected signals were calculated in **Figure 8**. The centralized frequencies of the coherent structures are found in the range 0 � 230 *Hz*, 0 � 240 *Hz*, 0 � 230 *Hz*. The rapid attenuation of PSD demonstrates the low noise of our experimental system. Multiresolution analysis of OWT was used to extract the

*Continuous wavelet transform coefficients at (a) y*<sup>þ</sup> ¼ 20*:*8*, (b) y*<sup>þ</sup> ¼ 33*:*5*, (c) y*<sup>þ</sup> ¼ 42*:*6*.*

*Extracting Coherent Structures in Near-Wall Turbulence Based on Wavelet Analysis*

WT of a signal is equivalent to local cross-correlation analysis between the signal

Using the multiresolution analysis of OWT, the turbulence signal was split into seven scales as in **Table 1**, which eliminates most of the redundant signals. The frequency range of the approximate signal is mainly in the range 0 � 260 *Hz*, which covers most of the coherent structures. In **Table 1**, the coherent structures are

The extracted signals of each level are shown in **Figure 9**, where "A7" is the approximate signal, i.e., the coherent structures; where "sD7" is the incoherent

and where "s" is the original signal. "D1 � D7" are the detailed signals of each level.

To characterize the properties of the extracted signals, the probability density functions (PDFs) were analyzed in **Figure 10**. It can be observed that the incoherent structures are approximately Gaussian, demonstrating isotropic characteristics. The PDFs of coherent structures deviate from the Gaussian distribution, presenting strong anisotropic characteristics. And the PDFs of the coherent structures resemble that of the original turbulence signals. This means that coherent structures contrib-

sD7 ¼ s � A7 ¼ D1 þ D2 þ ⋯ þ D7, (8)

and wavelet function. OWT carries out the multi-resolution analysis for both decomposition and reconstruction of the original turbulence signal. It is thought of the wavelet coefficients as digital filters as which the original signal is passed through low-pass filters to decompose into low-frequency components and passed

through high-pass filters to analyze into high-frequency components.

found to take almost 75 % of the whole energy in turbulence.

In the work, the 5th order of Daubechies wavelet was selected as the basis function, whose central frequency fc is 0.6667 *Hz*. The calculated CWT coefficients of the three signals are shown in **Figure 7**, where the quasi-periodic structures (coherent structures) of turbulence can be clearly observed. The modulus of the wavelet coefficients shows that during the vortex breakdown, which is caused by the strong nonlinear flow instability, energy is spread over a wide range of scales. Large-scale structures exhibit anisotropic properties in the flow. Their peaks and troughs appear at the scale about 300, corresponding with a frequency of 0.6667/ 300 = 0.0022 *Hz*. These dominant scales will have the highest level of energy in turbulence. At the smaller scales, the vortices break up into intermittent small-scale features. Some organizations are evident here, with periodical and intermittent turbulent bursts. The low-frequency structures will decay to isotropic structures and dissipate in turbulence at last.

**Figure 6.** *Fluctuating velocity signals at three positions (a) y*<sup>þ</sup> ¼ 20*:*8*, (b) y*<sup>þ</sup> ¼ 33*:*5*, (c) y*<sup>þ</sup> ¼ 42*:*6*.*

*Extracting Coherent Structures in Near-Wall Turbulence Based on Wavelet Analysis DOI: http://dx.doi.org/10.5772/intechopen.92015*

**Figure 7.** *Continuous wavelet transform coefficients at (a) y*<sup>þ</sup> ¼ 20*:*8*, (b) y*<sup>þ</sup> ¼ 33*:*5*, (c) y*<sup>þ</sup> ¼ 42*:*6*.*

#### **4.2 Extraction of coherent structures**

**4. Extraction and verification of turbulent structures**

ating velocity signals are shown in **Figure 6**.

*Advances in Complex Analysis and Applications*

coefficients at last.

**Figure 6.**

**106**

and dissipate in turbulence at last.

**4.1 Preliminary evaluation of coherent structures**

To extract the coherent structures in turbulence, the signals at the central area of turbulence should be selected. According to previous studies [20–22], the formation of the coherent structures in turbulence is formed in the area of 0 <*y*<sup>þ</sup> <30, and the self-sustaining of the coherent structures is in the area of 20<*y*<sup>þ</sup> <60. As a result, three testing positions with the *y*<sup>þ</sup> 20.8, 33.5, and 42.6 were selected, whose fluctu-

For preliminary evaluations of the coherent structures, CWT is first utilized for

the analysis. CWT is a mathematical mapping similar to the Fourier transform [23, 24]. It is linear, invertible, and orthogonal. However, the Fourier transform uses basis functions, including the sines and cosines, which extend to infinity in time, while wavelet basis functions drop towards zero outside a finite domain (compact support). This allows for an effective localization in both time and frequency. CWT uses inner products to measure the similarity between the turbulence signal and the wavelet function, which defines a mapping between the two. CWT compares the turbulence signal to shifted and compressed/stretched versions of the wavelet function. Compressing/stretching is also referred to as dilation or scaling and corresponds to the physical notion of scale. By continuously varying the values of the scale parameter, *a*, and the position parameter, *b*, one can obtain the CWT

In the work, the 5th order of Daubechies wavelet was selected as the basis function, whose central frequency fc is 0.6667 *Hz*. The calculated CWT coefficients of the three signals are shown in **Figure 7**, where the quasi-periodic structures (coherent structures) of turbulence can be clearly observed. The modulus of the wavelet coefficients shows that during the vortex breakdown, which is caused by the strong nonlinear flow instability, energy is spread over a wide range of scales. Large-scale structures exhibit anisotropic properties in the flow. Their peaks and troughs appear at the scale about 300, corresponding with a frequency of 0.6667/ 300 = 0.0022 *Hz*. These dominant scales will have the highest level of energy in turbulence. At the smaller scales, the vortices break up into intermittent small-scale features. Some organizations are evident here, with periodical and intermittent turbulent bursts. The low-frequency structures will decay to isotropic structures

*Fluctuating velocity signals at three positions (a) y*<sup>þ</sup> ¼ 20*:*8*, (b) y*<sup>þ</sup> ¼ 33*:*5*, (c) y*<sup>þ</sup> ¼ 42*:*6*.*

To obtain the frequency range of coherent structures in turbulence, power spectrum densities of the three selected signals were calculated in **Figure 8**. The centralized frequencies of the coherent structures are found in the range 0 � 230 *Hz*, 0 � 240 *Hz*, 0 � 230 *Hz*. The rapid attenuation of PSD demonstrates the low noise of our experimental system. Multiresolution analysis of OWT was used to extract the coherent structures in turbulence [9, 25, 26].

WT of a signal is equivalent to local cross-correlation analysis between the signal and wavelet function. OWT carries out the multi-resolution analysis for both decomposition and reconstruction of the original turbulence signal. It is thought of the wavelet coefficients as digital filters as which the original signal is passed through low-pass filters to decompose into low-frequency components and passed through high-pass filters to analyze into high-frequency components.

Using the multiresolution analysis of OWT, the turbulence signal was split into seven scales as in **Table 1**, which eliminates most of the redundant signals. The frequency range of the approximate signal is mainly in the range 0 � 260 *Hz*, which covers most of the coherent structures. In **Table 1**, the coherent structures are found to take almost 75 % of the whole energy in turbulence.

The extracted signals of each level are shown in **Figure 9**, where "A7" is the approximate signal, i.e., the coherent structures; where "sD7" is the incoherent structures, which is calculated by:

$$\mathbf{sD7} = \mathbf{s} - \mathbf{A7} = \mathbf{D1} + \mathbf{D2} + \dots + \mathbf{D7},\tag{8}$$

and where "s" is the original signal. "D1 � D7" are the detailed signals of each level.

#### **4.3 Verification of extracted signals**

To characterize the properties of the extracted signals, the probability density functions (PDFs) were analyzed in **Figure 10**. It can be observed that the incoherent structures are approximately Gaussian, demonstrating isotropic characteristics. The PDFs of coherent structures deviate from the Gaussian distribution, presenting strong anisotropic characteristics. And the PDFs of the coherent structures resemble that of the original turbulence signals. This means that coherent structures contribute the most to turbulence entrainment.

For further validation of the extracted coherent and incoherent structures, correlation analysis was carried out here. A correlation parameter *β* between these structures was defined [27]:

$$\beta = \frac{\overline{\upsilon\_d \upsilon\_d}}{\overline{\upsilon^2}} = \frac{\frac{1}{N} \sum \upsilon\_{di} \upsilon\_{di}}{\frac{1}{N} \sum \upsilon\_i^2} \tag{9}$$

Value of *β* represents the correlation between coherent and incoherent structures. A large value of *β* means that the coherent structures are divided into detailed signals as incoherent structures, denoting an inappropriate selection of the decomposition level. If the coherent structures are correctly extracted, the correlation parameter *β* should be 0. According to Eq. (9), *β* of our three selected signals at the level 7 are �3*:*<sup>8444</sup> � <sup>10</sup>�4, 7*:*<sup>2638</sup> � <sup>10</sup>�4, and 3*:*<sup>2677</sup> � <sup>10</sup>�4, respectively, demonstrating the low correlation between the extracted coherent and incoherent structures. This

*Extracted signals in turbulence. (a) Incoherent structure (y*<sup>þ</sup> ¼ 20*:*8*); (b) coherent structure (y*<sup>þ</sup> ¼ 20*:*8*); (c) incoherent structure (y*<sup>þ</sup> ¼ 33*:*5*); (d) coherent structure (y*<sup>þ</sup> ¼ 33*:*5*); (e) incoherent structure (y*<sup>þ</sup> ¼ 42*:*6*);*

**Signal Frequency/***Hz* **Energy/%**

*Extracting Coherent Structures in Near-Wall Turbulence Based on Wavelet Analysis*

*Frequency and energy distribution of seven level decompositions.*

*DOI: http://dx.doi.org/10.5772/intechopen.92015*

**Table 1.**

**Figure 9.**

**109**

*(f) coherent structure (y*<sup>þ</sup> ¼ 42*:*6*).*

*s* 0 � 33,335 100 100 100 *A*7 0 � 260 85.6498 77.0677 81.3847 *D*7 260 � 520 0.0147 0.0200 0.0259 *D*6 520 � 1042 0.0062 0.0086 0.0129 *D*5 1042 � 2083 0.0019 0.0036 0.0042 *D*4 2083 � 4167 0.0464 0.0703 0.0868 *D*3 4167 � 83,334 0.3957 0.8099 0.8505 *D*2 83,334 � 16,668 3.5993 5.5974 5.2322 *D*1 16,668 � 33,335 10.2411 16.4225 12.4028

*y*<sup>þ</sup> ¼ **20***:***8** *y*<sup>þ</sup> ¼ **33***:***5** *y*<sup>þ</sup> ¼ **42***:***6**

proves the validity of the proposed extraction process.

where *v* is the fluctuating velocity signal. The subscripts "*a*" and "*d*" represent the coherent and incoherent structures. *N* is the number of the sampling points. *vavd* can be regarded as the stress between the coherent and incoherent structures.


*Extracting Coherent Structures in Near-Wall Turbulence Based on Wavelet Analysis DOI: http://dx.doi.org/10.5772/intechopen.92015*

#### **Table 1.**

*Frequency and energy distribution of seven level decompositions.*

#### **Figure 9.**

For further validation of the extracted coherent and incoherent structures, correlation analysis was carried out here. A correlation parameter *β* between these

> 1 *N*

where *v* is the fluctuating velocity signal. The subscripts "*a*" and "*d*" represent the coherent and incoherent structures. *N* is the number of the sampling points. *vavd* can be regarded as the stress between the coherent and incoherent structures.

P*vaivdi* 1 *N* P*v*<sup>2</sup> *i*

(9)

*<sup>β</sup>* <sup>¼</sup> *vavd <sup>v</sup>*<sup>2</sup> <sup>¼</sup>

*Power spectrum densities at the three positions (a) y*<sup>þ</sup> ¼ 20*:*8*, (b) y*<sup>þ</sup> ¼ 33*:*5*, (c) y*<sup>þ</sup> ¼ 42*:*6*.*

structures was defined [27]:

*Advances in Complex Analysis and Applications*

**Figure 8.**

**108**

*Extracted signals in turbulence. (a) Incoherent structure (y*<sup>þ</sup> ¼ 20*:*8*); (b) coherent structure (y*<sup>þ</sup> ¼ 20*:*8*); (c) incoherent structure (y*<sup>þ</sup> ¼ 33*:*5*); (d) coherent structure (y*<sup>þ</sup> ¼ 33*:*5*); (e) incoherent structure (y*<sup>þ</sup> ¼ 42*:*6*); (f) coherent structure (y*<sup>þ</sup> ¼ 42*:*6*).*

Value of *β* represents the correlation between coherent and incoherent structures. A large value of *β* means that the coherent structures are divided into detailed signals as incoherent structures, denoting an inappropriate selection of the decomposition level. If the coherent structures are correctly extracted, the correlation parameter *β* should be 0. According to Eq. (9), *β* of our three selected signals at the level 7 are �3*:*<sup>8444</sup> � <sup>10</sup>�4, 7*:*<sup>2638</sup> � <sup>10</sup>�4, and 3*:*<sup>2677</sup> � <sup>10</sup>�4, respectively, demonstrating the low correlation between the extracted coherent and incoherent structures. This proves the validity of the proposed extraction process.

coherent structures deviate from it. The similarity of the PDFs of the coherent structures and the original turbulence signal demonstrate that the coherent structures make most contributions to turbulence. A correlation parameter between coherent and incoherent structures was defined, which proves the successful sepa-

*Extracting Coherent Structures in Near-Wall Turbulence Based on Wavelet Analysis*

The authors acknowledge the support from the National Natural Science Foundation of China (Grant No. 51879218, 51679203) and Fundamental Research

Reprinted (adapted) with permission from Chinese Physics B, 2013, 22(7): 074703.

Funds for the Central Universities (Grant No. 3102018gxc007, 3102020H

ration of coherent structure from turbulence.

*DOI: http://dx.doi.org/10.5772/intechopen.92015*

The authors declare no conflict of interest.

**Acknowledgements**

**Conflict of interest**

HZY030004).

**Notes**

**Abbreviations**

**Nomenclature**

*s*

**111**

WT wavelet transform PSD power spectrum density OWT orthogonal wavelet transform PDF probability density function CWT continuous wavelet transform DWT discrete wavelet transform DNS direct numerical simulation

*s t*ð Þ original turbulence signal

~*s* coherent part of the signal

*fs* sampling frequency

<sup>0</sup> incoherent part of the signal

*y*<sup>þ</sup> dimensionless wall distance *A*7, *D*1 � *D*7 detailed signal of each level *v* fluctuating velocity signal *β* correlation parameter

*fc* central frequency of particular wavelet basis

*φ<sup>m</sup>*,*<sup>n</sup>*ð Þ*t* scaling function *a* scale parameter *b* position parameter *m*<sup>0</sup> critical scale *ψ <sup>m</sup>*,*<sup>n</sup>*ð Þ*t* wavelet function

*f* frequency

**Figure 10.** *Probability density functions at three testing positions (a) y*<sup>þ</sup> ¼ 20*:*8*, (b) y*<sup>þ</sup> ¼ 33*:*5*, (c) y*<sup>þ</sup> ¼ 42*:*6*.*

#### **5. Conclusion**

The flow field of the turbulence boundary layer was measured using hot-film anemometer in a gravitational low-speed water tunnel. The coherent and incoherent structures in turbulence were separated successfully with an extraction method based on WT. With CWT, the turbulent structures can be observed in various scales. With DWT, multiresolution analysis can be carried out for the decomposition and reconstruction of vortical structures in different scales. The PDF of the incoherent structures was found to obey the Gaussian distribution, while that of the *Extracting Coherent Structures in Near-Wall Turbulence Based on Wavelet Analysis DOI: http://dx.doi.org/10.5772/intechopen.92015*

coherent structures deviate from it. The similarity of the PDFs of the coherent structures and the original turbulence signal demonstrate that the coherent structures make most contributions to turbulence. A correlation parameter between coherent and incoherent structures was defined, which proves the successful separation of coherent structure from turbulence.

#### **Acknowledgements**

The authors acknowledge the support from the National Natural Science Foundation of China (Grant No. 51879218, 51679203) and Fundamental Research Funds for the Central Universities (Grant No. 3102018gxc007, 3102020H HZY030004).

#### **Conflict of interest**

The authors declare no conflict of interest.

#### **Notes**

Reprinted (adapted) with permission from Chinese Physics B, 2013, 22(7): 074703.

#### **Abbreviations**


#### **Nomenclature**


**5. Conclusion**

*Advances in Complex Analysis and Applications*

**Figure 10.**

**110**

The flow field of the turbulence boundary layer was measured using hot-film anemometer in a gravitational low-speed water tunnel. The coherent and incoherent structures in turbulence were separated successfully with an extraction method based on WT. With CWT, the turbulent structures can be observed in various scales. With DWT, multiresolution analysis can be carried out for the decomposition and reconstruction of vortical structures in different scales. The PDF of the incoherent structures was found to obey the Gaussian distribution, while that of the

*Probability density functions at three testing positions (a) y*<sup>þ</sup> ¼ 20*:*8*, (b) y*<sup>þ</sup> ¼ 33*:*5*, (c) y*<sup>þ</sup> ¼ 42*:*6*.*

#### **Appendix I: complex wavelet transform**

The continuous wavelet transform (CWT) has the drawback of redundancy. As the dilation parameter a and the shift parameter b take continuous values, the resulting CWT is a very redundant representation. Therefore, the discrete wavelet transform was proposed to overcome this problem by setting the scale and shift parameters on a discrete set of basis functions. Their discretization is performed by:

$$a = a\_0^j \cdots b = k a\_0^j b\_0 \cdots \text{for} \cdot j, k \in \mathbb{Z} \tag{10}$$

where *a*<sup>0</sup> >1 is the dilation and *b*<sup>0</sup> 6¼ 0 is the translation. The family of wavelets can be expressed as:

$$
\Psi \,\, \_{j,k}(t) = a\_0^{-j/2} \Psi \left( a\_0^{-j} t - kb\_0 \right) \tag{11}
$$

and the discrete wavelet decomposition of a signal *f t*ð Þ is:

$$f(t) = \sum\_{j} \sum\_{k} D\_{f}(j,k) \boldsymbol{\nu}\_{j,k}(t) \tag{12}$$

where *Df*ð Þ *j*, *k* is the DWT of the signal *f t*ð Þ. The most widely used dilation and shift parameters are *a* ¼ 2 and *b* ¼ 1.

The basis function set *ψ <sup>j</sup>*,*<sup>k</sup>* n o should be orthonormal such that:

$$D\_f(j,k) = \int\_{-\infty}^{\infty} \left. \nu \right|\_{j,k}^\*(t) f(t) dt = \left\langle \nu \right|\_{j,k}(t) f(t) \right\rangle \tag{13}$$

The advantage of the DWT is the multi-resolution analysis ability. Although the standard DWT is powerful, it has three major disadvantages that undermine its applications: shift sensitivity, poor directionality, and absence of phase information.

Complex wavelet transform can be used to overcome these drawbacks. It uses complex-valued filtering and decomposes the signal into real and imaginary parts, which can be used to calculate the amplitude and phase information.

For turbulence analysis, the complex wavelet transform should be used since the modulus of the wavelet coefficients allows characterizing the evolution of the turbulent energy in both the time and frequency domains. The real-valued wavelets will make it difficult to sort out the features of the signal or the wavelet. On the contrary, the complex-valued wavelets can eliminate these spurious oscillations. The complex extension of a real signal *f t*ð Þ can be expressed as:

$$\mathbf{x}(t) = f(t) + j\mathbf{g}(t) \tag{14}$$

**Author details**

Xi'an, P.R. China

**113**

Peng Du†, Haibao Hu\*† and Xiao Huang

† These authors are contributed equally.

provided the original work is properly cited.

\*Address all correspondence to: huhaibao@nwpu.edu.cn

School of Marine Science and Technology, Northwestern Polytechnical University,

*Extracting Coherent Structures in Near-Wall Turbulence Based on Wavelet Analysis*

*DOI: http://dx.doi.org/10.5772/intechopen.92015*

© 2020 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium,

where *g t*ð Þ is the Hilbert transform of *f t*ð Þ and is denoted as *H ft* f g ð Þ and *j* ¼ ð Þ �<sup>1</sup> <sup>1</sup>*=*<sup>2</sup> . The instantaneous frequency and amplitude of the signal *x t*ð Þ can then be calculated as:

$$\begin{aligned} \text{Magnitude} & \text{of } x(t) = \sqrt{\left(f(t)^2 + \mathbf{g}(t)^2\right)} \\ \text{Angle of } x(t) &= \tan^{-1}[\mathbf{g}(t)/f(t)] \end{aligned} \tag{15}$$

The complex wavelet transform is able to remove the redundancy for turbulence analysis where the directionality and phase information play important roles.

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### **Author details**

**Appendix I: complex wavelet transform**

*Advances in Complex Analysis and Applications*

can be expressed as:

ð Þ �<sup>1</sup> <sup>1</sup>*=*<sup>2</sup>

**112**

calculated as:

shift parameters are *a* ¼ 2 and *b* ¼ 1. The basis function set *ψ <sup>j</sup>*,*<sup>k</sup>*

*<sup>a</sup>* <sup>¼</sup> *<sup>a</sup> <sup>j</sup>*

The continuous wavelet transform (CWT) has the drawback of redundancy. As

where *a*<sup>0</sup> >1 is the dilation and *b*<sup>0</sup> 6¼ 0 is the translation. The family of wavelets

where *Df*ð Þ *j*, *k* is the DWT of the signal *f t*ð Þ. The most widely used dilation and

The advantage of the DWT is the multi-resolution analysis ability. Although the standard DWT is powerful, it has three major disadvantages that undermine its applications: shift sensitivity, poor directionality, and absence of phase information. Complex wavelet transform can be used to overcome these drawbacks. It uses complex-valued filtering and decomposes the signal into real and imaginary parts,

For turbulence analysis, the complex wavelet transform should be used since the

where *g t*ð Þ is the Hilbert transform of *f t*ð Þ and is denoted as *H ft* f g ð Þ and *j* ¼

. The instantaneous frequency and amplitude of the signal *x t*ð Þ can then be

The complex wavelet transform is able to remove the redundancy for turbulence

analysis where the directionality and phase information play important roles.

modulus of the wavelet coefficients allows characterizing the evolution of the turbulent energy in both the time and frequency domains. The real-valued wavelets will make it difficult to sort out the features of the signal or the wavelet. On the contrary, the complex-valued wavelets can eliminate these spurious oscillations.

should be orthonormal such that:

*<sup>j</sup>*,*<sup>k</sup>*ð Þ*t f t*ð Þ*dt* ¼ *ψ <sup>j</sup>*,*<sup>k</sup>*ð Þ*t f t*ð Þ

D E

*x t*ðÞ¼ *f t*ðÞþ *jg t*ð Þ (14)

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi *f t*ð Þ<sup>2</sup> <sup>þ</sup> *g t*ð Þ<sup>2</sup> r� �

½ � *g t*ð Þ*=f t*ð Þ

�*j* <sup>0</sup> *t* � *kb*<sup>0</sup> � �

�*j=*2 <sup>0</sup> *ψ a*

X *k*

<sup>0</sup>*b*0⋯for � *j*, *k*∈*Z* (10)

*Df*ð Þ *j*, *k ψ <sup>j</sup>*,*<sup>k</sup>*ð Þ*t* (12)

(11)

(13)

(15)

the dilation parameter a and the shift parameter b take continuous values, the resulting CWT is a very redundant representation. Therefore, the discrete wavelet transform was proposed to overcome this problem by setting the scale and shift parameters on a discrete set of basis functions. Their discretization is performed by:

<sup>0</sup>⋯*<sup>b</sup>* <sup>¼</sup> *ka <sup>j</sup>*

*ψ <sup>j</sup>*,*k*ðÞ¼ *t a*

and the discrete wavelet decomposition of a signal *f t*ð Þ is:

*f t*ðÞ¼ <sup>X</sup> *j*

> ð<sup>∞</sup> �∞ *ψ* <sup>∗</sup>

which can be used to calculate the amplitude and phase information.

The complex extension of a real signal *f t*ð Þ can be expressed as:

Magnitudeof *x t*ðÞ¼

Angle of *x t*ðÞ¼ tan �<sup>1</sup>

n o

*Df*ð Þ¼ *j*, *k*

Peng Du†, Haibao Hu\*† and Xiao Huang School of Marine Science and Technology, Northwestern Polytechnical University, Xi'an, P.R. China

\*Address all correspondence to: huhaibao@nwpu.edu.cn

† These authors are contributed equally.

© 2020 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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[13] Kim YH, Cierpka C, Wereley ST. Flow field around a vibrating cantilever: Coherent structure education by continuous wavelet transform and proper orthogonal decomposition. Journal of Fluid Mechanics. 2011;**669**:584-606

[14] Yoshimatsu K, Schneider K, Okamoto N, et al. Intermittency and geometrical statistics of threedimensional homogeneous magnetohydrodynamic turbulence: A wavelet viewpoint. Physics of Plasmas. 2011; **18**(9):092304

[15] Baars WJ, Talluru KM, Hutchins N, et al. Wavelet analysis of wall turbulence to study large-scale modulation of small scales. Experiments in Fluids. 2015;**56**(10):188

[16] Camussi R. Coherent structure identification from wavelet analysis of particle image velocimetry data. Experiments in Fluids. 2002;**32**(1):76-86 *Extracting Coherent Structures in Near-Wall Turbulence Based on Wavelet Analysis DOI: http://dx.doi.org/10.5772/intechopen.92015*

[17] Longo S. Turbulence under spilling breakers using discrete wavelets. Experiments in Fluids. 2003;**34**(2): 181-191

**References**

[1] Kowal G, Lazarian A. Velocity field of compressible magnetohydrodynamic turbulence: Wavelet decomposition and mode scalings. The Astrophysical

*Advances in Complex Analysis and Applications*

[9] Khujadze G, Schneider K,

022011

Oberlack M, et al. Coherent vorticity extraction in turbulent boundary layers using orthogonal wavelets. Journal of Physics: Conference Series. 2011;**318**(2):

[10] Liandrat J, Moret-Bailly F. The wavelet transform-some applications to

[11] Nan J, Jin Z. Detecting multi-scale

[12] Jian-hua L, Nan J, Zhen-dong W, et al. Multi-scale coherent structures in turbulent boundary layer detected by locally averaged velocity structure functions. Applied Mathematics and Mechanics. 2005;**26**(4):495-504

[13] Kim YH, Cierpka C, Wereley ST. Flow field around a vibrating cantilever: Coherent structure education by

[14] Yoshimatsu K, Schneider K, Okamoto N, et al. Intermittency and geometrical statistics of threedimensional homogeneous magnetohydrodynamic turbulence: A wavelet viewpoint. Physics of Plasmas. 2011;

**18**(9):092304

continuous wavelet transform and proper orthogonal decomposition. Journal of Fluid Mechanics. 2011;**669**:584-606

[15] Baars WJ, Talluru KM, Hutchins N,

modulation of small scales. Experiments

Experiments in Fluids. 2002;**32**(1):76-86

[16] Camussi R. Coherent structure identification from wavelet analysis of particle image velocimetry data.

et al. Wavelet analysis of wall turbulence to study large-scale

in Fluids. 2015;**56**(10):188

fluid dynamics and turbulence. European Journal of Mechanics -

coherent eddy structures and intermittency in turbulent boundary layer by wavelet analysis. Chinese Physics Letters. 2005;**22**(8):1968

B/Fluids. 1990;**9**:1-19

[2] Xiao-Bing L, Zheng-Qing C, Chao-Qun L. Late-stage vertical structures and eddy motions in a transitional boundary layer. Chinese Physics Letters.

[3] Rinoshika A, Omori H. Orthogonal wavelet analysis of turbulent wakes

Orthogonal wavelet decomposition of turbulent structures behind a vehicle external mirror. Experimental Thermal

Schneider K, et al. Coherent vortices in high resolution direct numerical simulation of homogeneous isotropic turbulence: A wavelet viewpoint. Physics of Fluids. 2007;**19**(11):115109

[6] De Stefano G, Vasilyev OV. A fully adaptive wavelet-based approach to homogeneous turbulence simulation. Journal of Fluid Mechanics. 2012;**695**:

[7] Futatani S, Bos WJT, del-Castillo-Negrete D, et al. Coherent vorticity extraction in resistive drift-wave turbulence: Comparison of orthogonal wavelets versus proper orthogonal decomposition. Comptes Rendus Physique. 2011;**12**(2):123-131

[8] de la Llave PM, Cant S, Prosser R. On the use of biorthogonal interpolating wavelets for large-eddy simulation of turbulence. Journal of Computational Physics. 2012;**231**(20):6754-6769

behind various bluff bodies. Experimental Thermal and Fluid Science. 2011;**35**(7):1231-1238

[4] Rinoshika A, Watanabe S.

and Fluid Science. 2010;**34**(8):

[5] Okamoto N, Yoshimatsu K,

1389-1397

149-172

**114**

Journal. 2010;**720**(1):742

2010;**27**(2):024706

[18] Arimitsu T, Arimitsu N. Analysis of PDFs for energy transfer rates from 40963 DNS-verification of the scaling relation within MPDFT. Journal of Turbulence. 2011;**12**:N1

[19] Gang D, Shi-Sheng Z, Yang L. Time series prediction using wavelet process neural network. Chinese Physics B. 2008;**17**(6):1998

[20] Asai M, Minagawa M, Nishioka M. The instability and breakdown of a near-wall low-speed streak. Journal of Fluid Mechanics. 2002;**455**:289-314

[21] Chen L, Tang DB. Study on turbulent spots in plane Couette flow. Transactions of Nanjing University of Aeronautics & Astronautics. 2007; **24**(3):211-217

[22] Hu HB, Du P, Huang SH, Wang Y. Extraction and verification of coherent structures in near-wall turbulence. Chinese Physics B. 2013;**22**(7):074703

[23] Huang L, Kemao Q, Pan B, et al. Comparison of Fourier transform, windowed Fourier transform, and wavelet transform methods for phase extraction from a single fringe pattern in fringe projection profilometry. Optics and Lasers in Engineering. 2010;**48**(2):141-148

[24] Canal MR. Comparison of wavelet and short time Fourier transform methods in the analysis of EMG signals. Journal of Medical Systems. 2010;**34**(1): 91-94

[25] Du P, Wen J, Zhang Z, et al. Maintenance of air layer and drag reduction on superhydrophobic surface. Ocean Engineering. 2017;**130**:328-335

[26] Haibao H, Peng D, Feng Z, et al. Effect of hydrophobicity on turbulent boundary layer under water. Experimental Thermal and Fluid Science. 2015;**60**:148-156

[27] Sang YF. A review on the applications of wavelet transform in hydrology time series analysis. Atmospheric Research. 2013;**122**:8-15

**Chapter 8**

**Abstract**

**1. Introduction**

**117**

An Efficient Approach Based on

EMI Problems: Application to an

AC/DC Flyback Converter

*Bessem Zitouna and Jaleleddine Ben Hadj Slama*

the Near-Field Technique to Solve

Flyback converters have been widely used in low- and high-power applications

because of their simplicity and low cost. However, they incur electromagnetic compatibility problems which are more difficult to control. The present chapter proposes an efficient modeling method based on the near-field technique to solve real-world radiation problems of the power electronics circuits. Firstly, for the characterization of an AC/DC flyback converter, several experimental measurements of the magnetic near field are performed in the time domain over the converter. Subsequently, we have applied the time domain electromagnetic inverse method based on the genetic algorithms on the measured signals to find the equivalent radiating sources of the studied circuit. The accuracy and the efficiency of the proposed approach have been demonstrated by the good agreement between cartographies of the near magnetic field components calculated using the developed model and those measured. Finally, the developed equivalent model has been used to predict cartographies of other components of the magnetic field which will be compared to measured cartographies. This confirms that the identified equivalent sources can represent real sources in the studied structure. The proposed method could be used for diagnosis and fault location in power electronics systems.

**Keywords:** electromagnetic compatibility (EMC), flyback converter, inverse method, near-field measurement, time-domain analysis, genetic algorithms

To respond to the necessity and simplicity of marketed converter applications, innovation in power electronics is based on the growing constraints imposed by the exponential rise of our needs: better management of primary energy sources while offering increasingly sophisticated functions at reduced cost. This cost function can obviously take different forms depending on the context: weight, volume, efficiency, disturbances on the converter and its environment, technological cost, economic cost, protection and reliability, and lifetime of devices. No device can be excluded today from environmental constraints. At this stage, a lot of energy conversion structures are now very efficient. For example, the flyback converter is probably the most widely used structure in the electronics industry and is

#### **Chapter 8**

## An Efficient Approach Based on the Near-Field Technique to Solve EMI Problems: Application to an AC/DC Flyback Converter

*Bessem Zitouna and Jaleleddine Ben Hadj Slama*

### **Abstract**

Flyback converters have been widely used in low- and high-power applications because of their simplicity and low cost. However, they incur electromagnetic compatibility problems which are more difficult to control. The present chapter proposes an efficient modeling method based on the near-field technique to solve real-world radiation problems of the power electronics circuits. Firstly, for the characterization of an AC/DC flyback converter, several experimental measurements of the magnetic near field are performed in the time domain over the converter. Subsequently, we have applied the time domain electromagnetic inverse method based on the genetic algorithms on the measured signals to find the equivalent radiating sources of the studied circuit. The accuracy and the efficiency of the proposed approach have been demonstrated by the good agreement between cartographies of the near magnetic field components calculated using the developed model and those measured. Finally, the developed equivalent model has been used to predict cartographies of other components of the magnetic field which will be compared to measured cartographies. This confirms that the identified equivalent sources can represent real sources in the studied structure. The proposed method could be used for diagnosis and fault location in power electronics systems.

**Keywords:** electromagnetic compatibility (EMC), flyback converter, inverse method, near-field measurement, time-domain analysis, genetic algorithms

#### **1. Introduction**

To respond to the necessity and simplicity of marketed converter applications, innovation in power electronics is based on the growing constraints imposed by the exponential rise of our needs: better management of primary energy sources while offering increasingly sophisticated functions at reduced cost. This cost function can obviously take different forms depending on the context: weight, volume, efficiency, disturbances on the converter and its environment, technological cost, economic cost, protection and reliability, and lifetime of devices. No device can be excluded today from environmental constraints. At this stage, a lot of energy conversion structures are now very efficient. For example, the flyback converter is probably the most widely used structure in the electronics industry and is

frequently utilized for various applications (battery chargers, energy saving lamps, photovoltaic systems, LCD monitors, DVD players, …). Integration and coexistence in the same housing, as well as the increase in switching frequencies and carried powers, cause frequent and unwanted electromagnetic interference in the vicinity of these systems. These interferences present important sources of malfunction of the device itself and neighboring systems. For this reason, electro-magnetic compatibility (EMC) must be addressed from the starting process of design and manufacture of high-tech electronic devices. Indeed, the study of electromagnetic compatibility of a card gives quantitative tools to the power electronics to influence the radiated and conducted disturbances. It is essential to know the electromagnetic disturbances emitted by the equipment under test at different distances and to verify that they do not exceed a certain limit in order to avoid disturbing the neighboring equipment. Therefore, the EMC characterization tests have become an essential step in the various phases of the development of power electronics products. Unfortunately, these tests remain costly because we often need heavy, precise and very expensive equipment (anechoic chamber). To circumvent these limits, it is essential to develop radiating models that allow us to estimate the electromagnetic emissions of these systems before the prototype is realized (during the phase of virtual prototyping). The objective of this work is to elicit the electromagnetic inverse method in the time domain based on the near-field technique as an innovative and important solution that allows characterizing and modeling the radiation of the products of the power electronics. Thus, the added value of this chapter is to show the implementation of the inverse electromagnetic method in the time domain for the field of power electronics. In fact, the proposed approach will be of some use to people who make the development of power electronics systems. Actually, identifying all the sources of radiation in an electronic cartography and developing an equivalent radiation model that will allow us to estimate the electromagnetic emissions of these systems help solve the problems of interferences encountered in power electronics. The proposed method can be used for example as a method of analysis, fault detection and diagnosis of power electronic circuits. Indeed, the electromagnetic signature of a component could make it possible to verify its proper functioning. In order to find solutions related to the problems encountered in power electronics, the inverse electromagnetic method has been applied to different types of circuits.

radiation frequencies. Consequently, the models obtained by the electromagnetic inverse method in the frequency domain have several limitations and remain inapplicable for power electronics systems which are excited by different types of nonsinusoidal signals and which consequently emit on one broad band of frequencies. To this end, the added value of our approach is to propose an efficient technique based on the inverse electromagnetic method in the time domain which takes into account the temporal offset at the levels of the various radiated emissions of the circuit, which permits the Identification of the contribution of each radiating

*An Efficient Approach Based on the Near-Field Technique to Solve EMI Problems: Application…*

Until now, few studies have been performed and published on the time-domain inverse method. The authors in some study tried to combine the frequency inverse method with the time-frequency computation method [10, 11]. However, the developed modeling method was limited because it always depended on the parameters of the time intervals that mainly depended on the selected frequency band, the considered initial frequency and the used computer performance. Other approaches have been performed [12–15]. These studies were dedicated to the development of new computational techniques and analysis of an electromagnetic near field in the time domain. To develop a resolution method in the time domain, enabled by passing this kind of problem, we put forward the development and implementation of the temporal inverse electromagnetic method based on

The methodology of the time-domain electromagnetic inverse method has been presented [16]. This method was applied to identify the equivalent radiation model in a fairly simple structure, which was unfortunately not representative enough for all power electronic circuits. To demonstrate again the robustness and efficiency of the electromagnetic inverse method developed in the time domain when applied to complicated real cards composed of several bulky components, where the radiating sources are usually very close to each other, we propose to study in this chapter a complete application case (PCB and power electronic components). Hence, we will focus much more on the application of the temporal inverse electromagnetic method to find an equivalent model of an industrial system that emits nonsinusoidal radiations. This will make it possible to calculate the field emitted by the studied system for various distances and to study the coupling with the

In this chapter, the time domain electromagnetic inverse method is tested and used in an AC-DC flyback converter application. First, we describe the device under test. Then, we will present how the suggested approach is implemented and what these advantages can be. To do this, several works will be performed. We will firstly present the test bench of the near field in the time domain. Second, we will present the measurement results of the vertical component of the radiated magnetic near field (Hz) performed over an AC/DC converter, so as to model the radiation of this converter by the electromagnetic inverse method in the time domain. In the following part, the modeling methodology and its resolution based on the genetic algorithms are described. To validate the identified model using our suggested approach, a comparison is made between the tangential components (Hx) of the magnetic field, calculated using the optimized model parameters and the one

The system studied in this chapter is an AC-DC converter based on the principle

source of the circuit over time.

*DOI: http://dx.doi.org/10.5772/intechopen.89332*

elementary dipoles [16].

neighboring systems.

**119**

measured over the studied converter.

**2. Description of the studied flyback converter**

of switched-mode power supplies (**Figure 1**).

In the literature, a lot of work has been already achieved on the modeling of electronic components [1–3]. The inverse method based on the elementary dipoles was used to extract the equivalent model from PCB circuits [4–6]. In frequency domain, the inverse method was also used in order to model electronics cards [7–9].

These various approaches have been developed in the frequency domain and have been utilized to characterize and to model the radiated emissions on only one frequency. It seems that there is no time. This supposes that the studied systems have had sinusoidal radiations that do not correspond to the reality of power electronic systems where the electromagnetic disturbances are very important over a broad band of frequencies. For this effect, modeling this kind of radiation by the frequency inverse method requires to repeating the identification of equivalent sources for each signal frequency. This causes a very high processing time. However, from the results found in the frequency domain, we cannot know at what time of the cycle of operation the radiating sources intervene, and we thus suppose that the radiations of the sources contribute simultaneously, which does not correspond to reality. Nevertheless, the switching in power electronics often presents offsets which will be found at the level of radiated emissions. The spectral distribution therefore evolves as a function of time and the components do not all have the same

#### *An Efficient Approach Based on the Near-Field Technique to Solve EMI Problems: Application… DOI: http://dx.doi.org/10.5772/intechopen.89332*

radiation frequencies. Consequently, the models obtained by the electromagnetic inverse method in the frequency domain have several limitations and remain inapplicable for power electronics systems which are excited by different types of nonsinusoidal signals and which consequently emit on one broad band of frequencies. To this end, the added value of our approach is to propose an efficient technique based on the inverse electromagnetic method in the time domain which takes into account the temporal offset at the levels of the various radiated emissions of the circuit, which permits the Identification of the contribution of each radiating source of the circuit over time.

Until now, few studies have been performed and published on the time-domain inverse method. The authors in some study tried to combine the frequency inverse method with the time-frequency computation method [10, 11]. However, the developed modeling method was limited because it always depended on the parameters of the time intervals that mainly depended on the selected frequency band, the considered initial frequency and the used computer performance. Other approaches have been performed [12–15]. These studies were dedicated to the development of new computational techniques and analysis of an electromagnetic near field in the time domain. To develop a resolution method in the time domain, enabled by passing this kind of problem, we put forward the development and implementation of the temporal inverse electromagnetic method based on elementary dipoles [16].

The methodology of the time-domain electromagnetic inverse method has been presented [16]. This method was applied to identify the equivalent radiation model in a fairly simple structure, which was unfortunately not representative enough for all power electronic circuits. To demonstrate again the robustness and efficiency of the electromagnetic inverse method developed in the time domain when applied to complicated real cards composed of several bulky components, where the radiating sources are usually very close to each other, we propose to study in this chapter a complete application case (PCB and power electronic components). Hence, we will focus much more on the application of the temporal inverse electromagnetic method to find an equivalent model of an industrial system that emits nonsinusoidal radiations. This will make it possible to calculate the field emitted by the studied system for various distances and to study the coupling with the neighboring systems.

In this chapter, the time domain electromagnetic inverse method is tested and used in an AC-DC flyback converter application. First, we describe the device under test. Then, we will present how the suggested approach is implemented and what these advantages can be. To do this, several works will be performed. We will firstly present the test bench of the near field in the time domain. Second, we will present the measurement results of the vertical component of the radiated magnetic near field (Hz) performed over an AC/DC converter, so as to model the radiation of this converter by the electromagnetic inverse method in the time domain. In the following part, the modeling methodology and its resolution based on the genetic algorithms are described. To validate the identified model using our suggested approach, a comparison is made between the tangential components (Hx) of the magnetic field, calculated using the optimized model parameters and the one measured over the studied converter.

#### **2. Description of the studied flyback converter**

The system studied in this chapter is an AC-DC converter based on the principle of switched-mode power supplies (**Figure 1**).

frequently utilized for various applications (battery chargers, energy saving lamps, photovoltaic systems, LCD monitors, DVD players, …). Integration and coexistence in the same housing, as well as the increase in switching frequencies and carried powers, cause frequent and unwanted electromagnetic interference in the vicinity of these systems. These interferences present important sources of malfunction of the device itself and neighboring systems. For this reason, electro-magnetic compatibility (EMC) must be addressed from the starting process of design and manufacture of high-tech electronic devices. Indeed, the study of electromagnetic compatibility of a card gives quantitative tools to the power electronics to influence the radiated and conducted disturbances. It is essential to know the electromagnetic disturbances emitted by the equipment under test at different distances and to verify that they do not exceed a certain limit in order to avoid disturbing the neighboring equipment. Therefore, the EMC characterization tests have become an essential step in the various phases of the development of power electronics products. Unfortunately, these tests remain costly because we often need heavy, precise and very expensive equipment (anechoic chamber). To circumvent these limits, it is essential to develop radiating models that allow us to estimate the electromagnetic emissions of these systems before the prototype is realized (during the phase of virtual prototyping). The objective of this work is to elicit the electromagnetic inverse method in the time domain based on the near-field technique as an innovative and important solution that allows characterizing and modeling the radiation of the products of the power electronics. Thus, the added value of this chapter is to show the implementation of the inverse electromagnetic method in the time domain for the field of power electronics. In fact, the proposed approach will be of some use to people who make the development of power electronics systems. Actually, identifying all the sources of radiation in an electronic cartography and developing an equivalent radiation model that will allow us to estimate the electromagnetic emissions of these systems help solve the problems of interferences encountered in power electronics. The proposed method can be used for example as a method of analysis, fault detection and diagnosis of power electronic circuits. Indeed, the electromagnetic signature of a component could make it possible to verify its proper functioning. In order to find solutions related to the problems encountered in power electronics, the inverse electromagnetic method has been

*Advances in Complex Analysis and Applications*

In the literature, a lot of work has been already achieved on the modeling of

These various approaches have been developed in the frequency domain and have been utilized to characterize and to model the radiated emissions on only one frequency. It seems that there is no time. This supposes that the studied systems have had sinusoidal radiations that do not correspond to the reality of power electronic systems where the electromagnetic disturbances are very important over a broad band of frequencies. For this effect, modeling this kind of radiation by the frequency inverse method requires to repeating the identification of equivalent sources for each signal frequency. This causes a very high processing time. However, from the results found in the frequency domain, we cannot know at what time of the cycle of operation the radiating sources intervene, and we thus suppose that the radiations of the sources contribute simultaneously, which does not correspond to reality. Nevertheless, the switching in power electronics often presents offsets which will be found at the level of radiated emissions. The spectral distribution therefore evolves as a function of time and the components do not all have the same

electronic components [1–3]. The inverse method based on the elementary dipoles was used to extract the equivalent model from PCB circuits [4–6]. In frequency domain, the inverse method was also used in order to model electronics

applied to different types of circuits.

cards [7–9].

**118**

Switching power supplies are much smaller and lighter than linear power supplies, which explain their increasingly widespread use in various fields, particularly in those of embedded systems. Also, switched-mode power supplies have a good efficiency until 90%. Contrarily, they have regulatory problems which are more difficult to control [17–19]. Due to the rectangular signal to the switching frequency, these power supplies produce a relatively large noise. This problem makes them unsuitable for certain applications. To this end, it is essential to characterize and control the EMC behavior of this type of converters.

The electrical characteristics of the studied flyback converter are illustrated in

*An Efficient Approach Based on the Near-Field Technique to Solve EMI Problems: Application…*

In our study, the measurement of the studied converter radiation is carried out in one single measurement for different radiation frequencies. This presents the benefits of a near field in the time domain compared to the frequency test bench. **Figure 3** presents the different parts of the near-field test bench in time domain. The temporal measurement method uses a high-precision oscilloscope that presents a wide bandwidth. This oscilloscope is utilized for displaying and recording temporal signals picked up by the measurement of the magnetic field probe. To sweep the probe above the studied structure, a plotting table is used. The displacement of the probe over the device under test is manually performed

**3. Description of magnetic near-field test bench in time domain**

The displacement of the probe above the studied structure is manually performed. The displacement procedure of the probe from measurement point to

*Detailed structure of magnetic probes: (a) probe of tangential components Hx and Hy, (b) probe of vertical*

*component, (c) measurement principle of electronic probe Hz.*

another on the sweeping surface is described in **Figure 5**.

**Table 1**.

*DOI: http://dx.doi.org/10.5772/intechopen.89332*

(**Figure 4**).

**Figure 3.**

**Figure 4.**

**121**

*Near-field test bench in time domain.*

The studied converter in this chapter is of a flyback type of a low power (5 W). The basic schema of a studied converter is illustrated in **Figure 2**.

**Figure 1.** *Flyback AC/DC converter.*

#### **Figure 2.**

*Schematic of AC/DC flyback converter.*


#### **Table 1.**

*Characteristics of studied converter.*

*An Efficient Approach Based on the Near-Field Technique to Solve EMI Problems: Application… DOI: http://dx.doi.org/10.5772/intechopen.89332*

The electrical characteristics of the studied flyback converter are illustrated in **Table 1**.

#### **3. Description of magnetic near-field test bench in time domain**

In our study, the measurement of the studied converter radiation is carried out in one single measurement for different radiation frequencies. This presents the benefits of a near field in the time domain compared to the frequency test bench.

**Figure 3** presents the different parts of the near-field test bench in time domain.

The temporal measurement method uses a high-precision oscilloscope that presents a wide bandwidth. This oscilloscope is utilized for displaying and recording temporal signals picked up by the measurement of the magnetic field probe.

To sweep the probe above the studied structure, a plotting table is used. The displacement of the probe over the device under test is manually performed (**Figure 4**).

The displacement of the probe above the studied structure is manually performed. The displacement procedure of the probe from measurement point to another on the sweeping surface is described in **Figure 5**.

**Figure 3.** *Near-field test bench in time domain.*

**Figure 4.**

*Detailed structure of magnetic probes: (a) probe of tangential components Hx and Hy, (b) probe of vertical component, (c) measurement principle of electronic probe Hz.*

Switching power supplies are much smaller and lighter than linear power supplies, which explain their increasingly widespread use in various fields, particularly in those of embedded systems. Also, switched-mode power supplies have a good efficiency until 90%. Contrarily, they have regulatory problems which are more difficult to control [17–19]. Due to the rectangular signal to the switching frequency, these power supplies produce a relatively large noise. This problem makes them unsuitable for certain applications. To this end, it is essential to characterize and control the EMC behavior of this type of converters.

The studied converter in this chapter is of a flyback type of a low power (5 W).

**Type Input voltage Output voltage Maximum current Efficiency** Flyback 220 V AC 5 V DC 1 A 79%

The basic schema of a studied converter is illustrated in **Figure 2**.

*Advances in Complex Analysis and Applications*

**Figure 1.**

**Figure 2.**

**Table 1.**

**120**

*Schematic of AC/DC flyback converter.*

*Characteristics of studied converter.*

*Flyback AC/DC converter.*

**Figure 5.** *Probe displacement in measurement surface.*

In this study and according to our needs, we utilize two coil probes: one for measuring the normal component of the magnetic field (Hz) and the other for measuring the two tangential components (Hx and Hy). Each utilized probe is shielded. The shielding of the used probe is made by copper. It consists of a loop connected to the central conductor on one side and to the external shield of a coaxial cable on the other side, as shown in **Figure 4**. The probe is connected to the oscilloscope through a highly shielded coaxial cable.

To measure the magnetic fields around the components and systems, we use magnetic probes; and to capture the different components of the magnetic field H, it is necessary to place the normal to the surface of the collinearly loop of the desired component. It is possible to utilize a single probe and to orient it differently according to the component to be measured. The principle of the measurement protocol is based on the Faraday and Lenz law, which stipulates that if a variable magnetic field crosses through a closed circuit, then the resulting flux variation will cause an electromotive force at the terminals of the loop (**Figure 4c**). Accordingly, we have:

$$e(t) = -\frac{d\rho(t)}{dt} \tag{1}$$

In fact, extracting the magnetic field H(t) from the measured voltage at the

*An Efficient Approach Based on the Near-Field Technique to Solve EMI Problems: Application…*

*μ*<sup>0</sup> � *S*

This suggested approach is based on time domain measurements of the near field above the studied structure. The distribution of the near field on the chosen scanned surface is presented in the form of cartography. The cartography is a matrix representation of the amplitude and the phase of the measured near field at each point and at each instant over the studied system. It is characterized by the dimensions of the measurement surface (Xmin:Xmax; Ymin:Ymax), the distance between two measuring points (ΔX et ΔY), and the measuring height (Z) (**Figure 6**).

**Figure 7b** represents the chosen face used for the time domain measurements to identify the equivalent radiating sources. This face enables a good detection of the

*(a) The studied card and (b) the selected studied card face for measuring the magnetic near field used to*

ð*t* 0

*V t*ð Þ *dt* (4)

*H t*ðÞ¼� <sup>1</sup>

**4. Measurement of near field emitted by studied converter**

magnetic near field radiated by all components of the studied card.

probe terminals can be defined as follows:

*DOI: http://dx.doi.org/10.5772/intechopen.89332*

**Figure 6.**

**Figure 7.**

**123**

*identify the model.*

*Cartography parameters of near-field measurements.*

Finally, and through a transformation equation, we can deduce the magnetic field. The magnetic probe allows transforming the variable magnetic flux which crosses through it into a voltage at its terminals. Subsequently, we deduce the magnetic field at the center of the probe. Indeed, for a circular conductive loop of a radius R dived in a magnetic field B(t), the induced potential difference by the variable magnetic field is given by the following equation:

$$dV(t) = -\oint \mu\_0 \frac{\partial H(t)}{\partial t} \quad \overrightarrow{\text{n}} \quad d\textbf{S} \tag{2}$$

where n! is the probe surface normal.

Since the probe surface is small enough, we can consider the magnetic field Hz(t) constant over the entire loop area S. As a result, Eq. (5) becomes:

$$V(t) = -\mu\_0 \times \mathcal{S} \frac{\partial H(t)}{\partial t} \tag{3}$$

where *S* = *π* � *r* <sup>2</sup> is the probe surface (r = 1.6 mm) and *<sup>μ</sup>*<sup>0</sup> = 4 � *<sup>π</sup>* � <sup>10</sup>�<sup>7</sup> H/m is the permeability in the free space.

*An Efficient Approach Based on the Near-Field Technique to Solve EMI Problems: Application… DOI: http://dx.doi.org/10.5772/intechopen.89332*

In fact, extracting the magnetic field H(t) from the measured voltage at the probe terminals can be defined as follows:

$$H(t) = -\frac{1}{\mu\_0 \times \mathcal{S}} \int\_0^t V(t) \, dt \tag{4}$$

#### **4. Measurement of near field emitted by studied converter**

This suggested approach is based on time domain measurements of the near field above the studied structure. The distribution of the near field on the chosen scanned surface is presented in the form of cartography. The cartography is a matrix representation of the amplitude and the phase of the measured near field at each point and at each instant over the studied system. It is characterized by the dimensions of the measurement surface (Xmin:Xmax; Ymin:Ymax), the distance between two measuring points (ΔX et ΔY), and the measuring height (Z) (**Figure 6**).

**Figure 7b** represents the chosen face used for the time domain measurements to identify the equivalent radiating sources. This face enables a good detection of the magnetic near field radiated by all components of the studied card.

**Figure 6.** *Cartography parameters of near-field measurements.*

*(a) The studied card and (b) the selected studied card face for measuring the magnetic near field used to identify the model.*

In this study and according to our needs, we utilize two coil probes: one for measuring the normal component of the magnetic field (Hz) and the other for measuring the two tangential components (Hx and Hy). Each utilized probe is shielded. The shielding of the used probe is made by copper. It consists of a loop connected to the central conductor on one side and to the external shield of a coaxial

To measure the magnetic fields around the components and systems, we use magnetic probes; and to capture the different components of the magnetic field H, it is necessary to place the normal to the surface of the collinearly loop of the desired

*e t*ðÞ¼� *<sup>d</sup>φ*ð Þ*<sup>t</sup>*

Finally, and through a transformation equation, we can deduce the magnetic field. The magnetic probe allows transforming the variable magnetic flux which crosses through it into a voltage at its terminals. Subsequently, we deduce the magnetic field at the center of the probe. Indeed, for a circular conductive loop of a radius R dived in a magnetic field B(t), the induced potential difference by the

> *<sup>∂</sup>H t*ð Þ *∂t*

Since the probe surface is small enough, we can consider the magnetic field

*V t*ðÞ¼�*μ*<sup>0</sup> � *<sup>S</sup> <sup>∂</sup>H t*ð Þ

n

<sup>2</sup> is the probe surface (r = 1.6 mm) and *<sup>μ</sup>*<sup>0</sup> = 4 � *<sup>π</sup>* � <sup>10</sup>�<sup>7</sup> H/m is

*dt* (1)

! *d*S (2)

*<sup>∂</sup><sup>t</sup>* (3)

cable on the other side, as shown in **Figure 4**. The probe is connected to the

component. It is possible to utilize a single probe and to orient it differently according to the component to be measured. The principle of the measurement protocol is based on the Faraday and Lenz law, which stipulates that if a variable magnetic field crosses through a closed circuit, then the resulting flux variation will cause an electromotive force at the terminals of the loop (**Figure 4c**). Accordingly,

oscilloscope through a highly shielded coaxial cable.

variable magnetic field is given by the following equation:

where n! is the probe surface normal.

where *S* = *π* � *r*

**122**

the permeability in the free space.

*V t*ðÞ¼�∮ *μ*<sup>0</sup>

Hz(t) constant over the entire loop area S. As a result, Eq. (5) becomes:

we have:

**Figure 5.**

*Probe displacement in measurement surface.*

*Advances in Complex Analysis and Applications*

By analogy with the frequency inverse method, in this work, the identification of the equivalent radiating sources is based on the measurements of the magnetic near field [8, 9, 20]. This demonstrates its effectiveness in guaranteeing the uniqueness of the equivalent radiating model [9]. To bypass the complexities to measure the electrical near field, it is possible to extract the values of the electric field by exploiting the measurements of the magnetic field [21, 22].

For the magnetic dipole:

*Hx Hy Hz* 3 7 7 <sup>5</sup> ¼ � *<sup>r</sup>*<sup>2</sup> 4*R* �

*DOI: http://dx.doi.org/10.5772/intechopen.89332*

sin *θ* sin *θ* cos *θ*

þ 3 *<sup>R</sup>*<sup>2</sup> *I t*<sup>0</sup> ð Þ�

∧

tt0

q

where R ¼

algorithms.

**125**

chart given in **Figure 9**.

delay time variable, *R* ¼

q

1 *c*2 �� *d*<sup>2</sup>

> > sin *θ* sin *θ* cos *θ*

> > > *i* !

> > > *j* !

> > > *k* !

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

the optimization phase of the equivalent model parameters by the genetic

At this stage, the problem of identifying the source dipoles is processed. The identification procedure of the radiating sources is presented in the following flow-

In order to identify accurately the shape of the first dipole excitation signal, we will apply to the initially measured cartography a subprogram to seek the measured signal which has the maximum amplitude relative to other measured maximum temporal signals in this cartography. And on the basis of the measurement point that corresponds to the most intense radiation, we extract a limited part of the initial cartography and we propose a single elementary radiating dipole (electric or magnetic dipole) for the extracted cartography. In fact, the searched current form flowing in the first dipole to identify is well known; it is a normalized temporal signal relative to this measured maximum signal in this local cartography. Subsequently, an optimization method based on the genetic algorithms is applied to identify the parameters of the proposed dipole in this selected cartography. At each iteration, the genetic algorithms shall modify all the parameters of the dipole (a constant k which represents the amplitude of the excitation current circulating in

??ð Þ Xd � Xo <sup>2</sup> <sup>þ</sup> ð Þ Yd � Yo <sup>2</sup> <sup>þ</sup> ð Þ Zd � Zo <sup>2</sup>

As the temporal measured signals above the studied system are not sinusoidal well as the excitations currents are not unique. In this proposed method, it is not necessary to initially propose a number of elementary dipoles during the equivalent model identification. Thus, the proposed method consists in identifying at each time a single particular dipole, until the identification of all dipoles that corresponds the equivalent model. Because the measures are performed at a very close distance above the studied structure, a hypothesis is adopted; it is based on the linearity between the radiated magnetic field by an elementary dipole and the current flowing in this dipole. This assumption is adopted for guessing the currents forms on the various dipoles, and is very helpful when simplifying the resolution during

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð Þ *Xd* � *<sup>X</sup>*<sup>0</sup> <sup>2</sup> <sup>þ</sup> ð Þ *Yd* � *<sup>Y</sup>*<sup>0</sup> <sup>2</sup> <sup>þ</sup> ð Þ *Zd* � *<sup>Z</sup>*<sup>0</sup> <sup>2</sup>

BB@ *:*

1

CCCA

0

BBB@

0

*xd* � *x*0 *yd* � *y*0 *zd* � *z*0

sin *ϕ* cos *ϕ*

*I t*<sup>0</sup> ð Þ *dt*<sup>2</sup> <sup>þ</sup>

*An Efficient Approach Based on the Near-Field Technique to Solve EMI Problems: Application…*

1 *R c*

1 *c*2 *d*2 *I t*<sup>0</sup> ð Þ *dt*2

sin *ϕ* cos *ϕ*

*dI t*<sup>0</sup> ð Þ *dt* þ

> þ 3 *R c*

*xd* � *x*0 *yd* � *y*0 *zd* � *z*0

1 *<sup>R</sup>*<sup>2</sup> *I t*<sup>0</sup> ð Þ�

> *dI t*<sup>0</sup> ð Þ *dt*

> > > t

<sup>0</sup> = t � (R/c) is the

.

1

(6)

CCA

#### **5. Equivalent radiating model of studied flyback converter**

In this study, the proposed temporal electromagnetic inverse method is based on the genetic algorithms. To apply our approach, a near-field measurements characterized by a very high signal-to-noise ratio are needed. The proposed method consists in identifying an equivalent model that emits the same radiation of the studied structure to be modeled.

The optimized equivalent model is based on a network of electric or magnetic dipoles (**Figure 8**).

The dipole may be represented by a d vector that comprises all the parameters: d = (Md, xd, yd, zd, θd, φd), where Md is the magnetic moment of the dipole which varies against time. The xd, yd and zd represent the dipole's position, and their orientations are represented by θ<sup>d</sup> and φd. In the time domain, the components of the magnetic field emitted by the elementary electric and magnetic dipoles in an observation point M (xo, yo, zo) can be deduced from the analytical expressions in the frequency domain using the frequency-time transformation operator. In the Cartesian coordinate system, the magnetic field components are expressed by the following equations [16]:

For the electric dipole:

$$
\begin{aligned}
\begin{bmatrix} H\_x \\ H\_y \\ H\_z \end{bmatrix} &= -\frac{\Delta l}{4\pi R^2} \times \left(\frac{1}{c}\frac{dI(t')}{dt} + \frac{1}{R}I(t')\right) \\
&\quad \left(\begin{bmatrix} \sin\theta & \sin\phi \\ \sin\theta & \cos\phi \\ \cos\theta \end{bmatrix}, \begin{bmatrix} xd-x0 \\ yd-y0 \\ zd-z0 \end{bmatrix}\right) \wedge \begin{bmatrix} xd-x0 \\ yd-y0 \\ zd-z0 \end{bmatrix} \begin{pmatrix} \overrightarrow{i} \\ \overrightarrow{j} \\ \overrightarrow{k} \end{pmatrix}
\end{aligned} \tag{5}
$$

**Figure 8.** *Presentation of the elementary dipoles.*

*An Efficient Approach Based on the Near-Field Technique to Solve EMI Problems: Application… DOI: http://dx.doi.org/10.5772/intechopen.89332*

For the magnetic dipole:

By analogy with the frequency inverse method, in this work, the identification of the equivalent radiating sources is based on the measurements of the magnetic near field [8, 9, 20]. This demonstrates its effectiveness in guaranteeing the uniqueness of the equivalent radiating model [9]. To bypass the complexities to measure the electrical near field, it is possible to extract the values of the electric

In this study, the proposed temporal electromagnetic inverse method is based on the genetic algorithms. To apply our approach, a near-field measurements characterized by a very high signal-to-noise ratio are needed. The proposed method consists in identifying an equivalent model that emits the same radiation of the

The optimized equivalent model is based on a network of electric or magnetic

The dipole may be represented by a d vector that comprises all the parameters: d = (Md, xd, yd, zd, θd, φd), where Md is the magnetic moment of the dipole which varies against time. The xd, yd and zd represent the dipole's position, and their orientations are represented by θ<sup>d</sup> and φd. In the time domain, the components of the magnetic field emitted by the elementary electric and magnetic dipoles in an observation point M (xo, yo, zo) can be deduced from the analytical expressions in the frequency domain using the frequency-time transformation operator. In the Cartesian coordinate system, the magnetic field components are expressed by the

> þ 1 *<sup>R</sup> I t*<sup>0</sup> ð Þ �

*xd* � *x*0 *yd* � *y*0 *zd* � *z*0

1

*xd* � *x*0 *yd* � *y*0 *zd* � *z*0

*i* ! 1

(5)

CCCA

0

BBB@

*j* !

*k* !

CCA ∧

field by exploiting the measurements of the magnetic field [21, 22].

**5. Equivalent radiating model of studied flyback converter**

studied structure to be modeled.

*Advances in Complex Analysis and Applications*

dipoles (**Figure 8**).

following equations [16]: For the electric dipole:

> *Hx Hy Hz*

3 7 7 <sup>5</sup> ¼ � <sup>Δ</sup>*<sup>l</sup>*

<sup>4</sup>*<sup>π</sup> <sup>R</sup>*<sup>2</sup> �

sin *ϕ* cos *ϕ*

sin *θ* sin *θ* cos *θ*

BB@ *:*

0

1 *c dI t*<sup>0</sup> ð Þ *dt*

�

**Figure 8.**

**124**

*Presentation of the elementary dipoles.*

$$
\begin{bmatrix} H\_x \\ H\_y \\ H\_z \end{bmatrix} = -\frac{r^2}{4R} \times \left[ \left( \frac{1}{c^2} \frac{d^2I(t')}{dt^2} + \frac{1}{Rc} \frac{dI(t')}{dt} + \frac{1}{R^2} I(t') \right) \right.
$$

$$
\begin{bmatrix} \sin\theta & \sin\phi \\ \sin\theta & \cos\phi \\ \cos\theta & \cos\theta \end{bmatrix} - \frac{1}{R^2} \left( \frac{1}{c^2} \frac{d^2I(t')}{dt^2} + \frac{3}{Rc} \frac{dI(t')}{dt} \right. \\
\begin{aligned}
& \left. \frac{3}{R^2} I(t') \right) \left( \begin{bmatrix} \sin\theta & \sin\phi \\ \sin\theta & \cos\phi \\ \cos\theta & \cos\theta \end{bmatrix} \cdot \begin{bmatrix} zd - x0 \\ yd - y0 \\ zd - z0 \end{bmatrix} \right) \\
& \land \begin{bmatrix} xd - x0 \\ yd - y0 \\ zd - z0 \end{bmatrix} \begin{bmatrix} \bar{i} \\ \bar{j} \\ \bar{k} \end{bmatrix} \end{aligned} \tag{6}
$$

where R ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi tt0 ??ð Þ Xd � Xo <sup>2</sup> <sup>þ</sup> ð Þ Yd � Yo <sup>2</sup> <sup>þ</sup> ð Þ Zd � Zo <sup>2</sup> q t <sup>0</sup> = t � (R/c) is the delay time variable, *R* ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð Þ *Xd* � *<sup>X</sup>*<sup>0</sup> <sup>2</sup> <sup>þ</sup> ð Þ *Yd* � *<sup>Y</sup>*<sup>0</sup> <sup>2</sup> <sup>þ</sup> ð Þ *Zd* � *<sup>Z</sup>*<sup>0</sup> <sup>2</sup> q .

As the temporal measured signals above the studied system are not sinusoidal well as the excitations currents are not unique. In this proposed method, it is not necessary to initially propose a number of elementary dipoles during the equivalent model identification. Thus, the proposed method consists in identifying at each time a single particular dipole, until the identification of all dipoles that corresponds the equivalent model. Because the measures are performed at a very close distance above the studied structure, a hypothesis is adopted; it is based on the linearity between the radiated magnetic field by an elementary dipole and the current flowing in this dipole. This assumption is adopted for guessing the currents forms on the various dipoles, and is very helpful when simplifying the resolution during the optimization phase of the equivalent model parameters by the genetic algorithms.

At this stage, the problem of identifying the source dipoles is processed. The identification procedure of the radiating sources is presented in the following flowchart given in **Figure 9**.

In order to identify accurately the shape of the first dipole excitation signal, we will apply to the initially measured cartography a subprogram to seek the measured signal which has the maximum amplitude relative to other measured maximum temporal signals in this cartography. And on the basis of the measurement point that corresponds to the most intense radiation, we extract a limited part of the initial cartography and we propose a single elementary radiating dipole (electric or magnetic dipole) for the extracted cartography. In fact, the searched current form flowing in the first dipole to identify is well known; it is a normalized temporal signal relative to this measured maximum signal in this local cartography. Subsequently, an optimization method based on the genetic algorithms is applied to identify the parameters of the proposed dipole in this selected cartography. At each iteration, the genetic algorithms shall modify all the parameters of the dipole (a constant k which represents the amplitude of the excitation current circulating in

**Figure 9.** *Chart of proposed method based on optimization algorithm.*

this proposed dipole, its geometry, its coordinates, and its orientations) in order to minimize the error calculated by the fitness function. If the error is unacceptable and the genetic algorithms cannot converge after a certain number of iterations, the optimization cycle repeats itself several times until obtaining a new composed generation of best solutions who is the calculated field coincides with the measured one on all points of the cartography and at all moments (even in the case of the small values of the measured field). Thus, the radiating dipole and its corresponding parameters are identified.

A supplementary explanation of the proposed approach is illustrated in **Figure 10**. At each identification, this procedure will be repeated until the extrac-

Population size (Np) Np = 20 � number of parameters of dipoles

*An Efficient Approach Based on the Near-Field Technique to Solve EMI Problems: Application…*

Selection function Roulette Crossover rate 0.8

The superposition method is applied in a near-field zone where it is assumed that in each area of the local cartography we identify a relatively intense field. The emission is mainly caused by a very close source and the contributions from the other relatively distant sources are very low compared with that of the intense source. Consequently, the search for the model is carried out dipole by dipole until identifying all radiating sources in the initially measured cartography. Indeed, the fitness function is used by the genetic algorithms for identifying the existing source in the each local cartography. Thus, the error of the global cartography is the sum of

t*K*¼1 P<sup>M</sup> i¼1 � � �

HZð Þ i,tK measured‐HZði,tKÞestimated<sup>j</sup>

HZð Þ i,tK measured � <sup>100</sup>

To guarantee and accelerate the convergence of the proposed temporal inverse method, we take into account the study of [23] to choose the optimal parameters of

By applying the proposed method to identify the radiating sources, we have found seven magnetic dipoles as a number of sources. Despite the high density of

tion of all radiating sources in this measured magnetic fields cartography.

the errors of different local cartographies.

*Optimum parameters of genetic algorithms.*

**Table 2.**

**127**

**Figure 10.**

Fitness function *Error* <sup>¼</sup> <sup>P</sup><sup>T</sup>

*DOI: http://dx.doi.org/10.5772/intechopen.89332*

*Methodology of proposed electromagnetic inverse method in time domain.*

the genetic algorithms. These parameters are given in **Table 2**.

By using the analytical equations, the parameters of the first identified dipole are used to calculate the near magnetic field radiated in order to delete it from the initial measured cartography. This simplifies the resolution of the remaining cartography. The procedure is repeated until the scanning of all the measured cartography and the different dipole sources are identified.

*An Efficient Approach Based on the Near-Field Technique to Solve EMI Problems: Application… DOI: http://dx.doi.org/10.5772/intechopen.89332*

#### **Figure 10.**

*Methodology of proposed electromagnetic inverse method in time domain.*


#### **Table 2.**

this proposed dipole, its geometry, its coordinates, and its orientations) in order to minimize the error calculated by the fitness function. If the error is unacceptable and the genetic algorithms cannot converge after a certain number of iterations, the optimization cycle repeats itself several times until obtaining a new composed generation of best solutions who is the calculated field coincides with the measured one on all points of the cartography and at all moments (even in the case of the small values of the measured field). Thus, the radiating dipole and its corresponding

By using the analytical equations, the parameters of the first identified dipole are used to calculate the near magnetic field radiated in order to delete it from the initial measured cartography. This simplifies the resolution of the remaining cartography. The procedure is repeated until the scanning of all the measured cartography and

parameters are identified.

**Figure 9.**

**126**

the different dipole sources are identified.

*Chart of proposed method based on optimization algorithm.*

*Advances in Complex Analysis and Applications*

*Optimum parameters of genetic algorithms.*

A supplementary explanation of the proposed approach is illustrated in **Figure 10**. At each identification, this procedure will be repeated until the extraction of all radiating sources in this measured magnetic fields cartography.

The superposition method is applied in a near-field zone where it is assumed that in each area of the local cartography we identify a relatively intense field. The emission is mainly caused by a very close source and the contributions from the other relatively distant sources are very low compared with that of the intense source. Consequently, the search for the model is carried out dipole by dipole until identifying all radiating sources in the initially measured cartography. Indeed, the fitness function is used by the genetic algorithms for identifying the existing source in the each local cartography. Thus, the error of the global cartography is the sum of the errors of different local cartographies.

To guarantee and accelerate the convergence of the proposed temporal inverse method, we take into account the study of [23] to choose the optimal parameters of the genetic algorithms. These parameters are given in **Table 2**.

By applying the proposed method to identify the radiating sources, we have found seven magnetic dipoles as a number of sources. Despite the high density of

#### *Advances in Complex Analysis and Applications*

components in the converter and knowing that the measurements of radiated disturbances are performed at a very close distance above these components, and on the basis of research work on the modeling of the radiation of electronic components and systems, we have carried out the analysis of the radiations of the studied converter. This permits us to identify the original components of the equivalent radiating dipoles. Unfortunately, the modeling of systems by the inverse method in the frequency domain has several limitations and remains not applicable for systems of power electronics operating over a wide frequency band and excited by different types of nonsinusoidal signals. At this stage, it is the interest of the temporal inverse method proposed in this chapter. Thus, without modeling of the radiated emissions for each frequency of the measured signals (which causes a very significant calculation time). The proposed approach has the advantage of finding

in a reasonable processing time a single model of radiation valid over the whole band of frequencies in measured signal. From the results obtained by the inverse method proposed in the time domain, we can see that the first two identified magnetic dipoles represent the equivalent models of two diodes. The third identified source represents the equivalent radiation of the transformer. In fact, this last source has a center situated at coordinates Xd = 0.10 cm, Yd = 0.21 cm and Zd = 1.21 cm of the studied structure. It presents the coordinates of the trans-

*An Efficient Approach Based on the Near-Field Technique to Solve EMI Problems: Application…*

**Figure 12** gives a spatial representation of the equivalent dipoles identified by our proposed approach as well as the different shapes of the excitation currents of

The parameters of the equivalent model obtained by the proposed method are

Hence, it presents the case where the emitted disturbances of the tracks are very low compared to that of the components. Finally, we notice that the components of the power electronics card represent the majority of the radiation of the studied

In order to study these electromagnetic disturbances, we present in **Figure 14**

#1 3.12 e 7 1.46/1.33/0.645 0.57 7.45 #2 4.42 e 7 1.42/0.32/0.79 58.47 132.99 #3 1.12 e 6 0.10/0.21/1.21 9.17 9.74 #4 2.96 e 7 0.96/0.20/0.61 11.46 17.77 #5 3.86 e 7 1.34/1.19/0.65 43.56 128.98 #6 1.48 e 7 0.35/1.08/0.37 72.80 179.42 #7 1.90 e 8 1.36/0.74/0.62 83.12 8.85

**) [Xdi Ydi Zdi] (cm) θi(0) φi(0)**

the measured magnetic near-field cartographies at different moments.

*Internal structure of studied converter and radiations emitted by different radiating components.*

former position in the studied card, as depicted in **Figure 11**.

the source dipoles.

shown in **Table 3**.

**Table 3.**

**Figure 13.**

**129**

structure as illustrated in **Figure 13**.

*DOI: http://dx.doi.org/10.5772/intechopen.89332*

**Identified dipoles Mdi (A m<sup>2</sup>**

*Model parameters obtained by the proposed method.*

**Figure 11.** *Equivalent radiating source of transformer determined by proposed method.*

**Figure 12.** *Identified equivalent model of studied structure and forms of excitation signals.*

*An Efficient Approach Based on the Near-Field Technique to Solve EMI Problems: Application… DOI: http://dx.doi.org/10.5772/intechopen.89332*

in a reasonable processing time a single model of radiation valid over the whole band of frequencies in measured signal. From the results obtained by the inverse method proposed in the time domain, we can see that the first two identified magnetic dipoles represent the equivalent models of two diodes. The third identified source represents the equivalent radiation of the transformer. In fact, this last source has a center situated at coordinates Xd = 0.10 cm, Yd = 0.21 cm and Zd = 1.21 cm of the studied structure. It presents the coordinates of the transformer position in the studied card, as depicted in **Figure 11**.

**Figure 12** gives a spatial representation of the equivalent dipoles identified by our proposed approach as well as the different shapes of the excitation currents of the source dipoles.

The parameters of the equivalent model obtained by the proposed method are shown in **Table 3**.

Hence, it presents the case where the emitted disturbances of the tracks are very low compared to that of the components. Finally, we notice that the components of the power electronics card represent the majority of the radiation of the studied structure as illustrated in **Figure 13**.

In order to study these electromagnetic disturbances, we present in **Figure 14** the measured magnetic near-field cartographies at different moments.


**Table 3.**

components in the converter and knowing that the measurements of radiated disturbances are performed at a very close distance above these components, and on the basis of research work on the modeling of the radiation of electronic components and systems, we have carried out the analysis of the radiations of the studied converter. This permits us to identify the original components of the equivalent radiating dipoles. Unfortunately, the modeling of systems by the inverse method in the frequency domain has several limitations and remains not applicable for systems of power electronics operating over a wide frequency band and excited by different types of nonsinusoidal signals. At this stage, it is the interest of the temporal inverse method proposed in this chapter. Thus, without modeling of the radiated emissions for each frequency of the measured signals (which causes a very significant calculation time). The proposed approach has the advantage of finding

*Equivalent radiating source of transformer determined by proposed method.*

*Advances in Complex Analysis and Applications*

*Identified equivalent model of studied structure and forms of excitation signals.*

**Figure 11.**

**Figure 12.**

**128**

*Model parameters obtained by the proposed method.*

**Figure 13.**

*Internal structure of studied converter and radiations emitted by different radiating components.*

By analyzing **Figure 14**, we notice that the all sources do not appear at the same time. This is due to the time lag at the level of the electrical quantities passing through the different elements of the studied structure. This kind of transient phenomena is not easy to identify in the frequency analysis where the equivalent sources appear at the time such as presented in [8]. For a diagnostic study, where it is desired to control the good functioning of a sensitive element in a power electronics card, this type of time analysis becomes an effective alternative that enables

us to use the electromagnetic signature based on the temporal near field for the disgnosis of some components of the card. The electromagnetic inverse method in the time domain can therefore be used as a nonintrusive method of diagnosing and detecting defects in converter circuits. We cite for example the case of clampingdiode faults in a three-level NPC inverter circuit using the temporal analysis of the near magnetic field. Actually, the failure of a diode results in a modification during a delimited sequence at each period, from the temporal cartography of the magnetic

*An Efficient Approach Based on the Near-Field Technique to Solve EMI Problems: Application…*

In **Figure 15**, we present at various moments a comparison between the measured cartographies of the normal magnetic field component (Hz) and the calculated cartographies using the parameters of the equivalent model obtained by the

By examining the results got, we notice a good agreement between these cartographies. This demonstrates that the suggested method can identify with very

In order to validate the found equivalent model, we measured the tangential component of the magnetic field (Hx measured) and subsequently we compared it with that is calculated (Hx estimated) by exploiting the optimized parameters of the

*Comparison of estimated near-field cartographies (Hx) to those measured at (a) t = 10.06 μs and*

field measured in a zone close to the circuit.

*DOI: http://dx.doi.org/10.5772/intechopen.89332*

good accuracy all radiating sources of the circuit.

**6. Validation of identified equivalent model**

**Figure 16.**

**131**

*(b) t = 10.47 μs.*

electromagnetic inverse method proposed in the time domain.

**Figure 14.**

*Cartographies of the magnetic near field at various moments.*

**Figure 15.** *Measured magnetic field and estimated one (a) t = 9.95 μs, (b) t = 10.06 μs, (c) t = 10.56 μs.*

*An Efficient Approach Based on the Near-Field Technique to Solve EMI Problems: Application… DOI: http://dx.doi.org/10.5772/intechopen.89332*

us to use the electromagnetic signature based on the temporal near field for the disgnosis of some components of the card. The electromagnetic inverse method in the time domain can therefore be used as a nonintrusive method of diagnosing and detecting defects in converter circuits. We cite for example the case of clampingdiode faults in a three-level NPC inverter circuit using the temporal analysis of the near magnetic field. Actually, the failure of a diode results in a modification during a delimited sequence at each period, from the temporal cartography of the magnetic field measured in a zone close to the circuit.

In **Figure 15**, we present at various moments a comparison between the measured cartographies of the normal magnetic field component (Hz) and the calculated cartographies using the parameters of the equivalent model obtained by the electromagnetic inverse method proposed in the time domain.

By examining the results got, we notice a good agreement between these cartographies. This demonstrates that the suggested method can identify with very good accuracy all radiating sources of the circuit.

#### **6. Validation of identified equivalent model**

In order to validate the found equivalent model, we measured the tangential component of the magnetic field (Hx measured) and subsequently we compared it with that is calculated (Hx estimated) by exploiting the optimized parameters of the

**Figure 16.** *Comparison of estimated near-field cartographies (Hx) to those measured at (a) t = 10.06 μs and (b) t = 10.47 μs.*

By analyzing **Figure 14**, we notice that the all sources do not appear at the same

time. This is due to the time lag at the level of the electrical quantities passing through the different elements of the studied structure. This kind of transient phenomena is not easy to identify in the frequency analysis where the equivalent sources appear at the time such as presented in [8]. For a diagnostic study, where it is desired to control the good functioning of a sensitive element in a power electronics card, this type of time analysis becomes an effective alternative that enables

**Figure 14.**

**Figure 15.**

**130**

*Cartographies of the magnetic near field at various moments.*

*Advances in Complex Analysis and Applications*

*Measured magnetic field and estimated one (a) t = 9.95 μs, (b) t = 10.06 μs, (c) t = 10.56 μs.*

equivalent radiating sources. In two distinct instants (t1 = 10.06 μs and t2 = 10.47 μs), a comparison between the measured and estimated of the near-field cartographies is presented in **Figure 16**.

**References**

2014. pp. 1-6

[1] Saidi S, Ben Hadj Slama J. Analysis and modeling of power MOSFET radiation. Progress In Electromagnetics

*DOI: http://dx.doi.org/10.5772/intechopen.89332*

[8] Saidi S, Ben Hadj Slama J. A nearfield technique based on PZMI, GA, and ANN: Application to power electronics

[9] Benyoubi F, Pichon L, Bensetti M, Le Bihan Y, Feliach M. An efficient method

[10] Liu Y, Ravelo B, Fernandez-Lopez P. Modeling of magnetic near-field radiated by electronic devices disturbed by transient signals with complex form. Applied Physics Research. 2012;**4**(1):

[11] Liu Y, Ravelo B. Application of nearfield emission processing for microwave circuits under ultra-short duration

[12] Liu Y, Ravelo B. Fully time-domain scanning of EM near-field radiated by

Electromagnetics Research B. 2014;**57**:

Heddebaut M. Novel electromagnetic field measuring instrument with realtime visualization. Proc IEEE Int. Symp. EMC, Austin, Texas, USA. 2009:133-138

[14] Ravelo B, Liu Y, Slama JBH. Timedomain planar near-field/near-field transform with PWS operations. The European Physical Journal Applied

[15] Liu Y, Ravelo B, Jastrzebski AK, Hadj Slama JB. Calculation of the time domain z-component of the EM-nearfield from the x- and y-components. In:

perturbations. Advanced Electromagnetics. 2012;**1**(3)

RF circuits. Progress In

[13] Rioult J, Seethramdoo D,

Physics. 2011;**53**(03):1-8

systems. IEEE Transactions on Electromagnetic Compatibility. 2014;

for modeling the magnetic field emissions of power electronic equipment from magnetic near field measurements. IEEE Transactions on Electromagnetic Compatibility. 2017;

**56**(4):784-791

*An Efficient Approach Based on the Near-Field Technique to Solve EMI Problems: Application…*

**59**(2):609-617

3-18

21-46

Research M. 2013;**31**:247-262

[2] Labiedh W, Ben Hadj Slama J. Analysis and modeling of the magnetic near fields emitted by an IGBT and by a power diode generic radiating model for active components. In: International Conference on Electrical Sciences and Technologies in Maghreb (CISTEM).

[3] Levy P-E, Gautier C, Costa F, Revol B, Labarre C. Accurate modeling of radiated electromagnetic field by a coil with atoroidalferromagnetic core. IEEE Transactions on Electromagnetic Compatibility. 2013;**55**(5):825-833

[4] Shim HW, Hubing T. A closed-form expression for estimating radiated emissions from the power planes in a populated printed circuit board. IEEE Transactions on Electromagnetic Compatibility. 2006;**48**(1):74-81

[5] Tong X, Thomas DWP, Nothofer A, Sewell P, Christopoulos C. Modeling electromagnetic emissions from printed circuit boards in closed environments using equivalent dipoles. IEEE Transactions on Electromagnetic Compatibility. 2010;**52**(2):462-470

[6] Beghou L, Liu Y, Pichon L, Costa F. Synthesis of equivalent 3-D models from near field measurements

application to the EMC of power printed circuits boards. IEEE Transactions on Magnetics. 2009;**45**(3):1650-1653

Detection of electromagnetic radiations sources at the switching time scale using an inverse problem-based resolution method—Application to power

electronic circuits. IEEE Transactions on Electromagnetic Compatibility. 2015;

[7] Beghou L, Costa F, Pichon L.

**57**(1)

**133**

According to the results presented in the previous figure, the magnetic near field calculated using the identified model parameters is in good agreement with the measured magnetic near field. These results confirm the existence of the equivalent sources network found by the electromagnetic inverse method in the time domain. Thus, the results showed that the proposed method guarantees the uniqueness of the solution. This confirms the effectiveness of the proposed approach for model the high-power structures where the radiated emissions are very high.

#### **7. Conclusion**

The frequency domain electromagnetic inverse method has several disadvantages when modeling radiation of power electronics systems. These disadvantages are mainly related to the radiation over a broad frequency band. To overcome these drawbacks, we have proposed in this chapter the use of the time domain electromagnetic inverse method. This novel method has been applied to identify the equivalent radiating sources of an AC-DC converter. The radiating sources are identified by exploiting the measurements of the vertical component (Hz) of the magnetic field performed over the studied converter. This equivalent model has been validated by using the measurements of the tangential component of the near magnetic field (Hx). The good agreement between the calculated results by using the optimized parameters of the equivalent model and the experimental results measured above the studied structure highlights the robustness of the suggested inverse problem when seeking equivalent radiating sources for industrial systems creating transient signals.

#### **Author details**

Bessem Zitouna\* and Jaleleddine Ben Hadj Slama Laboratory of Advanced Technology and Intelligent Systems (LATIS), National Engineering School of Sousse (ENISo), University of Sousse, Tunisia

\*Address all correspondence to: bessem.zitouna@yahoo

© 2020 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

*An Efficient Approach Based on the Near-Field Technique to Solve EMI Problems: Application… DOI: http://dx.doi.org/10.5772/intechopen.89332*

### **References**

equivalent radiating sources. In two distinct instants (t1 = 10.06 μs and

the high-power structures where the radiated emissions are very high.

cartographies is presented in **Figure 16**.

*Advances in Complex Analysis and Applications*

**7. Conclusion**

creating transient signals.

**Author details**

**132**

Bessem Zitouna\* and Jaleleddine Ben Hadj Slama

provided the original work is properly cited.

\*Address all correspondence to: bessem.zitouna@yahoo

Laboratory of Advanced Technology and Intelligent Systems (LATIS), National

© 2020 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium,

Engineering School of Sousse (ENISo), University of Sousse, Tunisia

t2 = 10.47 μs), a comparison between the measured and estimated of the near-field

calculated using the identified model parameters is in good agreement with the measured magnetic near field. These results confirm the existence of the equivalent sources network found by the electromagnetic inverse method in the time domain. Thus, the results showed that the proposed method guarantees the uniqueness of the solution. This confirms the effectiveness of the proposed approach for model

According to the results presented in the previous figure, the magnetic near field

The frequency domain electromagnetic inverse method has several disadvantages when modeling radiation of power electronics systems. These disadvantages are mainly related to the radiation over a broad frequency band. To overcome these drawbacks, we have proposed in this chapter the use of the time domain electromagnetic inverse method. This novel method has been applied to identify the equivalent radiating sources of an AC-DC converter. The radiating sources are identified by exploiting the measurements of the vertical component (Hz) of the magnetic field performed over the studied converter. This equivalent model has been validated by using the measurements of the tangential component of the near magnetic field (Hx). The good agreement between the calculated results by using the optimized parameters of the equivalent model and the experimental results measured above the studied structure highlights the robustness of the suggested inverse problem when seeking equivalent radiating sources for industrial systems

[1] Saidi S, Ben Hadj Slama J. Analysis and modeling of power MOSFET radiation. Progress In Electromagnetics Research M. 2013;**31**:247-262

[2] Labiedh W, Ben Hadj Slama J. Analysis and modeling of the magnetic near fields emitted by an IGBT and by a power diode generic radiating model for active components. In: International Conference on Electrical Sciences and Technologies in Maghreb (CISTEM). 2014. pp. 1-6

[3] Levy P-E, Gautier C, Costa F, Revol B, Labarre C. Accurate modeling of radiated electromagnetic field by a coil with atoroidalferromagnetic core. IEEE Transactions on Electromagnetic Compatibility. 2013;**55**(5):825-833

[4] Shim HW, Hubing T. A closed-form expression for estimating radiated emissions from the power planes in a populated printed circuit board. IEEE Transactions on Electromagnetic Compatibility. 2006;**48**(1):74-81

[5] Tong X, Thomas DWP, Nothofer A, Sewell P, Christopoulos C. Modeling electromagnetic emissions from printed circuit boards in closed environments using equivalent dipoles. IEEE Transactions on Electromagnetic Compatibility. 2010;**52**(2):462-470

[6] Beghou L, Liu Y, Pichon L, Costa F. Synthesis of equivalent 3-D models from near field measurements application to the EMC of power printed circuits boards. IEEE Transactions on Magnetics. 2009;**45**(3):1650-1653

[7] Beghou L, Costa F, Pichon L. Detection of electromagnetic radiations sources at the switching time scale using an inverse problem-based resolution method—Application to power electronic circuits. IEEE Transactions on Electromagnetic Compatibility. 2015; **57**(1)

[8] Saidi S, Ben Hadj Slama J. A nearfield technique based on PZMI, GA, and ANN: Application to power electronics systems. IEEE Transactions on Electromagnetic Compatibility. 2014; **56**(4):784-791

[9] Benyoubi F, Pichon L, Bensetti M, Le Bihan Y, Feliach M. An efficient method for modeling the magnetic field emissions of power electronic equipment from magnetic near field measurements. IEEE Transactions on Electromagnetic Compatibility. 2017; **59**(2):609-617

[10] Liu Y, Ravelo B, Fernandez-Lopez P. Modeling of magnetic near-field radiated by electronic devices disturbed by transient signals with complex form. Applied Physics Research. 2012;**4**(1): 3-18

[11] Liu Y, Ravelo B. Application of nearfield emission processing for microwave circuits under ultra-short duration perturbations. Advanced Electromagnetics. 2012;**1**(3)

[12] Liu Y, Ravelo B. Fully time-domain scanning of EM near-field radiated by RF circuits. Progress In Electromagnetics Research B. 2014;**57**: 21-46

[13] Rioult J, Seethramdoo D, Heddebaut M. Novel electromagnetic field measuring instrument with realtime visualization. Proc IEEE Int. Symp. EMC, Austin, Texas, USA. 2009:133-138

[14] Ravelo B, Liu Y, Slama JBH. Timedomain planar near-field/near-field transform with PWS operations. The European Physical Journal Applied Physics. 2011;**53**(03):1-8

[15] Liu Y, Ravelo B, Jastrzebski AK, Hadj Slama JB. Calculation of the time domain z-component of the EM-nearfield from the x- and y-components. In: 41st European Microwave Conference. 2011

[16] Zitouna B, Ben Hadj Slama J. Enhancement of time-domain electromagnetic inverse method for modeling circuits radiations. IEEE Transactions on Electromagnetic Compatibility. 2016;**58**(2):534-542

[17] Kim Y-H, Jang J-W, Shin S-C, Chung-Yuen W. Weighted-efficiency enhancement control for a photovoltaic AC module interleaved flyback inverter using a synchronous rectifier. IEEE Transactions on Power Electronics. 2014;**29**(12)

[18] Chen Y, Chang C, Yang P. A novel primary-side controlled universal-input AC-DC LED driver based on a sourcedriving control scheme. IEEE Transactions on Power Electronics. 2015;**30**(8)

[19] Zhu Z, Wu Q, Wang Z. Selfcompensating OCP control scheme for primary-side controlled flyback AC/DC converters. IEEE Transactions on Power Electronics. 2017;**32**(5)

[20] Gao X, Fan J, Zhang Y, Kajbaf H, Pommerenke D. Far-field prediction using only magnetic near-field scanning for EMI test. IEEE Transactions on Electromagnetic Compatibility. 2014; **56**(6):1335-1343

[21] Ravelo B. E-field extraction from Hnear-field in time-domain by using PWS method. Progress In Electromagnetics Research B. 2010;**25**:171-189

[22] Liu Y, Ravelo B, Jastrzebski AK. Calculation of time-domain near field Ex,y,z(t) from Hx,y(t) with PWS and FFT transforms. In: Proc. Int. Symp. Electromagn. Compat.; 17-20 September 2012; Roma, Italy. 2012. pp. 1-6

[23] Saidi S, Slama JBH. Effect of genetic algorithm parameters on convergence of the electromagnetic inverse method. In:

8th International Multi-Conference on Systems, Signals & Devices (SSD); 22-25 March 2011; Sousse, Tunisia. 2011. pp. 1-5

**Chapter 9**

**Abstract**

Mathematical Fundamentals

We have proven that the long wave with finite amplitude responds to the structure of the medium. The heterogeneity in a medium structure always introduces additional nonlinearity in comparison with the homogeneous medium. At the same time, a question appears on the inverse problem, namely, is there sufficient information in the wave field to reconstruct the structure of the medium? It turns out that the knowledge on the evolution of nonlinear waves enables us to form the theoretical fundamentals of the diagnostic method to define the characteristics of a heterogeneous medium using the long waves of finite amplitudes (inverse problem). The mass contents of the particular components can be denoted with specified

**Keywords:** diagnostics, nonlinear waves, inverse problem, asymptotic model,

Natural media, in the general case, should not be treated as structureless. The experiments have shown that the intrinsic structure of a medium influences the wave motions [1–8]. Existing inhomogeneities complicate the problem and, at the same time, are fully manifested under the propagation of nonlinear waves. The principal part of the problem is associated with the phenomena caused by the nonlinear behavior of natural media, in particular, more substantial increase of nonlinear effects in structured media than that in homogeneous ones [1–5, 7].

The wave processes in heterogeneous media are usually described in terms of more or less complicated models. Under the conditions of local equilibrium, the media are traditionally modelled irrespective of their structure. In the framework of continuum mechanics, the known idealization of a real medium as a homogeneous

**2. The model notion of the medium with structure**

of a Diagnostic Method

by Long Nonlinear Waves

for the Structured Media

*Vyacheslav Vakhnenko, Dmitri Vengrovich*

*and Alexandre Michtchenko*

accuracy by this diagnostic method.

structured medium

**1. Introduction**

**135**

#### **Chapter 9**

41st European Microwave Conference.

*Advances in Complex Analysis and Applications*

8th International Multi-Conference on Systems, Signals & Devices (SSD); 22-25 March 2011; Sousse, Tunisia. 2011.

pp. 1-5

[16] Zitouna B, Ben Hadj Slama J. Enhancement of time-domain electromagnetic inverse method for modeling circuits radiations. IEEE Transactions on Electromagnetic Compatibility. 2016;**58**(2):534-542

[17] Kim Y-H, Jang J-W, Shin S-C, Chung-Yuen W. Weighted-efficiency enhancement control for a photovoltaic AC module interleaved flyback inverter using a synchronous rectifier. IEEE Transactions on Power Electronics.

[18] Chen Y, Chang C, Yang P. A novel primary-side controlled universal-input AC-DC LED driver based on a source-

driving control scheme. IEEE Transactions on Power Electronics.

[19] Zhu Z, Wu Q, Wang Z. Selfcompensating OCP control scheme for primary-side controlled flyback AC/DC converters. IEEE Transactions on Power

[20] Gao X, Fan J, Zhang Y, Kajbaf H, Pommerenke D. Far-field prediction using only magnetic near-field scanning for EMI test. IEEE Transactions on Electromagnetic Compatibility. 2014;

[21] Ravelo B. E-field extraction from Hnear-field in time-domain by using PWS method. Progress In Electromagnetics

[22] Liu Y, Ravelo B, Jastrzebski AK. Calculation of time-domain near field Ex,y,z(t) from Hx,y(t) with PWS and FFT

Electromagn. Compat.; 17-20 September

[23] Saidi S, Slama JBH. Effect of genetic algorithm parameters on convergence of the electromagnetic inverse method. In:

transforms. In: Proc. Int. Symp.

2012; Roma, Italy. 2012. pp. 1-6

Research B. 2010;**25**:171-189

Electronics. 2017;**32**(5)

**56**(6):1335-1343

**134**

2011

2014;**29**(12)

2015;**30**(8)

## Mathematical Fundamentals of a Diagnostic Method by Long Nonlinear Waves for the Structured Media

*Vyacheslav Vakhnenko, Dmitri Vengrovich and Alexandre Michtchenko*

#### **Abstract**

We have proven that the long wave with finite amplitude responds to the structure of the medium. The heterogeneity in a medium structure always introduces additional nonlinearity in comparison with the homogeneous medium. At the same time, a question appears on the inverse problem, namely, is there sufficient information in the wave field to reconstruct the structure of the medium? It turns out that the knowledge on the evolution of nonlinear waves enables us to form the theoretical fundamentals of the diagnostic method to define the characteristics of a heterogeneous medium using the long waves of finite amplitudes (inverse problem). The mass contents of the particular components can be denoted with specified accuracy by this diagnostic method.

**Keywords:** diagnostics, nonlinear waves, inverse problem, asymptotic model, structured medium

#### **1. Introduction**

Natural media, in the general case, should not be treated as structureless. The experiments have shown that the intrinsic structure of a medium influences the wave motions [1–8]. Existing inhomogeneities complicate the problem and, at the same time, are fully manifested under the propagation of nonlinear waves. The principal part of the problem is associated with the phenomena caused by the nonlinear behavior of natural media, in particular, more substantial increase of nonlinear effects in structured media than that in homogeneous ones [1–5, 7].

#### **2. The model notion of the medium with structure**

The wave processes in heterogeneous media are usually described in terms of more or less complicated models. Under the conditions of local equilibrium, the media are traditionally modelled irrespective of their structure. In the framework of continuum mechanics, the known idealization of a real medium as a homogeneous

one has been relatively successful in the description of wave processes (see, e.g., [9–11]). The continuum models are commonly applied to the mixtures whose dispersive dissipative properties are treated with regard to the interactions between the components [12–15]. On this level, the media are modelled in the framework of a homogeneous elastic, viscous elastic, and elastic-plastic media. In this case, the features of the medium structure are taken into account indirectly through the kinetic parameters (relaxation time, viscous coefficients, etc.) [4–6, 10, 12–15].

The model of multi-velocity interpenetratable continua was developed in terms of classical continuum mechanics [16] and statistical physics in order to describe the dynamical behavior of multicomponent media. A fundamental assumption in the theory of mixtures [15] reproduces the assumption in the model of multi-velocity interpenetratable continua [16], namely, that the particles of the existing components occupy each microvolume. The equations of motion for each component involve the terms describing the mass, force, and energy interactions between the components. The problem is complicated by the necessity to employ, in the general case, the experimental data for establishing theoretical relations between the macro parameters at the component interaction level. Moreover, if the component interaction is determined, then these models would be indispensable in the theory of multicomponent media.

In all the models mentioned, the formalism of continuum mechanics is based on the principle of local action as well as on the generalization of the mechanics laws relating the point mass to the continuum [11].

Traditionally, it was considered that in heterogeneous media with wavelength appreciably exceeding the size of the structural heterogeneities, the perturbations propagate in the same way as inhomogeneous media [9, 10, 15]. However, this statement should be proven, and we shall show that this approximation is not universally correct. In general case, the wave propagation cannot be described in

The properties of a medium deviate from the equilibrium state under the propagation of intensive waves. Moreover, an unperturbed medium can be in one of the unstable stationary states. So, a geophysical medium, within a current physical concept, is an open thermodynamic system, which substantially influences the exchanges of energy and mass. Thus, a description of open systems should take into account the peculiarities of their inner structure, dynamical processes occurring on the level of structural elements. What is more, the state of media under the action of high-frequency wave perturbations departs from equilibrium, and, thus, the behavior of media cannot be described in the framework of equilibrium thermodynamics. Consequently, there is a necessity to develop new mathematical models to take into account the nonlinear wave perturbations and irreversible inner exchange

terms of the average characteristics (continuum model).

*Model of the layered medium with two homogeneous components in the period.*

*Mathematical Fundamentals of a Diagnostic Method by Long Nonlinear Waves…*

*DOI: http://dx.doi.org/10.5772/intechopen.91462*

**3. Asymptotic averaged model for structured medium**

We describe the wave processes in nonequilibrium heterogeneous media in terms of an asymptotic averaged model [18–22]. The obtained integral differential system of equations cannot be reduced to the average terms (pressure, mass velocity, specific volume) and contains the terms with characteristic sizes of individual

The most straightforward heterogeneous media for which the effect of the structure can be analyzed are media with a regular structure. Features of the propagation of long-wave perturbations will be investigated by using, as an example, a periodic medium under conditions of equality of stresses and mass velocities on the boundaries of neighboring components. It is supposed that the microstructure elements of medium *dv* (see **Figure 1**) are large enough that it is possible to submit to

processes.

**Figure 1.**

components.

**137**

**3.1 Background and initial equations**

When going from the integral equations to differential balance equations, the existence of a differentially small microvolume *dv* is assumed. On the one hand, this volume is so small that the mechanics laws of the point can be extended to the whole microvolume. On the other hand, the volume contains so many structural elements of the medium that, in this sense, it can be regarded as macroscopic one despite its smallness as compared to the entire volume occupied by the medium. So, the passage to the differential balance equations is based on the assumption that microstructural scales *ε* are small as compared to the characteristic macroscopic scale of the *λ* and the passage should be made to the limiting case *ε=λ* ! 0. Contraction of the volume *dv* to the point is in the general case correct for continuous functions [11, 15]. This means that all points within the differentially small volume are equivalent. Hence, for the case of a mixture, the equivalence of the points implies that field characteristics should be averaged over *dv*. Hence, it is assumed that the equations of motion can be written in terms of average density, mass velocity, and pressure of each component. We note that these models do not contain the exact sizes of components.

The application of the models of a homogeneous medium to the description of the dynamical wave processes in a structured natural medium is associated with specific fundamental difficulties [3, 4, 6, 8]. In what follows, we treat the medium structure at the macrolevel. We abandon the assumption that the differentially small volume *dv* contains all the components of the medium. Nevertheless, we consider the long-wave approach with the wavelength *λ* much higher than the characteristic length of the medium structure *ε*. We consider a structured medium (**Figure 1**) in which separated components are considered as a homogeneous medium (the differentially small volume *dv* is much smaller than the characteristic size of a particular component *ε*).

Within continuum mechanics [17], the known idealization of a real medium as homogeneous is widely used for modelling their dynamic behavior. In these models, the effect of heterogeneity is taken into account indirectly throughout the kinetic parameters such as a viscous coefficient and relaxation time. The inner processes, in this case, are manifested through dispersive dissipative properties of a medium.

*Mathematical Fundamentals of a Diagnostic Method by Long Nonlinear Waves… DOI: http://dx.doi.org/10.5772/intechopen.91462*

**Figure 1.**

*Model of the layered medium with two homogeneous components in the period.*

Traditionally, it was considered that in heterogeneous media with wavelength appreciably exceeding the size of the structural heterogeneities, the perturbations propagate in the same way as inhomogeneous media [9, 10, 15]. However, this statement should be proven, and we shall show that this approximation is not universally correct. In general case, the wave propagation cannot be described in terms of the average characteristics (continuum model).

The properties of a medium deviate from the equilibrium state under the propagation of intensive waves. Moreover, an unperturbed medium can be in one of the unstable stationary states. So, a geophysical medium, within a current physical concept, is an open thermodynamic system, which substantially influences the exchanges of energy and mass. Thus, a description of open systems should take into account the peculiarities of their inner structure, dynamical processes occurring on the level of structural elements. What is more, the state of media under the action of high-frequency wave perturbations departs from equilibrium, and, thus, the behavior of media cannot be described in the framework of equilibrium thermodynamics. Consequently, there is a necessity to develop new mathematical models to take into account the nonlinear wave perturbations and irreversible inner exchange processes.

#### **3. Asymptotic averaged model for structured medium**

We describe the wave processes in nonequilibrium heterogeneous media in terms of an asymptotic averaged model [18–22]. The obtained integral differential system of equations cannot be reduced to the average terms (pressure, mass velocity, specific volume) and contains the terms with characteristic sizes of individual components.

#### **3.1 Background and initial equations**

The most straightforward heterogeneous media for which the effect of the structure can be analyzed are media with a regular structure. Features of the propagation of long-wave perturbations will be investigated by using, as an example, a periodic medium under conditions of equality of stresses and mass velocities on the boundaries of neighboring components. It is supposed that the microstructure elements of medium *dv* (see **Figure 1**) are large enough that it is possible to submit to

the laws of classical continuum mechanics for each component. At the same time, the inner processes in each component will be considered within a relaxation approach. The notions based on the relaxation nature of a phenomenon are regarded to be promising and fruitful. We consider that the properties of the medium, such as density, sound velocity, and relaxation time, vary periodically (although this assumption is unessential in the final result).

#### *3.1.1 Motion equations for individual component*

The analysis of wave motions is based on the hydrodynamic approach. This restriction can be imposed for the modelling of nonlinear waves in water-saturated soils, bubble media, aerosols, etc. [13]. The set of acceptable media could be extended to solid media where the powerful loads are studied in the condition that the strength and plasticity of the material can be neglected [23]. In the hydrodynamic approach, we have considered the media without tangential stresses, while there are equalities of the stresses as well as of mass velocities on boundaries of neighboring components. Also, we assume that the medium is barotropic. The individual components of the medium are considered to be described by the classical equations of hydrodynamics. In the Lagrangian coordinate system ð Þ *l*, *t* , the equations of one-dimensional motion for each component have the form

$$\begin{aligned} \frac{\partial r^{\nu}}{\partial l^{\nu}} &= \frac{V}{V\_{0}}, \quad u = \frac{\partial r}{\partial t}, \\\frac{\partial u}{\partial t} + V\_{0} \left(\frac{r}{l}\right)^{\nu - 1} \frac{\partial p}{\partial l} &= \mathbf{0}. \end{aligned} \tag{1}$$

models of a relaxing medium are more general than the equilibrium models for

the state equation. It defines the completeness of the relaxation process:

*Mathematical Fundamentals of a Diagnostic Method by Long Nonlinear Waves…*

An equilibrium state equation of a barotropic medium is one-parameter equation. As a result of relaxation, an additional variable *ξ* (inner parameter) appears in

i. The lack of relaxation (inner interaction processes are frozen) *ξ* ¼ 1,

ii. The relaxation complete (there is the local thermodynamic equilibrium)

The equations of state (5) and (6) are considered to be known. These relation-

ships enable us to introduce the sound velocities for fast processes:

*c* 2

*c* 2

The equilibrium equations of state are considered to be known:

[17]). We note that the mechanisms of the exchange processes are not explicitly defined when deriving Eq. (9), and the thermodynamic and kinetic parameters appear only in this equation. These characteristics can be found

*ρ<sup>e</sup>* � *ρ*<sup>0</sup> ¼

ð *p*

*p*0 *c* �2

Clearly, for the fast processes *ωτ<sup>p</sup>* ≫ 1 � �, we have the relation (5), and for the

The substantiation of Eq. (9) within the framework of the thermodynamics of irreversible processes has been given in [17, 24–26]. As far as we know, the first work in this field was paper by Mandelshtam and Leontovich (see Section 81 in

The phenomenological approach for describing the relaxation processes in hydrodynamics has been developed in many publications [13, 17, 26]. The dynamic equation of state was used (a) for describing the propagation of sound waves in a

*τp dρ dt* � *<sup>c</sup>* �2 *f dp*

The slow and fast processes are compared, utilizing relaxation time *τp*. The dynamic state equation is written down in the form of the first-order differential

*p* ¼ *p*ð Þ *ρ*, *ξ :* (4)

*p* ¼ *p*ð Þ¼ *ρ*, 1 *p <sup>f</sup>*ð Þ*ρ :* (5)

*p* ¼ *p*ð Þ¼ *ρ*, 0 *pe*ð Þ*ρ :* (6)

*<sup>f</sup>* ¼ *dp <sup>f</sup> =dρ* (7)

*<sup>e</sup>* ¼ *dpe=dρ:* (8)

*<sup>e</sup> dp:* (10)

*dt* � � <sup>þ</sup> *<sup>ρ</sup>* � *<sup>ρ</sup><sup>e</sup>* ð Þ¼ <sup>0</sup>*:* (9)

describing the wave propagation.

*DOI: http://dx.doi.org/10.5772/intechopen.91462*

There are two limiting cases:

*ξ* ¼ 0,

and for slow processes

slow ones *ωτ<sup>p</sup>* ≪ 1 � �, we obtain (6).

experimentally.

**139**

equation:

The equation of continuity can also be used in the alternative form

$$\frac{\partial V}{\partial t} - \nu V\_0 \frac{\partial r^{\nu - 1} u}{\partial l^{\nu}} = 0. \tag{2}$$

Here *<sup>V</sup>* <sup>¼</sup> *<sup>ρ</sup>*�<sup>1</sup> is the specific volume; *<sup>ν</sup>* is a parameter of symmetry, where *<sup>ν</sup>* <sup>¼</sup> <sup>1</sup> is planar symmetry, *ν* ¼ 2 is cylindrical one, and *ν* ¼ 3 is spherical one; and index 0 relates to the initial state. The other notations are those that are generally accepted.

Conditions for matching are the equality of mass velocities and pressures on the boundaries of the components:

$$\begin{bmatrix} \mathfrak{u} \end{bmatrix} = \mathbf{0}, \qquad \begin{bmatrix} p \end{bmatrix} = \mathbf{0}. \tag{3}$$

#### *3.1.2 Dynamic state equation*

Considering the models of a relaxing medium as more general than the equilibrium models for describing the evolution of high-gradient waves, we will take into account the relaxing processes for each component. Thermodynamic equilibrium is disturbed owing to the propagation of fast perturbations in a medium. There are processes of the interaction that tend to return the equilibrium. The parameters characterizing this interaction are referred to as the inner variables, unlike the macro parameters such as the pressure *p*, mass velocity *u*, and density *ρ*. In essence, the change of macro parameters caused by the changes of inner parameters is a relaxation process. From the nonequilibrium thermodynamics standpoint, the

*Mathematical Fundamentals of a Diagnostic Method by Long Nonlinear Waves… DOI: http://dx.doi.org/10.5772/intechopen.91462*

models of a relaxing medium are more general than the equilibrium models for describing the wave propagation.

An equilibrium state equation of a barotropic medium is one-parameter equation. As a result of relaxation, an additional variable *ξ* (inner parameter) appears in the state equation. It defines the completeness of the relaxation process:

$$p = p(\rho, \xi). \tag{4}$$

There are two limiting cases:

i. The lack of relaxation (inner interaction processes are frozen) *ξ* ¼ 1,

$$p = p(\rho, \mathbf{1}) = p\_f(\rho). \tag{5}$$

ii. The relaxation complete (there is the local thermodynamic equilibrium) *ξ* ¼ 0,

$$p = p(\rho, \mathbf{0}) = p\_\epsilon(\rho). \tag{6}$$

The equations of state (5) and (6) are considered to be known. These relationships enable us to introduce the sound velocities for fast processes:

$$
\sigma\_f^2 = d p\_f / d \rho \tag{7}
$$

and for slow processes

$$c\_{\epsilon}^{2} = dp\_{\epsilon}/d\rho. \tag{8}$$

The slow and fast processes are compared, utilizing relaxation time *τp*. The dynamic state equation is written down in the form of the first-order differential equation:

$$
\tau\_p \left( \frac{d\rho}{dt} - c\_f^{-2} \frac{dp}{dt} \right) + (\rho - \rho\_e) = \mathbf{0}.\tag{9}
$$

The equilibrium equations of state are considered to be known:

$$
\rho\_\epsilon - \rho\_0 = \int\_{p\_0}^p c\_\epsilon^{-2} dp. \tag{10}
$$

Clearly, for the fast processes *ωτ<sup>p</sup>* ≫ 1 � �, we have the relation (5), and for the slow ones *ωτ<sup>p</sup>* ≪ 1 � �, we obtain (6).

The substantiation of Eq. (9) within the framework of the thermodynamics of irreversible processes has been given in [17, 24–26]. As far as we know, the first work in this field was paper by Mandelshtam and Leontovich (see Section 81 in [17]). We note that the mechanisms of the exchange processes are not explicitly defined when deriving Eq. (9), and the thermodynamic and kinetic parameters appear only in this equation. These characteristics can be found experimentally.

The phenomenological approach for describing the relaxation processes in hydrodynamics has been developed in many publications [13, 17, 26]. The dynamic equation of state was used (a) for describing the propagation of sound waves in a

relaxing medium [17], (b) for taking into account the exchange processes within media (gas-solid particles) [26], and (c) for studying wave fields in gas-liquid media and in soil [13]. In most works, the equation of state has been derived from the concept of precise mechanism for the inner process. Within the context of mixture theory, Biot [12] attempted to account for the nonequilibrium in velocities between components directly in the equations of motion in the form of dissipative terms.

�*ε*�<sup>1</sup> *<sup>∂</sup> <sup>r</sup> <sup>ν</sup>* ð Þð Þ <sup>0</sup>

*<sup>ε</sup>*<sup>0</sup> *<sup>u</sup>*ð Þ <sup>0</sup> � *<sup>∂</sup>r*ð Þ <sup>0</sup> *∂t* � �

�*ε*�<sup>1</sup> *ν*

�*ε*�1*<sup>ν</sup> <sup>r</sup>ν*�<sup>1</sup> ð Þð Þ <sup>0</sup> *<sup>∂</sup>p*ð Þ <sup>0</sup>

*<sup>∂</sup> <sup>r</sup>ν*�<sup>1</sup> ð Þð Þ <sup>0</sup> *<sup>u</sup>*ð Þ <sup>0</sup>

*<sup>∂</sup> <sup>r</sup> <sup>ν</sup>* ð Þð Þ <sup>0</sup> *∂s*

*∂u*ð Þ <sup>0</sup> *∂t*

*∂V*ð Þ <sup>0</sup> *<sup>∂</sup><sup>t</sup>* � *<sup>ν</sup>*

*<sup>u</sup>*ð Þ <sup>0</sup> <sup>¼</sup> *<sup>∂</sup>r*ð Þ <sup>0</sup>

þ

*<sup>∂</sup><sup>t</sup>* ,

þ *ν r*

and perform the normalization Ð <sup>1</sup>

*<sup>∂</sup><sup>ξ</sup>* <sup>þ</sup> *<sup>ε</sup>*<sup>0</sup> *<sup>∂</sup> <sup>r</sup> <sup>ν</sup>* ð Þð Þ <sup>0</sup>

þ … ¼ 0,

*DOI: http://dx.doi.org/10.5772/intechopen.91462*

*<sup>∂</sup><sup>ξ</sup>* <sup>þ</sup> *<sup>ε</sup>*<sup>0</sup> *<sup>∂</sup>u*ð Þ <sup>0</sup>

*<sup>∂</sup><sup>ξ</sup>* <sup>þ</sup> *<sup>ε</sup>*<sup>0</sup> *<sup>∂</sup>V*ð Þ <sup>0</sup>

*<sup>∂</sup><sup>s</sup>* � *<sup>∂</sup> <sup>r</sup> <sup>ν</sup>* ð Þð Þ<sup>1</sup>

!

*<sup>∂</sup><sup>t</sup>* <sup>þ</sup> *<sup>ν</sup> <sup>r</sup>*

*<sup>∂</sup><sup>t</sup>* <sup>þ</sup> *<sup>ν</sup>*

*<sup>r</sup>*ð Þ <sup>0</sup> <sup>¼</sup> *<sup>r</sup>*ð Þ <sup>0</sup> ð Þ *<sup>s</sup>*, *<sup>t</sup>* , are independent of *<sup>ξ</sup>*. Furthermore

*<sup>∂</sup><sup>ξ</sup>* <sup>¼</sup> *<sup>V</sup>*ð Þ <sup>0</sup> ,

*∂s*

*u*ð Þ <sup>0</sup> *<sup>∂</sup><sup>s</sup>* � *<sup>ν</sup>*

þ *ν r*

*<sup>ν</sup>*�<sup>1</sup> � �ð Þ <sup>0</sup> *<sup>∂</sup>p*ð Þ<sup>1</sup>

Thus, we can average the equations during the period *<sup>ξ</sup>*. We define h i� <sup>¼</sup> <sup>Ð</sup> <sup>1</sup>

integrals can be calculated as *<sup>∂</sup>p*ð Þ<sup>1</sup> *<sup>=</sup>∂<sup>ξ</sup>* � � <sup>¼</sup> 0, *<sup>∂</sup>u*ð Þ<sup>1</sup> *<sup>=</sup>∂<sup>ξ</sup>* � � <sup>¼</sup> 0, and *<sup>∂</sup>r*ð Þ<sup>1</sup> *<sup>=</sup>∂<sup>ξ</sup>* � � <sup>¼</sup> 0. Moreover, as *<sup>u</sup>*ð Þ <sup>0</sup> � � <sup>¼</sup> *<sup>u</sup>*ð Þ <sup>0</sup> , *<sup>p</sup>*ð Þ <sup>0</sup> � � <sup>¼</sup> *<sup>p</sup>*ð Þ <sup>0</sup> than *<sup>∂</sup>p*ð Þ<sup>1</sup> *<sup>=</sup>∂<sup>ξ</sup>* <sup>¼</sup> 0. This means that *<sup>p</sup>*ð Þ<sup>1</sup> does not also depend on *ξ*. After integrating over the structure period the equations

containing the value of zero order of *ε*, we obtain the averaged system

*<sup>∂</sup><sup>s</sup>* <sup>¼</sup> *<sup>V</sup>*ð Þ <sup>0</sup> D E,

*<sup>∂</sup><sup>t</sup>* ,

*dp* � *<sup>V</sup>*ð Þ <sup>0</sup>

Hereafter, we will consider only the zero approximation of the equations, and,

*<sup>ν</sup>*�<sup>1</sup> � �ð Þ <sup>0</sup> *<sup>∂</sup>p*ð Þ <sup>0</sup>

*<sup>∂</sup> <sup>r</sup><sup>ν</sup>*�<sup>1</sup> ð Þð Þ <sup>0</sup>

Unlike the values *u*ð Þ <sup>0</sup> , *p*ð Þ <sup>0</sup> , *p*ð Þ<sup>1</sup> , and *r*ð Þ <sup>0</sup> , the specific volume *V*ð Þ <sup>0</sup> is a function of *ξ*.

*<sup>∂</sup><sup>s</sup>* <sup>¼</sup> 0,

*u*ð Þ <sup>0</sup> *<sup>∂</sup><sup>s</sup>* <sup>¼</sup> <sup>0</sup>

*<sup>τ</sup>pVe <sup>p</sup>*ð Þ <sup>0</sup> ð Þ *<sup>V</sup>*ð Þ <sup>0</sup> � *Ve <sup>p</sup>*ð Þ <sup>0</sup> � � � � \* +*dt:* (16)

þ *ν r*

*<sup>∂</sup> <sup>r</sup> <sup>ν</sup>* ð Þð Þ <sup>0</sup>

*∂u*ð Þ <sup>0</sup> *∂t*

with the averaged equation of state

*c*2 *f*

\* +

*d V*ð Þ <sup>0</sup> D E ¼ � *<sup>V</sup>*ð Þ <sup>0</sup> � �<sup>2</sup>

**141**

*∂ V*ð Þ <sup>0</sup> � � *<sup>∂</sup><sup>t</sup>* � *<sup>ν</sup>*

*<sup>u</sup>*ð Þ <sup>0</sup> <sup>¼</sup> *<sup>∂</sup>r*ð Þ <sup>0</sup>

*<sup>∂</sup> <sup>r</sup><sup>ν</sup>*�<sup>1</sup> ð Þð Þ<sup>1</sup>

*<sup>∂</sup><sup>ξ</sup>* <sup>¼</sup> 0,

*<sup>∂</sup> <sup>r</sup><sup>ν</sup>*�<sup>1</sup> ð Þð Þ <sup>0</sup>

*u*ð Þ<sup>1</sup> *<sup>∂</sup><sup>ξ</sup>* <sup>¼</sup> <sup>0</sup>*:*

and *r*ð Þ<sup>1</sup> are periodic, the

*u*ð Þ <sup>0</sup> *<sup>∂</sup><sup>ξ</sup>* � *<sup>ν</sup>*

<sup>0</sup>*d<sup>ξ</sup>* <sup>¼</sup> 1. Since *<sup>p</sup>*ð Þ<sup>1</sup> , *<sup>u</sup>*ð Þ<sup>1</sup> ,

*<sup>∂</sup> <sup>r</sup> <sup>ν</sup>* ð Þð Þ<sup>1</sup>

*<sup>ν</sup>*�<sup>1</sup> � �ð Þ <sup>0</sup> *<sup>∂</sup>p*ð Þ <sup>0</sup>

*<sup>∂</sup> <sup>r</sup><sup>ν</sup>*�<sup>1</sup> ð Þð Þ <sup>0</sup>

*<sup>∂</sup><sup>ξ</sup>* � *<sup>V</sup>*ð Þ <sup>0</sup>

*<sup>ν</sup>*�<sup>1</sup> � �ð Þ <sup>0</sup> *<sup>∂</sup>p*ð Þ <sup>0</sup>

*Mathematical Fundamentals of a Diagnostic Method by Long Nonlinear Waves…*

*<sup>∂</sup> <sup>r</sup>ν*�<sup>1</sup> ð Þð Þ <sup>0</sup> *<sup>u</sup>*ð Þ <sup>0</sup> *<sup>∂</sup><sup>s</sup>* �*<sup>ν</sup>*

þ … ¼ 0,

*<sup>∂</sup><sup>s</sup>* <sup>þ</sup>*<sup>ν</sup> <sup>r</sup>*

According to the general theory of the asymptotic method, the terms of equal

powers of *<sup>ε</sup>* should vanish independently of each other. Thus, *<sup>∂</sup>p*ð Þ <sup>0</sup> *<sup>=</sup>∂<sup>ξ</sup>* <sup>¼</sup> 0, *<sup>∂</sup>u*ð Þ <sup>0</sup> *<sup>=</sup>∂<sup>ξ</sup>* <sup>¼</sup> 0, and *<sup>∂</sup> <sup>r</sup><sup>ν</sup>*�<sup>1</sup> ð Þð Þ <sup>0</sup> *<sup>=</sup>∂<sup>ξ</sup>* <sup>¼</sup> 0, i.e., *<sup>p</sup>*ð Þ <sup>0</sup> <sup>¼</sup> *<sup>p</sup>*ð Þ <sup>0</sup> ð Þ *<sup>s</sup>*, *<sup>t</sup>* , *<sup>u</sup>*ð Þ <sup>0</sup> <sup>¼</sup> *<sup>u</sup>*ð Þ <sup>0</sup> ð Þ *<sup>s</sup>*, *<sup>t</sup>* , and

*<sup>ν</sup>*�<sup>1</sup> � �ð Þ<sup>1</sup> *<sup>∂</sup>p*ð Þ <sup>0</sup>

<sup>þ</sup> … <sup>¼</sup> 0, �

*<sup>∂</sup> <sup>r</sup>ν*�<sup>1</sup> ð Þð Þ<sup>1</sup> *<sup>u</sup>*ð Þ <sup>0</sup> *<sup>∂</sup><sup>ξ</sup>* � *<sup>ν</sup>*

<sup>þ</sup> … <sup>¼</sup> 0,

*<sup>∂</sup><sup>ξ</sup>* <sup>þ</sup> *<sup>ν</sup> <sup>r</sup>*

*<sup>ν</sup>*�<sup>1</sup> � �ð Þ <sup>0</sup> *<sup>∂</sup>p*ð Þ<sup>1</sup> *∂ξ* �

*<sup>∂</sup> <sup>r</sup>ν*�<sup>1</sup> ð Þð Þ <sup>0</sup> *<sup>u</sup>*ð Þ<sup>1</sup> *∂ξ*

!

(13)

(14)

<sup>0</sup>ð Þ� *dξ*

(15)

We assume that the relaxation time and sound velocities do not depend on time, but they are functions of pressure and the individual properties of the components. This means that in the process of a relaxation interaction, we can take into account the exchange of moment and heat but not that of mass. The dynamic equation of state determines the features of the intrastructural interaction for each component.

The equations of motion (1) have been written in the Lagrangian coordinate system. The necessity of such a description stems from the fact that the dynamic equation of state (9) has been written to the mass element of a medium. Besides, the use of the Lagrangian coordinates is essential for the application of the method of asymptotic averaging, since in these coordinates, the structure is independent of a wave process.

#### **3.2 Asymptotic averaged system of equations**

A regularity of structure and a nonlinearity of long-wave processes investigated here specify the choice of mathematical methods. One way of studying this heterogeneous medium is based on a method of asymptotic averaging of equations with high-oscillating coefficients [20, 27–30]. The essence of this method consists in the application of a multiscale method in combination with a space averaging. In accordance with this method, the mass space coordinate *m* ¼ *l ν =V*<sup>0</sup> is divided into two independent coordinates: slow coordinate *s* and fast one *ξ*, wherein

$$
\delta m = \mathfrak{s} + \varepsilon \mathfrak{s}, \qquad \frac{\partial}{\partial m} = \frac{\partial}{\partial \mathfrak{s}} + \varepsilon^{-1} \frac{\partial}{\partial \xi}. \tag{11}
$$

The slow coordinate *s* corresponds to a global change of the wave field and *s* is a constant value during a period, while the fast coordinate *ξ* traces the variations of a field in the structure period. The dependent functions are presented as a degree series over the structure period *ε*

$$\begin{split} V(m,t) &= V^{(0)}(s,t,\xi) + \varepsilon V^{(1)}(s,t,\xi) + \varepsilon^2 V^{(2)}(s,t,\xi) + \dots \\ p(m,t) &= p^{(0)}(s,t,\xi) + \varepsilon p^{(1)}(s,t,\xi) + \varepsilon^2 p^{(2)}(s,t,\xi) + \dots \\ u(m,t) &= u^{(0)}(s,t,\xi) + \varepsilon u^{(1)}(s,t,\xi) + \varepsilon^2 u^{(2)}(s,t,\xi) + \dots \\ r^{\nu}(m,t) &= (r^{\nu})^{(0)}(s,t,\xi) + \varepsilon (r^{\nu})^{(1)}(s,t,\xi) + \varepsilon^2 (r^{\nu})^{(2)}(s,t,\xi) + \dots \end{split} \tag{12}$$

where *p*ð Þ*<sup>i</sup>* , *u*ð Þ*<sup>i</sup>* , *V*ð Þ*<sup>i</sup>* , and *r*ð Þ*<sup>i</sup>* are defined as the one-period functions of *ξ*. In the Lagrangian mass coordinates the period is a constant which allows the averaging procedure to be performed.

We now will prove that *<sup>p</sup>*ð Þ <sup>0</sup> <sup>¼</sup> *<sup>p</sup>*ð Þ <sup>0</sup> ð Þ *<sup>s</sup>*, *<sup>t</sup>* , *<sup>p</sup>*ð Þ<sup>1</sup> <sup>¼</sup> *<sup>p</sup>*ð Þ<sup>1</sup> ð Þ *<sup>s</sup>*, *<sup>t</sup>* , *<sup>u</sup>*ð Þ <sup>0</sup> <sup>¼</sup> *<sup>u</sup>*ð Þ <sup>0</sup> ð Þ *<sup>s</sup>*, *<sup>t</sup>* , and *<sup>r</sup> <sup>ν</sup>* ð Þð Þ <sup>0</sup> <sup>¼</sup> *<sup>r</sup> <sup>ν</sup>* ð Þð Þ <sup>0</sup> ð Þ *<sup>s</sup>*, *<sup>t</sup>* are independent of the fast variable *<sup>ξ</sup>*. Indeed, after substitution of Eqs. (11) and (12) into the initial equations of motion, we obtain

*Mathematical Fundamentals of a Diagnostic Method by Long Nonlinear Waves… DOI: http://dx.doi.org/10.5772/intechopen.91462*

$$\begin{split} & -e^{-1} \frac{\partial (r^{\nu})^{(0)}}{\partial \xi} + e^{0} \left( \frac{\partial (r^{\nu})^{(0)}}{\partial t} - \frac{\partial (r^{\nu})^{(1)}}{\partial \xi} - V^{(0)} \right) + \dots = 0, \\ & \varepsilon^{0} \left( u^{(0)} - \frac{\partial r^{(0)}}{\partial t} \right) + \dots = 0, \\ & - e^{-1} \nu (r^{\nu-1})^{(0)} \frac{\partial p^{(0)}}{\partial \xi} + \varepsilon^{0} \left( \frac{\partial u^{(0)}}{\partial t} + \nu \left( r^{\nu-1} \right)^{(0)} \frac{\partial p^{(0)}}{\partial t} + \nu \left( r^{\nu-1} \right)^{(1)} \frac{\partial p^{(0)}}{\partial \xi} + \nu \left( r^{\nu-1} \right)^{(0)} \frac{\partial p^{(1)}}{\partial \xi} \right) + \dots = 0, \\ & - \varepsilon^{-1} \nu \frac{\partial (r^{\nu-1})^{(0)} u^{(0)}}{\partial \xi} + \varepsilon^{0} \left( \frac{\partial V^{(0)}}{\partial t} + \nu \frac{\partial (r^{\nu-1})^{(0)} u^{(0)}} - \nu \frac{\partial (r^{\nu-1})^{(1)} u^{(0)}}{\partial \xi} - \nu \frac{\partial (r^{\nu-1})^{(0)} u^{(0)}}{\partial \xi} \right) + \dots = 0, \end{split} \tag{13}$$

According to the general theory of the asymptotic method, the terms of equal powers of *<sup>ε</sup>* should vanish independently of each other. Thus, *<sup>∂</sup>p*ð Þ <sup>0</sup> *<sup>=</sup>∂<sup>ξ</sup>* <sup>¼</sup> 0, *<sup>∂</sup>u*ð Þ <sup>0</sup> *<sup>=</sup>∂<sup>ξ</sup>* <sup>¼</sup> 0, and *<sup>∂</sup> <sup>r</sup><sup>ν</sup>*�<sup>1</sup> ð Þð Þ <sup>0</sup> *<sup>=</sup>∂<sup>ξ</sup>* <sup>¼</sup> 0, i.e., *<sup>p</sup>*ð Þ <sup>0</sup> <sup>¼</sup> *<sup>p</sup>*ð Þ <sup>0</sup> ð Þ *<sup>s</sup>*, *<sup>t</sup>* , *<sup>u</sup>*ð Þ <sup>0</sup> <sup>¼</sup> *<sup>u</sup>*ð Þ <sup>0</sup> ð Þ *<sup>s</sup>*, *<sup>t</sup>* , and *<sup>r</sup>*ð Þ <sup>0</sup> <sup>¼</sup> *<sup>r</sup>*ð Þ <sup>0</sup> ð Þ *<sup>s</sup>*, *<sup>t</sup>* , are independent of *<sup>ξ</sup>*. Furthermore

$$\begin{split} \frac{\partial (r^{\nu})^{(0)}}{\partial \mathbf{s}} + \frac{\partial (r^{\nu})^{(1)}}{\partial \xi} &= V^{(0)}, \\ u^{(0)} &= \frac{\partial r^{(0)}}{\partial t}, \\ \frac{\partial u^{(0)}}{\partial t} + \nu (r^{\nu-1})^{(0)} \frac{\partial p^{(0)}}{\partial \mathbf{s}} + \nu (r^{\nu-1})^{(0)} \frac{\partial p^{(1)}}{\partial \xi} &= \mathbf{0}, \\ \frac{\partial V^{(0)}}{\partial t} - \nu \frac{\partial (r^{\nu-1})^{(0)} u^{(0)}}{\partial \mathbf{s}} - \nu \frac{\partial (r^{\nu-1})^{(1)} u^{(0)}}{\partial \xi} - \nu \frac{\partial (r^{\nu-1})^{(0)} u^{(1)}}{\partial \xi} &= \mathbf{0}. \end{split} \tag{14}$$

Thus, we can average the equations during the period *<sup>ξ</sup>*. We define h i� <sup>¼</sup> <sup>Ð</sup> <sup>1</sup> <sup>0</sup>ð Þ� *dξ* and perform the normalization Ð <sup>1</sup> <sup>0</sup>*d<sup>ξ</sup>* <sup>¼</sup> 1. Since *<sup>p</sup>*ð Þ<sup>1</sup> , *<sup>u</sup>*ð Þ<sup>1</sup> , and *r*ð Þ<sup>1</sup> are periodic, the integrals can be calculated as *<sup>∂</sup>p*ð Þ<sup>1</sup> *<sup>=</sup>∂<sup>ξ</sup>* � � <sup>¼</sup> 0, *<sup>∂</sup>u*ð Þ<sup>1</sup> *<sup>=</sup>∂<sup>ξ</sup>* � � <sup>¼</sup> 0, and *<sup>∂</sup>r*ð Þ<sup>1</sup> *<sup>=</sup>∂<sup>ξ</sup>* � � <sup>¼</sup> 0. Moreover, as *<sup>u</sup>*ð Þ <sup>0</sup> � � <sup>¼</sup> *<sup>u</sup>*ð Þ <sup>0</sup> , *<sup>p</sup>*ð Þ <sup>0</sup> � � <sup>¼</sup> *<sup>p</sup>*ð Þ <sup>0</sup> than *<sup>∂</sup>p*ð Þ<sup>1</sup> *<sup>=</sup>∂<sup>ξ</sup>* <sup>¼</sup> 0. This means that *<sup>p</sup>*ð Þ<sup>1</sup> does not also depend on *ξ*. After integrating over the structure period the equations containing the value of zero order of *ε*, we obtain the averaged system

$$\begin{aligned} \frac{\partial (r^{\nu})^{(0)}}{\partial t} &= \left\langle V^{(0)} \right\rangle, \\ u^{(0)} &= \frac{\partial r^{(0)}}{\partial t}, \\ \frac{\partial u^{(0)}}{\partial t} + \nu \left(r^{\nu-1}\right)^{(0)} \frac{\partial p^{(0)}}{\partial t} &= \mathbf{0}, \\ \frac{\partial \left\langle V^{(0)} \right\rangle}{\partial t} - \nu \frac{\partial (r^{\nu-1})^{(0)} u^{(0)}}{\partial t} &= \mathbf{0} \end{aligned} \tag{15}$$

with the averaged equation of state

$$d\left\langle V^{(0)}\right\rangle = -\left\langle \frac{\left(V^{(0)}\right)^2}{c\_f^2} \right\rangle dp - \left\langle \frac{V^{(0)}}{\tau\_p V\_\epsilon(p^{(0)})} \left(V^{(0)} - V\_\epsilon\left(p^{(0)}\right)\right) \right\rangle dt. \tag{16}$$

Unlike the values *u*ð Þ <sup>0</sup> , *p*ð Þ <sup>0</sup> , *p*ð Þ<sup>1</sup> , and *r*ð Þ <sup>0</sup> , the specific volume *V*ð Þ <sup>0</sup> is a function of *ξ*. Hereafter, we will consider only the zero approximation of the equations, and,

therefore, the upper index 0 is omitted. Choosing the wavelength *λ* to be large enough, we can always reduce the effect to zero from other approximation terms.

From the mathematical point of view, this means that in the transformation (19)

*<sup>∂</sup><sup>s</sup> :* (20)

*udt*, *tE* ¼ *t:* (21)

*ρ ξ*ð Þ*:* (24)

*<sup>d</sup><sup>ζ</sup>* <sup>¼</sup> <sup>~</sup>*ρ*�<sup>1</sup> (25)

(23)

*<sup>R</sup><sup>ν</sup>* <sup>¼</sup> *s V*h i*:* (22)

*<sup>∂</sup><sup>t</sup>* <sup>¼</sup> *<sup>∂</sup>νr<sup>ν</sup>*�<sup>1</sup>*<sup>u</sup>*

This condition is satisfied if *A* ¼ h i *V* , because the equation converts into the continuity Eq. (15). We obtain the following transformation between Lagrangian

*ν*�1

*<sup>∂</sup>r<sup>ν</sup>*�<sup>1</sup>*u V*h i�<sup>1</sup>

<sup>þ</sup> h i *<sup>V</sup> <sup>∂</sup><sup>p</sup>*

*<sup>∂</sup><sup>r</sup>* <sup>¼</sup> 0,

*<sup>∂</sup><sup>r</sup>* <sup>¼</sup> <sup>0</sup>*:*

It is reasonable to define the slow Lagrangian coordinate (non-mass one) as

the value *dr<sup>ν</sup>* is a total differential. Therefore, we must have

and Eulerian systems of coordinates:

*DOI: http://dx.doi.org/10.5772/intechopen.91462*

*∂A*

*Mathematical Fundamentals of a Diagnostic Method by Long Nonlinear Waves…*

*dr<sup>ν</sup>* <sup>¼</sup> h i *<sup>V</sup> ds* <sup>þ</sup> *<sup>ν</sup><sup>r</sup>*

Eq. (15) in the Eulerian system of coordinates then take the form

þ

It is convenient to determine the fast Eulerian coordinate *ζ* as

*∂ζ ∂ξ* � �

*t* <sup>¼</sup> <sup>~</sup>*<sup>ρ</sup>*

It should be noted that the average density ~*ρ* in the Eulerian coordinates is a

ð 1

*V ρ* ~*ρ*

0

proves that h i *<sup>V</sup>* �<sup>1</sup> is the average density of the medium in the Eulerian coordinates. Note that ~*ρ* 6¼ h i*ρ* . The value ~*ρ* is a real density. The value h i *V* is the specific volume averaged in units of mass over the period, and it is expressed as the ratio of the volume to the mass inside this volume. This value can be determined experimentally. At the same time, the averaged values *p* and *u* coincide in both Lagrangian and Eulerian systems of coordinates. Now the equations of motion (23) can be

The notation of the equations of motion in the averaged values enables us to suggest the method of the computer solution for the system of equations, where the integration step is restricted by the perturbation wavelength and not by the period of the structure [22]. Then the main computational problem associated with the smallness of the integration step can be avoided, and the equations of motion can be solved at a significant distance of wave propagation within a reasonable time.

We will analyze the propagation of nonlinear waves in a structured medium. To make the results more clear, we will restrict our consideration to a nonrelaxation

*V*ð Þ*ξ dξ* ¼

*<sup>∂</sup>*h i *<sup>V</sup>* �<sup>1</sup> *∂tE*

*∂u ∂tE* þ *u ∂u ∂r*

value usually used for density. A chain of identities

written in the usual form of the averaged density ~*ρ*.

**3.4 Nonlinear waves**

**143**

h i *V* ¼

ð 1

0

The averaged system of Eqs. (15) and (16) is an integrodifferential one and, in the general case, is not reduced to the averaged variables *p*, *u*, and *V*ð Þ <sup>0</sup> . The derivation of Eqs. (15) and (16) relates to a rigorous periodic medium. However, it may be shown that Eqs. (15) and (16) are also relevant to media with a quasiperiodic structure. Indeed, the pressure *p* and the mass velocity *u* are independent of the fast variable *ξ*. Hence on a microscale *ξ*, the action is statically uniform (waveless) over the whole period of the medium structure, while on the slow scale *s*, the action of perturbation is manifested by the wave motion of the medium. On a microlevel, the behavior of medium adheres only to the thermodynamic laws. There is a mechanical equilibrium. On a macrolevel, the motion of the medium is described by the wave dynamics laws for averaged variables. Mathematically, in the zero-order case of *ε*, the size of the period is infinitesimal ð Þ *ε* ! 0 . This signifies that the location of particular components in the period is irrelevant. Eqs. (15) and (16) do not change their form if the components are broken and/or change their location in an elementary cell. This means that Eqs. (15) and (16) describe the motion of any quasiperiodic (statistical heterogeneous) medium which has a constant mass content of components on the microlevel and the location of these components within the cell is not important.

In the case of nonlinear wave propagation, the individual components suffer different compressions. The structure of the medium is changed, with the result that the averaged specific volume h i *V* is changed. This change differs from the change of the specific volume for homogeneous medium under the same loading. Thus, the structure of medium is manifested in the wave motion, despite the fact that the equations of motion (15) (but not the equation of state) are written down for the averaged values *u*, *p*, and h i *V* only.

#### **3.3 System of equations in Eulerian coordinates**

In certain cases of theoretical analysis, it is more convenient to use the Eulerian coordinate system. The immediate employment of a method of the asymptotic averaging in Eulerian variables is impossible because of the variability of the microstructure sizes. However, from the zero approximation in the equations of motion (11), which are presented by the averaged values *p*, *u*, and h i *V* , the equations can be rewritten in the Eulerian system of coordinates ð Þ *r*, *tE* utilizing a transformation from the Lagrangian system ð Þ *s*, *t* [18–22]:

$$r = r(\mathfrak{s}, \mathfrak{t}), \quad \mathfrak{t}\_E = \mathfrak{t}.\tag{17}$$

There is an essential presumption that the velocity of the particle in the zero approximation is constant throughout of the structure, and, consequently, we can describe an averaged trajectory for the particle:

$$
\left(\frac{\partial r(s,t)}{\partial t}\right)\_s = u(s,t). \tag{18}
$$

From the physical point of view, it is clear that the position of the particle is unambiguously defined by its coordinate and time:

$$dr^{\nu} = A ds + \nu r^{\nu - 1} u dt, \qquad t\_{\mathcal{E}} = t. \tag{19}$$

*Mathematical Fundamentals of a Diagnostic Method by Long Nonlinear Waves… DOI: http://dx.doi.org/10.5772/intechopen.91462*

From the mathematical point of view, this means that in the transformation (19) the value *dr<sup>ν</sup>* is a total differential. Therefore, we must have

$$\frac{\partial A}{\partial t} = \frac{\partial \nu r^{\nu - 1}u}{\partial s}. \tag{20}$$

This condition is satisfied if *A* ¼ h i *V* , because the equation converts into the continuity Eq. (15). We obtain the following transformation between Lagrangian and Eulerian systems of coordinates:

$$dr^{\nu} = \langle V \rangle ds + \nu r^{\nu - 1} udt, \qquad t\_E = t. \tag{21}$$

It is reasonable to define the slow Lagrangian coordinate (non-mass one) as

$$
\mathcal{R}^{\nu} = \mathfrak{s} \langle \mathcal{V} \rangle. \tag{22}
$$

Eq. (15) in the Eulerian system of coordinates then take the form

$$\begin{aligned} \frac{\partial \langle V \rangle^{-1}}{\partial t\_E} + \frac{\partial r^{\nu - 1} u \langle V \rangle^{-1}}{\partial r} &= 0, \\ \frac{\partial u}{\partial t\_E} + u \frac{\partial u}{\partial r} + \langle V \rangle \frac{\partial p}{\partial r} &= 0. \end{aligned} \tag{23}$$

It is convenient to determine the fast Eulerian coordinate *ζ* as

$$
\left(\frac{\partial \zeta}{\partial \xi}\right)\_t = \frac{\tilde{\rho}}{\rho(\xi)}.\tag{24}
$$

It should be noted that the average density ~*ρ* in the Eulerian coordinates is a value usually used for density. A chain of identities

$$\langle V \rangle = \int\_0^1 V(\xi)d\xi = \int\_0^1 V \frac{\rho}{\tilde{\rho}} d\zeta = \tilde{\rho}^{-1} \tag{25}$$

proves that h i *<sup>V</sup>* �<sup>1</sup> is the average density of the medium in the Eulerian coordinates. Note that ~*ρ* 6¼ h i*ρ* . The value ~*ρ* is a real density. The value h i *V* is the specific volume averaged in units of mass over the period, and it is expressed as the ratio of the volume to the mass inside this volume. This value can be determined experimentally. At the same time, the averaged values *p* and *u* coincide in both Lagrangian and Eulerian systems of coordinates. Now the equations of motion (23) can be written in the usual form of the averaged density ~*ρ*.

The notation of the equations of motion in the averaged values enables us to suggest the method of the computer solution for the system of equations, where the integration step is restricted by the perturbation wavelength and not by the period of the structure [22]. Then the main computational problem associated with the smallness of the integration step can be avoided, and the equations of motion can be solved at a significant distance of wave propagation within a reasonable time.

#### **3.4 Nonlinear waves**

We will analyze the propagation of nonlinear waves in a structured medium. To make the results more clear, we will restrict our consideration to a nonrelaxation media *c* ¼ *cf* ¼ *ce* � �. The averaged equation of state in this case is simplified to the form

$$d\langle V\rangle = -\left\langle \frac{V^2}{c^2} \right\rangle dp,\tag{26}$$

The system of Eq. (15) is hyperbolic ones, and this specifies the breaking solutions, which are shock waves. For the analysis of such solutions, it is necessary to

Now we can quickly formulate the conditions on the shock front, when there is

where indexes 0 and 1 relate to the parameters of the flow before and after the front, respectively. Hence, the formula for the averaged velocity of the shock front in terms of the Lagrangian variable *D* (dimension ½ � *D* is kg/s) and the mass velocity

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

� �*=*ð Þ h i *<sup>V</sup>*<sup>0</sup> � h i *<sup>V</sup>*<sup>1</sup>

*p*<sup>1</sup> � *p*<sup>0</sup>

In the next section, it will be proven that the heterogeneity of the medium structure always introduces additional nonlinearity that does not arise in a homogeneous medium. This effect makes it possible to formulate the theoretical grounds of a new diagnostic method that determines the characteristics of a heterogeneous medium with the use of finite-amplitude long waves (inverse problem). This diagnostic method can also be employed to find the mass contents of individual

In the previous section, we proved that the long wave with finite amplitude responds to the structure of medium. At the same time, a question appears, namely, is there sufficient information in the wave field to reconstruct the structure of medium? It turns out that the knowledge on the evolution of nonlinear waves enables one to define with certain accuracy the concentrations of medium

In this section, we shall prove the statement that the structure of medium always exalts the nonlinear effects under the propagation of long waves. At first, let us consider the sound velocity in homogeneous *c*hom and heterogeneous *c*eff media. Now we will show that in the general case with pressure increase the velocity of the

*c*eff ≥*c*hom*:* (35)

sound becomes higher in a structured medium than in a homogeneous one:

These systems are not expressed in the average hydrodynamical terms; hence the dynamical behavior of the medium cannot be modelled by a homogeneous medium even for long waves, if they are nonlinear. The structure of the medium

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

� �ð Þ h i *<sup>V</sup>*<sup>0</sup> � h i *<sup>V</sup>*<sup>1</sup>

ð Þ h i *V*<sup>1</sup> � h i *V*<sup>0</sup> *D* þ *u*<sup>1</sup> � *u*<sup>0</sup> ¼ 0, ð Þ *u*<sup>1</sup> � *u*<sup>0</sup> *D* � *p*<sup>1</sup> þ *p*<sup>0</sup> ¼ 0, (33)

conservation of the fluxes of mass and impulse through the shock front:

*Mathematical Fundamentals of a Diagnostic Method by Long Nonlinear Waves…*

*p*<sup>1</sup> � *p*<sup>0</sup>

q

∮ ½ �¼ h i *V ds* þ *udt* 0, ∮ ½ �¼ *uds* � *pdt* 0*:* (32)

,

*:*

(34)

present Eq. (15) in the form of integral conservation laws:

*u* follow from the following relations:

*DOI: http://dx.doi.org/10.5772/intechopen.91462*

*D* ¼

influences the nonlinear wave propagation.

components.

components.

**145**

q

*u*<sup>1</sup> � *u*<sup>0</sup> ¼

**4. Diagnostics of a medium by long nonlinear waves**

**4.1 The increase of nonlinearity in medium with structure**

and we can introduce an effective sound velocity by the formula

$$
\omega\_{\rm eff} = \sqrt{\langle \mathbf{V} \rangle^2 / \left\langle \frac{\mathbf{V}^2}{c^2} \right\rangle}. \tag{27}
$$

We obtain a traditional representation of the system of Eqs. (15) and (26).

The system of the equations is concerned in the hyperbolic type of a system. Now we restrict ourselves to the plane symmetry ð Þ *ν* ¼ 1 . Substituting the equation of state (26) into the equation of the continuity (the last equation in (15)), we get

$$
\left<\frac{V^2}{c^2}\right>\frac{\partial p}{\partial t} + \frac{\partial u}{\partial t} = 0.\tag{28}
$$

The combination of this equation with the third equation in (15) ð Þ *ν* ¼ 1 leads to the relationships

$$
\left(\frac{\partial u}{\partial t} \pm \left\langle \frac{V^2}{c^2} \right\rangle^{1/2} \frac{\partial p}{\partial t} \right) \pm \left\langle \frac{V^2}{c^2} \right\rangle^{-1/2} \left(\frac{\partial u}{\partial t} \pm \left\langle \frac{V^2}{c^2} \right\rangle^{1/2} \frac{\partial p}{\partial t} \right) = \mathbf{0}. \tag{29}
$$

From this relationship, it is seen that the averaged system of the equations pertains to the hyperbolic system. The equations for the characteristic in Lagrangian coordinates (mass space coordinate) have the forms

$$\frac{d\mathbf{s}}{dt} = \pm \left\langle \frac{V^2}{c^2} \right\rangle^{-1/2}. \tag{30}$$

In characteristic, the relations are the following:

$$I\_{\pm} = u \pm \int \left\langle \frac{V^2}{c^2} \right\rangle^{1/2} dp. \tag{31}$$

Analogously to the homogeneous medium, we call these relations as the Riemann invariants. The value (30) has the physical meaning, namely, it is the averaged velocity of the wave propagation in the Lagrangian coordinates. This velocity depends on pressure and integrally on a structure. Note the particular case. It is known that in vacuum the wave does not propagate. This result also follows formally from Eq. (30). The hyperbolism of a system points up that this system can describe the shock wave. The equations for the characteristic (30) and the Riemann invariants (31) are the integrodifferential equations, since they retain the variable *V*2 *=c*<sup>2</sup> � �, which depends on the properties of the structural elements in medium.

It should be noted that *c*eff is not an averaged value, i.e., *c*<sup>2</sup> eff 6¼ *<sup>c</sup>*<sup>2</sup> � �. Evidently, the structure of the medium introduces a certain contribution to the nonlinearity. In fact, even if *cf* 6¼ *f p*ð Þ, then in the general case, the value of *c*eff is a function of pressure.

*Mathematical Fundamentals of a Diagnostic Method by Long Nonlinear Waves… DOI: http://dx.doi.org/10.5772/intechopen.91462*

The system of Eq. (15) is hyperbolic ones, and this specifies the breaking solutions, which are shock waves. For the analysis of such solutions, it is necessary to present Eq. (15) in the form of integral conservation laws:

$$
\oint [\langle V \rangle ds + \mathbf{u} dt] = \mathbf{0}, \quad \oint [\mathbf{u} ds - p dt] = \mathbf{0}. \tag{32}
$$

Now we can quickly formulate the conditions on the shock front, when there is conservation of the fluxes of mass and impulse through the shock front:

$$(\langle V\_1 \rangle - \langle V\_0 \rangle)D + \mathfrak{u}\_1 - \mathfrak{u}\_0 = \mathbf{0}, \quad (\mathfrak{u}\_1 - \mathfrak{u}\_0)D - p\_1 + p\_0 = \mathbf{0}, \tag{33}$$

where indexes 0 and 1 relate to the parameters of the flow before and after the front, respectively. Hence, the formula for the averaged velocity of the shock front in terms of the Lagrangian variable *D* (dimension ½ � *D* is kg/s) and the mass velocity *u* follow from the following relations:

$$\begin{aligned} D &= \sqrt{(p\_1 - p\_0)/(\langle V\_0 \rangle - \langle V\_1 \rangle)}, \\ u\_1 - u\_0 &= \sqrt{(p\_1 - p\_0)(\langle V\_0 \rangle - \langle V\_1 \rangle)}. \end{aligned} \tag{34}$$

These systems are not expressed in the average hydrodynamical terms; hence the dynamical behavior of the medium cannot be modelled by a homogeneous medium even for long waves, if they are nonlinear. The structure of the medium influences the nonlinear wave propagation.

In the next section, it will be proven that the heterogeneity of the medium structure always introduces additional nonlinearity that does not arise in a homogeneous medium. This effect makes it possible to formulate the theoretical grounds of a new diagnostic method that determines the characteristics of a heterogeneous medium with the use of finite-amplitude long waves (inverse problem). This diagnostic method can also be employed to find the mass contents of individual components.

#### **4. Diagnostics of a medium by long nonlinear waves**

In the previous section, we proved that the long wave with finite amplitude responds to the structure of medium. At the same time, a question appears, namely, is there sufficient information in the wave field to reconstruct the structure of medium? It turns out that the knowledge on the evolution of nonlinear waves enables one to define with certain accuracy the concentrations of medium components.

#### **4.1 The increase of nonlinearity in medium with structure**

In this section, we shall prove the statement that the structure of medium always exalts the nonlinear effects under the propagation of long waves. At first, let us consider the sound velocity in homogeneous *c*hom and heterogeneous *c*eff media. Now we will show that in the general case with pressure increase the velocity of the sound becomes higher in a structured medium than in a homogeneous one:

$$
\mathcal{L}\_{\rm eff} \ge \mathcal{c}\_{\rm hom}.\tag{35}
$$

For the sake of clarity, we consider a medium in which the sound velocities of individual components are independent of the pressure:

$$c \neq f(p), \quad dc/dp = 0. \tag{36}$$

In addition to the analysis of the sound velocity in homogeneous and heterogeneous media, we consider now the evolution equations with the nonlinear term and compare the coefficients of nonlinearity in these media. Let us derive the evolution equation with weak nonlinearity. First of all, we have to note that the mass velocity

> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi *V*2 *=c*<sup>2</sup>

Functional dependence of an average specific value on the pressure increment

In this case, the system of Eq. (23) for planar symmetry *ν* ¼ 1 can be written as

*∂p*0

1 2 *d*2 h i *V dp*<sup>2</sup>

Now let us consider the waves propagating in one direction, and then with the

*∂ ∂t* þ *∂ <sup>∂</sup><sup>x</sup>* ! <sup>2</sup> *<sup>∂</sup> ∂x*

(see, e.g., Section 93 in Ref. [17]). Thus, after factorization of Eq. (45) we get

The coefficient of nonlinearity *α<sup>p</sup>* for the structured medium, when the sound velocities in the individual components are independent of the pressure *c* 6¼ *f p*ð Þ,

h i *<sup>V</sup> <sup>V</sup>*<sup>2</sup> *c*2 � ��3*=*<sup>2</sup>

*dp*<sup>2</sup> <sup>¼</sup> *d u*ð Þ <sup>þ</sup> *<sup>c</sup>*eff

*p*<sup>0</sup> þ 1 2 *d*2 h i *V dp*<sup>2</sup>

> � � � � � *p*¼*p*<sup>0</sup>

q� �*dp:* (41)

� � � � � *p*¼*p*<sup>0</sup>

*∂p*0<sup>2</sup>

*<sup>∂</sup><sup>x</sup>* follows from Eq. (41) with the assumed accuracy

*∂*2 *p*02

� � � � � *p*¼*p*<sup>0</sup>

*d*2 h i *V dp*<sup>2</sup> *<sup>p</sup>*<sup>0</sup> *<sup>∂</sup>p*<sup>0</sup>

*dp* <sup>¼</sup> h i *<sup>V</sup> <sup>V</sup>*<sup>3</sup>

*c*4 � � *V*<sup>2</sup>

*c*2 � ��3*=*<sup>2</sup>

*p*02

*<sup>∂</sup><sup>x</sup>* <sup>¼</sup> <sup>0</sup>*:* (44)

*:* (42)

*<sup>∂</sup><sup>t</sup>* <sup>¼</sup> 0, (43)

*<sup>∂</sup>t*<sup>2</sup> <sup>¼</sup> <sup>0</sup>*:* (45)

*<sup>∂</sup><sup>x</sup>* <sup>¼</sup> <sup>0</sup>*:* (47)

*:* (48)

(46)

*u* is related to the pressure *p* by means of [22]

*DOI: http://dx.doi.org/10.5772/intechopen.91462*

*u* ¼ ð *p*

*Mathematical Fundamentals of a Diagnostic Method by Long Nonlinear Waves…*

*p*<sup>0</sup> ¼ *p* � *p*<sup>0</sup> with the accuracy *O p*<sup>0</sup> ð Þ2 can be presented as a series:

*V*2 *c*2 � �

0

0

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi *V*2 *=c*<sup>2</sup> q� �

h i *V*

indicated accuracy, we can write (hereinafter index 0 is omitted):

�

*∂p*0 *∂x* þ 1 2

*d*2 h i *V* *∂*2 *p*0 *∂t*<sup>2</sup> þ

*∂u ∂t*

*∂p*0 *<sup>∂</sup><sup>t</sup>* � <sup>1</sup> 2 *d*2 h i *V dp*<sup>2</sup>

þ h i *V* <sup>0</sup>

*O p*0<sup>2</sup> � � and was used for derivation of the first equation. The evolution equation for

h i *V* ð Þ¼ *p* h i *V* <sup>0</sup> þ

h i *V* <sup>0</sup> *∂u ∂x* þ

> *∂p*0 *<sup>∂</sup><sup>x</sup>* <sup>¼</sup> *<sup>p</sup>*<sup>0</sup> *<sup>∂</sup><sup>u</sup>*

h i *<sup>V</sup>* <sup>2</sup> 0 *∂*2 *p*0 *<sup>∂</sup>x*<sup>2</sup> � *<sup>V</sup>*<sup>2</sup> *c*2 � �

*∂p*0 *∂t* þ *c*e*ff*

h i *<sup>V</sup> <sup>V</sup>*<sup>2</sup> *c*2 � ��3*=*<sup>2</sup>

can be presented as

*<sup>α</sup><sup>p</sup>* � <sup>1</sup> 2

**147**

The relationship *u*

one variable assumes the form

*po*

*d V*h i *dp* � � � � *p*¼*p*<sup>0</sup>

The equality sign is fulfilled (a) for an initial pressure, by virtue of the normalization, and also (b) for a special structured medium in which the relation *<sup>V</sup>*ð Þ*<sup>ξ</sup> <sup>=</sup>c*<sup>2</sup>ð Þ*<sup>ξ</sup>* is not a function on the fast variable *<sup>ξ</sup>*. We must prove which case results in equality and which gives the inequality.

Let us write the relations (36) for homogeneous medium consisting only one component:

$$c\_{\text{hom}} \neq f(p), \quad dc\_{\text{hom}}/dp = 0. \tag{37}$$

For multicomponent medium, the derivative *dc*e*ff =dp* is defined from the relationship

$$\frac{dc\_{\rm eff}}{dp} = \frac{2\langle V\rangle}{\langle V^2/c^2\rangle} \left( \langle V \rangle \left\langle \frac{V^3}{c^4} \right\rangle - \left\langle \frac{V^2}{c^2} \right\rangle^2 \right) \ge 0. \tag{38}$$

This last inequality follows from the well-known Cauchy-Schwarz inequality (see, e.g., [31]). Therefore, with the increase of pressure, the sound velocity *c*eff increases. Consequently, we have the inequality (35) at *p* ≥*p*0.

Moreover, at *p*> *p*<sup>0</sup> the shock adiabatic curve for the medium with a structure always lies above that for the homogeneous medium (they touch only at the initial point *p* ¼ *p*0):

$$\frac{d^2p}{d\left^2} \ge \left(\frac{d^2p}{dV^2}\right)\_{\text{hom}}.\tag{39}$$

Indeed, a ratio of these derivatives is equal to

$$\begin{split} \left(\frac{d^2 p}{d \langle V \rangle^2} \Big/ \left(\frac{d^2 p}{dV^2}\right)\_{\text{hom}} = \frac{\langle V^3/c^4 \rangle \langle V^2/c^2 \rangle^{-3}}{c\_{\text{hom}}^2 \langle V \rangle^3} \\ = \frac{\langle V^3/c^4 \rangle \langle V \rangle c\_{\text{eff}}^2}{c\_{\text{hom}}^2 \langle V^2/c^2 \rangle^2} \ge \frac{\langle V^3/c^4 \rangle \langle V \rangle}{\left\langle V^2/c^2 \right\rangle^2} \ge 1. \end{split} \tag{40}$$

Hence, a long wave with a finite amplitude responds to the structure of the medium, and the nonlinear effects increase as compared with those in the homogeneous medium. The nonlinearity takes place even if individual components are described by the linear evolution equation (i.e., at condition (36)).

The exception, as it was noted already, is a medium with the properties of structure *<sup>V</sup>*ð Þ*<sup>ξ</sup> <sup>=</sup>c*<sup>2</sup>ð Þ*<sup>ξ</sup>* 6¼ *<sup>f</sup>*ð Þ*<sup>ξ</sup>* . For this medium, only the equality sign is correct in the inequalities (35) and (39). Particular elements of the structure respond to the pressure variations, but the relative structure does not change, i.e., the ratio *V*ð Þ *ξ*, *p =V ξ*, *p*<sup>0</sup> � � does not depend on *<sup>ξ</sup>*. In this case, the value *<sup>c</sup>*eff <sup>¼</sup> ffiffiffiffiffiffiffiffi *<sup>c</sup>*<sup>2</sup> h i <sup>p</sup> is an averaged characteristic (see Eq. (27)). Therefore, the system of equations may be presented using the averaged variables *<sup>p</sup>*, *<sup>u</sup>*, h i *<sup>V</sup>* , and *<sup>c</sup>*eff <sup>¼</sup> ffiffiffiffiffiffiffiffi *<sup>c</sup>*<sup>2</sup> h i <sup>p</sup> . Heterogeneity does not introduce an additional nonlinearity for this medium, and the structure of medium does not affect the wave motion.

*Mathematical Fundamentals of a Diagnostic Method by Long Nonlinear Waves… DOI: http://dx.doi.org/10.5772/intechopen.91462*

In addition to the analysis of the sound velocity in homogeneous and heterogeneous media, we consider now the evolution equations with the nonlinear term and compare the coefficients of nonlinearity in these media. Let us derive the evolution equation with weak nonlinearity. First of all, we have to note that the mass velocity *u* is related to the pressure *p* by means of [22]

$$
\mu = \int\_{p\_o}^{p} \sqrt{\langle \mathbf{V}^2 / c^2 \rangle} dp. \tag{41}
$$

Functional dependence of an average specific value on the pressure increment *p*<sup>0</sup> ¼ *p* � *p*<sup>0</sup> with the accuracy *O p*<sup>0</sup> ð Þ2 can be presented as a series:

$$
\langle V \rangle(p) = \langle V \rangle\_0 + \frac{d\langle V \rangle}{dp} \bigg|\_{p=p\_0} p' + \frac{1}{2} \frac{d^2 \langle V \rangle}{dp^2} \bigg|\_{p=p\_0} p'^2. \tag{42}
$$

In this case, the system of Eq. (23) for planar symmetry *ν* ¼ 1 can be written as

$$
\langle V \rangle\_0 \frac{\partial u}{\partial t} + \left\langle \frac{V^2}{c^2} \right\rangle\_0 \frac{\partial p'}{\partial t} - \frac{1}{2} \frac{d^2 \langle V \rangle}{dp^2} \bigg|\_{p=p\_0} \frac{\partial p'^2}{\partial t} = 0,\tag{43}
$$

$$\frac{\partial \mu}{\partial t} + \langle V \rangle\_0 \frac{\partial p'}{\partial \mathbf{x}} = \mathbf{0}. \tag{44}$$

The relationship *u ∂p*0 *<sup>∂</sup><sup>x</sup>* <sup>¼</sup> *<sup>p</sup>*<sup>0</sup> *<sup>∂</sup><sup>u</sup> <sup>∂</sup><sup>x</sup>* follows from Eq. (41) with the assumed accuracy *O p*0<sup>2</sup> � � and was used for derivation of the first equation. The evolution equation for one variable assumes the form

$$\left\langle \left< V \right>\_{0}^{2} \frac{\partial^{2} p'}{\partial \mathbf{x}^{2}} - \left< \frac{V^{2}}{c^{2}} \right>\_{0} \frac{\partial^{2} p'}{\partial t^{2}} + \frac{1}{2} \frac{d^{2} \langle V \rangle}{dp^{2}} \bigg|\_{p=p\_{0}} \frac{\partial^{2} p'^{2}}{\partial t^{2}} = \mathbf{0}. \tag{45}$$

Now let us consider the waves propagating in one direction, and then with the indicated accuracy, we can write (hereinafter index 0 is omitted):

$$-\frac{\sqrt{\langle \mathbf{V}^2/c^2 \rangle}}{\langle \mathbf{V} \rangle} \frac{\partial}{\partial t} + \frac{\partial}{\partial \mathbf{x}} \rightarrow 2\frac{\partial}{\partial \mathbf{x}}\tag{46}$$

(see, e.g., Section 93 in Ref. [17]). Thus, after factorization of Eq. (45) we get

$$\frac{\partial p'}{\partial t} + c\_{\text{eff}} \frac{\partial p'}{\partial \mathbf{x}} + \frac{1}{2} \langle V \rangle \left\langle \frac{V^2}{c^2} \right\rangle^{-3/2} \frac{d^2 \langle V \rangle}{dp^2} p' \frac{\partial p'}{\partial \mathbf{x}} = \mathbf{0}. \tag{47}$$

The coefficient of nonlinearity *α<sup>p</sup>* for the structured medium, when the sound velocities in the individual components are independent of the pressure *c* 6¼ *f p*ð Þ, can be presented as

$$a\_{\mathbb{P}} \equiv \frac{1}{2} \langle V \rangle \left\langle \frac{V^2}{c^2} \right\rangle^{-3/2} \frac{d^2 \langle V \rangle}{dp^2} = \frac{d(u + c\_{\rm eff})}{dp} = \langle V \rangle \left\langle \frac{V^3}{c^4} \right\rangle \left\langle \frac{V^2}{c^2} \right\rangle^{-3/2}. \tag{48}$$

For all cases we take *α<sup>p</sup>* >0. For a homogeneous medium with *dc=dp* ¼ 0, we have *α<sup>p</sup>* hom ¼ *V=c*.

In certain media the value *V=c*<sup>2</sup> does not change within the period. The individual elements of the structure respond to the pressure variations so that a relative structure does not change, i.e., the ratio *V*ð Þ *ξ*, *p =V ξ*, *p*<sup>0</sup> � � does not depend on *ξ*. In this case, the value *<sup>c</sup>*eff <sup>¼</sup> ffiffiffiffiffiffiffiffi *<sup>c</sup>*<sup>2</sup> h i <sup>p</sup> derived from Eq. (27) is the averaged characteristic. Consequently, the system of equations may be presented in the averaged variables *<sup>p</sup>*, *<sup>u</sup>*, h i *<sup>V</sup>* , and *<sup>c</sup>*eff <sup>¼</sup> ffiffiffiffiffiffiffiffi *<sup>c</sup>*<sup>2</sup> h i <sup>p</sup> . Heterogeneity does not introduce the additional nonlinearity for these media. Such media behave like the homogeneous media under the action of the nonlinear wave perturbations.

For media, when the sound velocity is independent of the pressure ð Þ *c* 6¼ *f p*ð Þ , it is possible to show that heterogeneity of the medium, in the general case, introduces the additional nonlinearity. Let us consider the ratio of the nonlinearity coefficients for heterogeneous and homogeneous media. In the space of dimensionless normalized variables, this implies that at *<sup>p</sup>* <sup>¼</sup> *<sup>p</sup>*<sup>0</sup> we have h i *<sup>V</sup>* <sup>0</sup> <sup>¼</sup> 1 as well as *<sup>V</sup>*<sup>2</sup> *=c*<sup>2</sup> � � <sup>0</sup> ¼ 1 for the compared media.

Using the conditions (27) we can obtain

$$\frac{a\_p}{a\_{p\text{ hom}}} = \langle V \rangle \left\langle \frac{V^3}{c^4} \right\rangle \left\langle \frac{V^2}{c^2} \right\rangle^{-2} \ge 1. \tag{49}$$

(see sections 14.2-6 in Ref. [31]), i.e., when the value *<sup>V</sup>=c*<sup>2</sup> <sup>¼</sup> const does not vary within the period (*V*ð Þ*<sup>ξ</sup> <sup>=</sup>*ð Þ *<sup>c</sup>*ð Þ*<sup>ξ</sup>* <sup>2</sup> 6¼ *<sup>f</sup>*ð Þ*<sup>ξ</sup>* ). This heterogeneous medium has been considered above. For all other heterogeneous media for which the value *V=c*<sup>2</sup> changes within period, the inequality is realized in Eq. (49). So, in a heterogeneous medium, the value *α<sup>p</sup>* is always greater than *α<sup>p</sup>* hom in a homogeneous medium. Thus, it is proved that, in the general case, the heterogeneities in a medium introduce the additional nonlinearity. This effect provides the basis for a new method of diagnostics to define the properties of multicomponent media using the propagation of long

*Mathematical Fundamentals of a Diagnostic Method by Long Nonlinear Waves…*

The structure of the medium affects the wave field. There are different methods which allow the detection of gas bubbles and/or cracks in liquid [32], concrete [33],

In this section, we describe our new diagnostic method for the properties of the medium. The features of the motion of finite-amplitude long waves and the effect of the increase of nonlinearity in the heterogeneous medium in comparison with homogeneous medium form the basis for the development of theoretical fundamentals of the diagnostic method. In this method, the properties of individual components are defined by long waves of finite amplitudes; more specifically, the dependence *<sup>V</sup>=c*<sup>2</sup> <sup>¼</sup> *<sup>V</sup>=c*<sup>2</sup>ð Þ*<sup>ζ</sup>* on the fast Eulerian coordinate *<sup>ζ</sup>* (see Eq. (24)) is

Thus, the nonlinear wave evolution allows one to obtain the structure of the medium with an inherent accuracy. As a final result, the mass concentrations of the

It should be kept in mind that the period of the structure of medium is infinitely small in the long-wave model, so it is not always possible to indicate the location of the structure elements inside the period reliably. Hence, the media with the different structures plotted in **Figure 2**, for example, affect identically on wave fields. These two media are indistinguishable in the framework of the suggested method. Taking into account this indefiniteness, we consider the function *<sup>V</sup>=c*<sup>2</sup> <sup>¼</sup> *<sup>V</sup>=c*<sup>2</sup>ð Þ*<sup>ζ</sup>* that is to be the decreasing, integrable, mutually one-valued function on the interval

Now we represent the theoretical fundamentals for new method of diagnostics of medium by means of the long nonlinear waves. Let us prove the principal relation which enables us to obtain the inverse function *<sup>ζ</sup>* <sup>¼</sup> *<sup>ζ</sup> <sup>V</sup>=c*<sup>2</sup> ð Þ for the desired function

*V Vc*�<sup>2</sup> � �*<sup>n</sup>*þ<sup>1</sup> D E

ð Þ *n* þ 1 !h i *V*

It is known from theory of probability that the distribution function *f x*ð Þ (any one-valued, integrable, positive function) can be expressed by its central moments:

i *nqn* 3

*χ*ð Þ¼ *q F fx* ½ � ð Þ ð Þ*q* , (55)

*<sup>x</sup>nf x*ð Þ*dx:* (54)

<sup>5</sup> *Vc*�<sup>2</sup> � �*:* (53)

*<sup>V</sup>=c*<sup>2</sup> <sup>¼</sup> *<sup>V</sup>=c*<sup>2</sup>ð Þ*<sup>ζ</sup>* through the inverse Fourier transformation [19, 35–37]:

2 4

*n*¼0

*α<sup>n</sup>* ¼

ð<sup>∞</sup> �∞

nonlinear waves in such media.

*DOI: http://dx.doi.org/10.5772/intechopen.91462*

defined.

**149**

**4.2 Fundamentals of new diagnostic method**

and ice cover [34] employing the nonlinear effects.

individual components can be found using this method.

*ζ* ∈ ½ � 0, 1 , and equal to zero outside of this interval.

*<sup>ζ</sup> Vc*�<sup>2</sup> � � <sup>¼</sup> *<sup>F</sup>*�<sup>1</sup> <sup>X</sup><sup>∞</sup>

Indeed, by using the characteristic function

This inequality is the well-known Cauchy-Schwarz inequality (see formula (15.2-3) in Ref. [31]). Since h i *<sup>V</sup>* <sup>≥</sup>0 and *<sup>V</sup>=c*<sup>2</sup> � � <sup>≥</sup>0, we prove

$$
\begin{split}
\langle V \rangle \langle V^{3}/c^{4} \rangle & \quad \equiv \int^{\infty} V d\xi \cdot \int^{\infty} \frac{V^{3}}{c^{4}} d\xi = \int^{\infty} \frac{V^{2}}{c^{2}} \left( \frac{V}{c^{2}} \right)^{-1} d\xi \cdot \int^{\infty} \frac{V^{2}}{c^{2}} \frac{V}{c^{2}} d\xi \\ & \ge \left( \int^{\infty} \sqrt{\frac{V^{2}}{c^{2}} \left( \frac{V}{c^{2}} \right)^{-1}} \cdot \sqrt{\frac{V^{2}}{c^{2}} \frac{V}{c^{2}}} d\xi \right)^{2} \\ & = \left( \int^{\infty} \frac{V^{2}}{c^{2}} d\xi \right)^{2} \equiv \left( V^{2}/c^{2} \right)^{2}.
\end{split} \tag{50}
$$

It only remains to find the condition for the equality sign in (49). For this purpose, we apply the Cauchy-Schwarz inequality in vector form (see formula (15.2-5) in Ref. [31]):

$$\left| \left( \overrightarrow{a}, \overrightarrow{b} \right) \right|^2 \le \left( \overrightarrow{a}, \overrightarrow{a} \right) \left( \overrightarrow{b}, \overrightarrow{b} \right). \tag{51}$$

However, the equality sign is realized if and only if the vectors *a* ! and *b* ! are linearly dependent, i.e., *a* ! <sup>¼</sup> *kb* ! (*k* ¼ const). By designating *a* !, *a* ! � � � *<sup>V</sup>=c*<sup>2</sup> and *b* ! , *b* � �! � *<sup>V</sup>*<sup>2</sup> *=c*2, it is easy to notice that the equality sign is realized if and only if

$$
\sqrt{\frac{V^2}{c^2} \left(\frac{V}{c^2}\right)^{-1}} \bigg/ \sqrt{\frac{V^2}{c^2} \frac{V}{c^2}} = \text{const.}\tag{52}
$$

*Mathematical Fundamentals of a Diagnostic Method by Long Nonlinear Waves… DOI: http://dx.doi.org/10.5772/intechopen.91462*

(see sections 14.2-6 in Ref. [31]), i.e., when the value *<sup>V</sup>=c*<sup>2</sup> <sup>¼</sup> const does not vary within the period (*V*ð Þ*<sup>ξ</sup> <sup>=</sup>*ð Þ *<sup>c</sup>*ð Þ*<sup>ξ</sup>* <sup>2</sup> 6¼ *<sup>f</sup>*ð Þ*<sup>ξ</sup>* ). This heterogeneous medium has been considered above. For all other heterogeneous media for which the value *V=c*<sup>2</sup> changes within period, the inequality is realized in Eq. (49). So, in a heterogeneous medium, the value *α<sup>p</sup>* is always greater than *α<sup>p</sup>* hom in a homogeneous medium. Thus, it is proved that, in the general case, the heterogeneities in a medium introduce the additional nonlinearity. This effect provides the basis for a new method of diagnostics to define the properties of multicomponent media using the propagation of long nonlinear waves in such media.

#### **4.2 Fundamentals of new diagnostic method**

The structure of the medium affects the wave field. There are different methods which allow the detection of gas bubbles and/or cracks in liquid [32], concrete [33], and ice cover [34] employing the nonlinear effects.

In this section, we describe our new diagnostic method for the properties of the medium. The features of the motion of finite-amplitude long waves and the effect of the increase of nonlinearity in the heterogeneous medium in comparison with homogeneous medium form the basis for the development of theoretical fundamentals of the diagnostic method. In this method, the properties of individual components are defined by long waves of finite amplitudes; more specifically, the dependence *<sup>V</sup>=c*<sup>2</sup> <sup>¼</sup> *<sup>V</sup>=c*<sup>2</sup>ð Þ*<sup>ζ</sup>* on the fast Eulerian coordinate *<sup>ζ</sup>* (see Eq. (24)) is defined.

Thus, the nonlinear wave evolution allows one to obtain the structure of the medium with an inherent accuracy. As a final result, the mass concentrations of the individual components can be found using this method.

It should be kept in mind that the period of the structure of medium is infinitely small in the long-wave model, so it is not always possible to indicate the location of the structure elements inside the period reliably. Hence, the media with the different structures plotted in **Figure 2**, for example, affect identically on wave fields. These two media are indistinguishable in the framework of the suggested method. Taking into account this indefiniteness, we consider the function *<sup>V</sup>=c*<sup>2</sup> <sup>¼</sup> *<sup>V</sup>=c*<sup>2</sup>ð Þ*<sup>ζ</sup>* that is to be the decreasing, integrable, mutually one-valued function on the interval *ζ* ∈ ½ � 0, 1 , and equal to zero outside of this interval.

Now we represent the theoretical fundamentals for new method of diagnostics of medium by means of the long nonlinear waves. Let us prove the principal relation which enables us to obtain the inverse function *<sup>ζ</sup>* <sup>¼</sup> *<sup>ζ</sup> <sup>V</sup>=c*<sup>2</sup> ð Þ for the desired function *<sup>V</sup>=c*<sup>2</sup> <sup>¼</sup> *<sup>V</sup>=c*<sup>2</sup>ð Þ*<sup>ζ</sup>* through the inverse Fourier transformation [19, 35–37]:

$$\zeta\left(\mathrm{Vc}^{-2}\right) = F^{-1}\left[\sum\_{n=0}^{\infty} \frac{\left\langle \mathrm{V}\left(\mathrm{Vc}^{-2}\right)^{n+1} \right\rangle}{(n+1)!\langle \mathrm{V}\rangle} \mathrm{i}^{n} q^{n}\right] \left(\mathrm{Vc}^{-2}\right). \tag{53}$$

It is known from theory of probability that the distribution function *f x*ð Þ (any one-valued, integrable, positive function) can be expressed by its central moments:

$$a\_n = \int\_{-\infty}^{\infty} \varkappa^n f(\varkappa) d\varkappa. \tag{54}$$

Indeed, by using the characteristic function

$$\chi(q) = F[f(\mathbf{x})](q),\tag{55}$$

**Figure 2.**

*The equivalent distributions of the specific volume in elementary sell for diagnostic method.*

any positive integrable function *f x*ð Þ can be written as follows:

$$f(\mathbf{x}) = F^{-1}[\chi(q)](\mathbf{x}),\tag{56}$$

In the geometric sense, this relation signifies that the integral (in our case, it is an area between the curve *<sup>V</sup>=c*<sup>2</sup> <sup>¼</sup> *<sup>V</sup>=c*<sup>2</sup>ð Þ*<sup>ζ</sup>* and axes *<sup>O</sup><sup>ζ</sup>* and *O V=c*<sup>2</sup> ð Þ) can be calculated either over *ζ* or over *V=c*<sup>2</sup> (see **Figure 2**). Whereas, the inequality is realized for the

For a function defined on a finite interval, if this function is positive and

*Mathematical Fundamentals of a Diagnostic Method by Long Nonlinear Waves…*

∞ð

*V c*2

∞

�∞

This relation provides the connection between the central moment *α<sup>n</sup>* and the

Then the characteristic function *<sup>χ</sup>*ð Þ*<sup>q</sup>* for the inverse function *<sup>ζ</sup>* <sup>¼</sup> *<sup>ζ</sup> <sup>V</sup>=c*<sup>2</sup> ð Þ is expressed through *V V=c*<sup>2</sup> ð Þ*<sup>n</sup>* � �. By applying the inverse Fourier transformation,

> *M* ð

*Vc*�<sup>2</sup> � �*<sup>n</sup>*

0

The coefficients *V V=c*<sup>2</sup> ð Þ*<sup>n</sup>* � � (*<sup>n</sup>* <sup>¼</sup> 3, 4, … ) in Eq. (53) can be easily calculated, if

that follows directly from the equation of state. With mentioned accuracy, it is

We have proven the principal relation (53) for the method of diagnostics that allows one to find the properties of the individual components in structured media

Diagnostics of the structured medium properties by the long nonlinear waves is connected with the definition of values *V V=c*<sup>2</sup> ð Þ*<sup>n</sup>* � �. As indicated above, there is a problem related to the accuracy of the description of the structure by finite

Now, we shall show that the partial sum of series (53) is a step-function and approximates the desired function *<sup>ζ</sup>* <sup>¼</sup> *<sup>ζ</sup> <sup>V</sup>=c*<sup>2</sup> ð Þ with certain accuracy, namely, the

*<sup>n</sup>*¼<sup>0</sup>*Mn*þ<sup>1</sup> *s*

� �*<sup>n</sup> dζ*

*V c*2 � �*<sup>n</sup>*�<sup>1</sup>

*dV=c*<sup>2</sup> ð Þ *d V=<sup>c</sup>*

*ζd V=c* <sup>2</sup> � �*:*

<sup>2</sup> � � � �*<sup>n</sup>* ¼ �*n V*h i*α<sup>n</sup>*�<sup>1</sup>*:* (62)

*d V=c*

*dp* ¼ �ð Þ *<sup>n</sup>* <sup>þ</sup> <sup>1</sup> *V Vc*�<sup>2</sup> � �*<sup>n</sup>*þ<sup>1</sup> D E, (64)

<sup>2</sup> � � <sup>¼</sup> *<sup>M</sup><sup>n</sup>*þ<sup>1</sup>

*<sup>n</sup>=*ð Þ *<sup>n</sup>* <sup>þ</sup> <sup>1</sup> ! converge at *<sup>s</sup>* <sup>&</sup>lt; *<sup>M</sup>*�<sup>1</sup>

*n* þ 1

*<sup>=</sup>c*<sup>2</sup> � �ð Þ *<sup>p</sup>* . Indeed, they can be

*:* (63)

. Conse-

<sup>2</sup> � �

(61)

�∞

¼ �*n V*h i <sup>ð</sup>

*V V=c*

The physical value *Vc*�<sup>2</sup> is bounded by some constant *M*, hence

*ζ d V=c* <sup>2</sup> � �≤

*<sup>n</sup>* ð Þ *<sup>=</sup>n*! <sup>≤</sup>P<sup>∞</sup>

monotonic decreasing function *<sup>V</sup>=c*<sup>2</sup> <sup>¼</sup> *<sup>V</sup>=c*<sup>2</sup>ð Þ*<sup>ζ</sup>* .

*DOI: http://dx.doi.org/10.5772/intechopen.91462*

*V V=c*

finally, we find the required relationship (53).

*Vc*�<sup>2</sup> � �*<sup>n</sup>*�<sup>1</sup>

quently, the power series (53) also converges.

successively defined by the recurrence relation

we know the functional dependence h i *<sup>V</sup>* ð Þ *<sup>p</sup>* or *<sup>V</sup>*<sup>2</sup>

*d V Vc*�<sup>2</sup> � � � �*<sup>n</sup>*

possible to diagnose the structural properties of the medium.

**4.3 Approximation of diagnosed medium by layer medium**

*<sup>n</sup>*¼<sup>0</sup>∣*αn*<sup>∣</sup> *<sup>s</sup>*

by means of the long nonlinear waves.

<sup>2</sup> � � � �*<sup>n</sup>* <sup>¼</sup> h i *<sup>V</sup>*

bounded above, we have

value *V V=c*<sup>2</sup> ð Þ*<sup>n</sup>* � �

*α<sup>n</sup>* ¼

The series P<sup>∞</sup>

series (53).

**151**

ð ∞

�∞

where *<sup>F</sup>*½ �� is the Fourier transformation and *<sup>F</sup>*�<sup>1</sup> ½ �� is the inverse Fourier transformation.

We take into account the important fact from the theory of probability: the characteristic function *χ*ð Þ*q* is uniquely determined by the central moments *αn*:

$$\chi(q) = \sum\_{n=0}^{\infty} a\_n \mathbf{i}^n \frac{q^n}{n!}. \tag{57}$$

Hence, the function *f x*ð Þ can be found by means of the inverse Fourier transform:

$$f(\mathbf{x}) = F^{-1} \left[ \sum\_{n=0}^{\infty} a\_n \mathbf{i}^n \frac{q^n}{n!} \right](\mathbf{x}),\tag{58}$$

if series P<sup>∞</sup> *<sup>n</sup>*¼<sup>0</sup>∣*αn*<sup>∣</sup> *<sup>s</sup> <sup>n</sup>* ð Þ *<sup>=</sup>n*! converges absolutely for some value *<sup>s</sup>*>0 (see Section 18.3.7 in Ref. [31]).

These facts from the theory of probability are used to prove such a statement: if *<sup>V</sup>=c*<sup>2</sup> <sup>¼</sup> *<sup>V</sup>=c*<sup>2</sup>ð Þ*<sup>ζ</sup>* is a decreasing positive integrable function on the interval *<sup>ζ</sup>* <sup>∈</sup>½ � 0, 1 and equals to zero outside of it, then the inverse function *<sup>ζ</sup>* <sup>¼</sup> *<sup>ζ</sup> <sup>V</sup>=c*<sup>2</sup> ð Þ for the required function *<sup>V</sup>=c*<sup>2</sup> <sup>¼</sup> *<sup>V</sup>=c*<sup>2</sup>ð Þ*<sup>ζ</sup>* can be written as (53) in the averaged values

$$
\langle \left( V \left( V / c^2 \right)^n \right) \equiv \int\_{-\infty}^{\infty} V \left( V / c^2 \right)^n d\xi. \tag{59}
$$

Indeed, for the monotonic one-valued function *<sup>V</sup>=c*<sup>2</sup> <sup>¼</sup> *<sup>V</sup>=c*<sup>2</sup>ð Þ*<sup>ζ</sup>* , we find the integral in (59) by integrating the inverse function *<sup>ζ</sup>* <sup>¼</sup> *<sup>ζ</sup> <sup>V</sup>=c*<sup>2</sup> ð Þ, since the transformation Jacobian is not equal to zero. We have the chain of identifies

$$\begin{split} \left< V \left( V/c^2 \right)^n \right> &= \int\_0^1 V(\xi) \left( \frac{V}{c^2} \right)^n d\xi = \left< V \right> \int\_0^1 V \left( \frac{V}{c^2} \right)^n \rho d\zeta \\ &= \left< V \right> \int\_{-\infty}^\infty \left( \frac{V}{c^2} \right)^n \frac{d\zeta}{d(V/c^2)} d\left( V/c^2 \right). \end{split} \tag{60}$$

*Mathematical Fundamentals of a Diagnostic Method by Long Nonlinear Waves… DOI: http://dx.doi.org/10.5772/intechopen.91462*

In the geometric sense, this relation signifies that the integral (in our case, it is an area between the curve *<sup>V</sup>=c*<sup>2</sup> <sup>¼</sup> *<sup>V</sup>=c*<sup>2</sup>ð Þ*<sup>ζ</sup>* and axes *<sup>O</sup><sup>ζ</sup>* and *O V=c*<sup>2</sup> ð Þ) can be calculated either over *ζ* or over *V=c*<sup>2</sup> (see **Figure 2**). Whereas, the inequality is realized for the monotonic decreasing function *<sup>V</sup>=c*<sup>2</sup> <sup>¼</sup> *<sup>V</sup>=c*<sup>2</sup>ð Þ*<sup>ζ</sup>* .

For a function defined on a finite interval, if this function is positive and bounded above, we have

$$
\begin{split}
\langle\langle V\left(V/c^{2}\right)^{n}\rangle\rangle &= \langle V\rangle \int\_{-\infty}^{\infty} \left(\frac{V}{c^{2}}\right)^{n} \frac{d\zeta}{(dV/c^{2})} d\left(V/c^{2}\right) \\ &= -n\langle V\rangle \int\_{-\infty}^{\infty} \left(\frac{V}{c^{2}}\right)^{n-1} \zeta d\left(V/c^{2}\right).
\end{split} \tag{61}
$$

This relation provides the connection between the central moment *α<sup>n</sup>* and the value *V V=c*<sup>2</sup> ð Þ*<sup>n</sup>* � �

$$
\langle \left( V \left( V / c^2 \right)^n \right) \rangle = -n \langle V \rangle a\_{n-1}. \tag{62}
$$

Then the characteristic function *<sup>χ</sup>*ð Þ*<sup>q</sup>* for the inverse function *<sup>ζ</sup>* <sup>¼</sup> *<sup>ζ</sup> <sup>V</sup>=c*<sup>2</sup> ð Þ is expressed through *V V=c*<sup>2</sup> ð Þ*<sup>n</sup>* � �. By applying the inverse Fourier transformation, finally, we find the required relationship (53).

The physical value *Vc*�<sup>2</sup> is bounded by some constant *M*, hence

$$a\_n = \int\_{-\infty}^{\infty} \left(\mathcal{V}c^{-2}\right)^{n-1} \zeta d\left(\mathcal{V}/c^2\right) \le \int\_0^M \left(\mathcal{V}c^{-2}\right)^n d\left(\mathcal{V}/c^2\right) = \frac{M^{n+1}}{n+1}.\tag{63}$$

The series P<sup>∞</sup> *<sup>n</sup>*¼<sup>0</sup>∣*αn*<sup>∣</sup> *<sup>s</sup> <sup>n</sup>* ð Þ *<sup>=</sup>n*! <sup>≤</sup>P<sup>∞</sup> *<sup>n</sup>*¼<sup>0</sup>*Mn*þ<sup>1</sup> *s <sup>n</sup>=*ð Þ *<sup>n</sup>* <sup>þ</sup> <sup>1</sup> ! converge at *<sup>s</sup>* <sup>&</sup>lt; *<sup>M</sup>*�<sup>1</sup> . Consequently, the power series (53) also converges.

The coefficients *V V=c*<sup>2</sup> ð Þ*<sup>n</sup>* � � (*<sup>n</sup>* <sup>¼</sup> 3, 4, … ) in Eq. (53) can be easily calculated, if we know the functional dependence h i *<sup>V</sup>* ð Þ *<sup>p</sup>* or *<sup>V</sup>*<sup>2</sup> *<sup>=</sup>c*<sup>2</sup> � �ð Þ *<sup>p</sup>* . Indeed, they can be successively defined by the recurrence relation

$$\frac{d\left}{dp} = -(n+1)\left,\tag{64}$$

that follows directly from the equation of state. With mentioned accuracy, it is possible to diagnose the structural properties of the medium.

We have proven the principal relation (53) for the method of diagnostics that allows one to find the properties of the individual components in structured media by means of the long nonlinear waves.

#### **4.3 Approximation of diagnosed medium by layer medium**

Diagnostics of the structured medium properties by the long nonlinear waves is connected with the definition of values *V V=c*<sup>2</sup> ð Þ*<sup>n</sup>* � �. As indicated above, there is a problem related to the accuracy of the description of the structure by finite series (53).

Now, we shall show that the partial sum of series (53) is a step-function and approximates the desired function *<sup>ζ</sup>* <sup>¼</sup> *<sup>ζ</sup> <sup>V</sup>=c*<sup>2</sup> ð Þ with certain accuracy, namely, the diagnosed medium can be approximated by a layer medium. Let us write down the chain of the identities for any integrable function:

$$\begin{split} 2\pi f(-\infty) &= F[F[f(\infty)](q)](\infty) = F\left[\sum\_{n=0}^{\infty} \frac{\mathbf{i}^n q^n}{n!} \alpha\_n\right] \\ &= \sum\_{n=0}^{\infty} \frac{\mathbf{i}^n \alpha\_n}{n!} 2\pi (-\mathbf{i})^n \delta^{(n)}(\infty). \end{split} \tag{65}$$

Here we used the known relationships for the Fourier transform [31]):

$$F[F[f(\mathbf{x})](q)](\mathbf{x}) = 2\mathfrak{gl}(-\mathbf{x}),\tag{66}$$

It is well-known that the derivative of Heavyside function Θð Þ *x* is *δ*ð Þ *x* -function, then <sup>Θ</sup>ð Þ *<sup>n</sup>*þ<sup>1</sup> ð Þ¼ *<sup>x</sup> <sup>δ</sup>*ð Þ *<sup>n</sup>* ð Þ *<sup>x</sup>* . We equate functions (68) and (71) and consider that the number of steps for function *f* <sup>1</sup>ð Þ *x* is infinitely larger, and in this case we obtain

> X∞ *n*¼0

ð Þ *<sup>n</sup>* <sup>þ</sup> <sup>1</sup> ! *<sup>δ</sup>*ð Þ *<sup>n</sup>* ð Þþ *<sup>x</sup>* … <sup>¼</sup> <sup>X</sup><sup>∞</sup>

This relationship shows that when we use the partial sum of series on the right-

side, then the desired function *f x*ð Þ is approximated by the step-function *f* <sup>1</sup>ð Þ *x* with *N* steps. In other words, if it is necessary to restore the structure of medium by means of *N* periodic repeated layers, then we need to know the 2*N* � 1 moments *αn*,

For the sake of convenience, we write down the relation (74) in the expanded form. For this purpose, we multiply it by *xn* and integrate over *x*. We obtain the nonlinear system of the equations in the unknowns *b*1, *b*2, … , *bN*, *φ*2, *φ*3, and … , *φ<sup>N</sup>*

*φ*1*b*<sup>1</sup> þ *φ*2ð Þþ *b*<sup>2</sup> � *b*<sup>1</sup> *φ*3ð Þþ *b*<sup>3</sup> � *b*<sup>2</sup> … þ *φN*ð Þ¼ *bN* � *bN*�<sup>1</sup> *α*0,

<sup>3</sup> � *<sup>b</sup>*<sup>2</sup> 2 � � <sup>þ</sup> … <sup>þ</sup> *<sup>φ</sup><sup>N</sup> <sup>b</sup>*<sup>2</sup>

� � � � � � � � � � � � � � � � ��

� � <sup>¼</sup> ð Þ <sup>2</sup>*<sup>N</sup>* � <sup>1</sup> *<sup>α</sup>*2*N*�<sup>2</sup>*:*

Now, if *bi* implies the partition of *<sup>V</sup>=c*<sup>2</sup> ð Þ*<sup>i</sup>* and *<sup>φ</sup><sup>i</sup>* implies the partition of *<sup>ζ</sup>i*, we can obtain the system of Eq. (75) to define the structure of medium. Solution of these equations gives the information about the component properties of the medium, namely, the value *<sup>V</sup>=c*<sup>2</sup> on the structure period *<sup>ζ</sup>* <sup>∈</sup> ½ � 0, 1 is found in the

Let us note the special case of a periodic medium for which the value *V=c*<sup>2</sup> is constant within the period. This medium, as we already know, does not differ from a homogeneous one for the propagation of the long nonlinear waves. The same result follows from a system (75). Indeed, for homogeneous media the moments *α<sup>n</sup>*

*V Vc*�<sup>2</sup> � �*<sup>n</sup>*þ<sup>1</sup> D E

*V*<sup>0</sup> ¼ 1 have been used as before. Therefore, the values in the right-hand side of Eq. (75) are equal to *<sup>b</sup>* � *Vc*�<sup>2</sup> <sup>¼</sup> const. It is easy to see that the solution of system is

corresponds to the layer medium, for which *<sup>V</sup>=c*<sup>2</sup> 6¼ *<sup>f</sup>*ð Þ*<sup>ζ</sup>* , in particular, this medium

According to the asymptotic averaged model of a structured medium, the period of the structure is infinitely small, and this diagnostic method cannot give the exact

*b*<sup>1</sup> ¼ *b*<sup>2</sup> ¼ … ¼ *bN* ¼ *b* ¼ 1 and *φ*<sup>1</sup> ¼ 1 (where *φ<sup>i</sup>* is any value for *i* ≥2). This

ð Þ *<sup>n</sup>* <sup>þ</sup> <sup>1</sup> h i *<sup>V</sup>* <sup>¼</sup> *<sup>b</sup><sup>n</sup>*þ<sup>1</sup>

*=c*<sup>2</sup> � �

*n* þ 1

<sup>0</sup> <sup>¼</sup> *<sup>V</sup>*<sup>2</sup> *=c*<sup>2</sup> � �

*:* (76)

<sup>0</sup> ¼ 1 and h i *V* <sup>0</sup> ¼

*b<sup>n</sup>*þ<sup>1</sup> <sup>2</sup> � *bn*þ<sup>1</sup> 1

ð Þ *<sup>n</sup>* <sup>þ</sup> <sup>1</sup> ! *<sup>δ</sup>*ð Þ *<sup>n</sup>* ð Þþ *<sup>x</sup>* … <sup>þ</sup>

*αn n*!

*<sup>δ</sup>*ð Þ *<sup>n</sup>* ð Þ *<sup>x</sup> :*

(74)

(75)

*n*¼0

*<sup>n</sup>*! *<sup>δ</sup>*ð Þ *<sup>n</sup>* ð Þ *<sup>x</sup>* and also the *<sup>N</sup>* leading terms on the left-hand

*<sup>N</sup>* � *<sup>b</sup>*<sup>2</sup> *N*�1 � � <sup>¼</sup> <sup>2</sup>*α*1,

<sup>3</sup> � *<sup>b</sup>*<sup>2</sup>*N*�<sup>1</sup> 2 � � <sup>þ</sup> … <sup>þ</sup>

*φ*1 X∞ *n*¼0

hand side of Eq. (74) P<sup>2</sup>*N*�<sup>1</sup>

i.e., the values *V Vc*�<sup>2</sup> � � � �*<sup>n</sup>* .

<sup>1</sup> <sup>þ</sup> *<sup>φ</sup>*<sup>2</sup> *<sup>b</sup>*<sup>2</sup>

form of the step-function.

can be a homogeneous one.

are equal to

**153**

*φ*1*b*<sup>2</sup>

*φ*1*b*<sup>2</sup>*N*�<sup>1</sup>

*bn*þ<sup>1</sup> 1 ð Þ *n* þ 1 !

> X∞ *n*¼0

*<sup>n</sup>*¼<sup>0</sup> *<sup>α</sup><sup>n</sup>*

*b<sup>n</sup>*þ<sup>1</sup> *<sup>N</sup>* � *bn*þ<sup>1</sup> *N*�1

þ*φ<sup>N</sup>*

*DOI: http://dx.doi.org/10.5772/intechopen.91462*

(variable *φ*<sup>1</sup> ¼ 1 owing to normalization):

<sup>2</sup> � *<sup>b</sup>*<sup>2</sup> 1 � � <sup>þ</sup> *<sup>φ</sup>*<sup>3</sup> *<sup>b</sup>*<sup>2</sup>

<sup>þ</sup>*φ<sup>N</sup> <sup>b</sup>*<sup>2</sup>*N*�<sup>1</sup>

<sup>2</sup> � *<sup>b</sup>*<sup>2</sup>*N*�<sup>1</sup> 1 � � <sup>þ</sup> *<sup>φ</sup>*<sup>3</sup> *<sup>b</sup>*<sup>2</sup>*N*�<sup>1</sup>

> *<sup>N</sup>* � *<sup>b</sup>*<sup>2</sup>*N*�<sup>1</sup> *N*�1

> > *α<sup>n</sup>* ¼

Here, the conditions of normalization *V*<sup>2</sup>

<sup>1</sup> <sup>þ</sup> *<sup>φ</sup>*<sup>2</sup> *<sup>b</sup>*<sup>2</sup>*N*�<sup>1</sup>

*<sup>δ</sup>*ð Þ *<sup>n</sup>* ð Þþ *<sup>x</sup> <sup>φ</sup>*<sup>2</sup>

*Mathematical Fundamentals of a Diagnostic Method by Long Nonlinear Waves…*

$$F[q^n](\mathbf{x}) = 2\pi(-\mathbf{i})^n \delta^{(n)}(\mathbf{x}).\tag{67}$$

Hence, any integrable function can be represented by a series:

$$f(-\infty) = \sum\_{n=0}^{\infty} \frac{a\_n}{n!} \delta^{(n)}(\infty). \tag{68}$$

We will prove that the finite series (53) approximates the desired function *f x*ð Þ by step-function. Consider the step-function *f* <sup>1</sup>ð Þ *x* consisting of *N* steps:

$$f\_1(\mathbf{x}) = \begin{cases} \rho\_1, & \mathbf{0} < \mathbf{x} \le b\_1, \\ \rho\_2, & b\_1 < \mathbf{x} \le b\_2, \\ \vdots & \vdots \\ \rho\_N, & b\_{N-1} < \mathbf{x} \le b\_N \end{cases} \tag{69}$$

in order to approximate the desired function *f x*ð Þ. The relation (69) can be written down through the Heavyside functions as follows:

$$\begin{aligned} f\_1(\mathbf{x}) &= \rho\_1[\Theta(\mathbf{x}) - \Theta(\mathbf{x} - b\_1)] + \rho\_2[\Theta(\mathbf{x} - b\_1) - \Theta(\mathbf{x} - b\_2)] + \dots \\ &+ \rho\_N[\Theta(\mathbf{x} - b\_{N-1}) - \Theta(\mathbf{x} - b\_N)], \end{aligned} \tag{70}$$

Evidently, by increasing the number of steps *N* and choosing the values *φ<sup>i</sup>* and *bi*, any integrable function *f x*ð Þ can be approximated by the step-function *f* <sup>1</sup>ð Þ *x* . It is convenient to use a notation

$$\begin{split} f\_1(-\mathbf{x}) &= \varrho\_1[\Theta(\mathbf{x} + b\_1) - \Theta(\mathbf{x})] + \varrho\_2[\Theta(\mathbf{x} + b\_2) - \Theta(\mathbf{x} + b\_1)] + \dots \\ &+ \varrho\_N[\Theta(\mathbf{x} + b\_N) - \Theta(\mathbf{x} + b\_{N-1})], \end{split} \tag{71}$$

that follows immediately from (70) after substitution:

$$\Theta(\mathfrak{x}) = \mathbf{1} - \Theta(-\mathfrak{x}).\tag{72}$$

The Heavyside function Θð Þ *x* þ *b* can be expanded into a Taylor series in the neighborhood of point *x*:

$$
\Theta(\varkappa + b) = \Theta(\varkappa) + \sum\_{n=1}^{\infty} \frac{b^n}{n!} \Theta^{(n)}(\varkappa). \tag{73}
$$

*Mathematical Fundamentals of a Diagnostic Method by Long Nonlinear Waves… DOI: http://dx.doi.org/10.5772/intechopen.91462*

It is well-known that the derivative of Heavyside function Θð Þ *x* is *δ*ð Þ *x* -function, then <sup>Θ</sup>ð Þ *<sup>n</sup>*þ<sup>1</sup> ð Þ¼ *<sup>x</sup> <sup>δ</sup>*ð Þ *<sup>n</sup>* ð Þ *<sup>x</sup>* . We equate functions (68) and (71) and consider that the number of steps for function *f* <sup>1</sup>ð Þ *x* is infinitely larger, and in this case we obtain

$$\begin{split} \rho\_1 \sum\_{n=0}^{\infty} \frac{b\_1^{n+1}}{(n+1)!} \delta^{(n)}(\mathbf{x}) + \rho\_2 \sum\_{n=0}^{\infty} \frac{b\_2^{n+1} - b\_1^{n+1}}{(n+1)!} \delta^{(n)}(\mathbf{x}) + \dots + \\ + \rho\_N \sum\_{n=0}^{\infty} \frac{b\_N^{n+1} - b\_{N-1}^{n+1}}{(n+1)!} \delta^{(n)}(\mathbf{x}) + \dots = \sum\_{n=0}^{\infty} \frac{a\_n}{n!} \delta^{(n)}(\mathbf{x}). \end{split} \tag{74}$$

This relationship shows that when we use the partial sum of series on the righthand side of Eq. (74) P<sup>2</sup>*N*�<sup>1</sup> *<sup>n</sup>*¼<sup>0</sup> *<sup>α</sup><sup>n</sup> <sup>n</sup>*! *<sup>δ</sup>*ð Þ *<sup>n</sup>* ð Þ *<sup>x</sup>* and also the *<sup>N</sup>* leading terms on the left-hand side, then the desired function *f x*ð Þ is approximated by the step-function *f* <sup>1</sup>ð Þ *x* with *N* steps. In other words, if it is necessary to restore the structure of medium by means of *N* periodic repeated layers, then we need to know the 2*N* � 1 moments *αn*, i.e., the values *V Vc*�<sup>2</sup> � � � �*<sup>n</sup>* .

For the sake of convenience, we write down the relation (74) in the expanded form. For this purpose, we multiply it by *xn* and integrate over *x*. We obtain the nonlinear system of the equations in the unknowns *b*1, *b*2, … , *bN*, *φ*2, *φ*3, and … , *φ<sup>N</sup>* (variable *φ*<sup>1</sup> ¼ 1 owing to normalization):

$$\begin{aligned} \rho\_1 b\_1 + \rho\_2 (b\_2 - b\_1) + \rho\_3 (b\_3 - b\_2) + \dots + \rho\_N (b\_N - b\_{N-1}) &= a\_0, \\ \rho\_1 b\_1^2 + \rho\_2 (b\_2^2 - b\_1^2) + \rho\_3 (b\_3^2 - b\_2^2) + \dots + \rho\_N (b\_N^2 - b\_{N-1}^2) &= 2a\_1, \\ \rho\_1 \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \\ \rho\_1 b\_1^{2N-1} + \rho\_2 (b\_2^{2N-1} - b\_1^{2N-1}) + \rho\_3 (b\_3^{2N-1} - b\_2^{2N-1}) + \dots + \\ \quad + \rho\_N (b\_N^{2N-1} - b\_{N-1}^{2N-1}) &= (2N - 1)a\_{2N-2}. \end{aligned} \tag{75}$$

Now, if *bi* implies the partition of *<sup>V</sup>=c*<sup>2</sup> ð Þ*<sup>i</sup>* and *<sup>φ</sup><sup>i</sup>* implies the partition of *<sup>ζ</sup>i*, we can obtain the system of Eq. (75) to define the structure of medium. Solution of these equations gives the information about the component properties of the medium, namely, the value *<sup>V</sup>=c*<sup>2</sup> on the structure period *<sup>ζ</sup>* <sup>∈</sup> ½ � 0, 1 is found in the form of the step-function.

Let us note the special case of a periodic medium for which the value *V=c*<sup>2</sup> is constant within the period. This medium, as we already know, does not differ from a homogeneous one for the propagation of the long nonlinear waves. The same result follows from a system (75). Indeed, for homogeneous media the moments *α<sup>n</sup>* are equal to

$$a\_n = \frac{\left\langle V \left( V c^{-2} \right)^{n+1} \right\rangle}{(n+1) \langle V \rangle} = \frac{b^{n+1}}{n+1}.\tag{76}$$

Here, the conditions of normalization *V*<sup>2</sup> *=c*<sup>2</sup> � � <sup>0</sup> <sup>¼</sup> *<sup>V</sup>*<sup>2</sup> *=c*<sup>2</sup> � � <sup>0</sup> ¼ 1 and h i *V* <sup>0</sup> ¼ *V*<sup>0</sup> ¼ 1 have been used as before. Therefore, the values in the right-hand side of Eq. (75) are equal to *<sup>b</sup>* � *Vc*�<sup>2</sup> <sup>¼</sup> const. It is easy to see that the solution of system is *b*<sup>1</sup> ¼ *b*<sup>2</sup> ¼ … ¼ *bN* ¼ *b* ¼ 1 and *φ*<sup>1</sup> ¼ 1 (where *φ<sup>i</sup>* is any value for *i* ≥2). This corresponds to the layer medium, for which *<sup>V</sup>=c*<sup>2</sup> 6¼ *<sup>f</sup>*ð Þ*<sup>ζ</sup>* , in particular, this medium can be a homogeneous one.

According to the asymptotic averaged model of a structured medium, the period of the structure is infinitely small, and this diagnostic method cannot give the exact

can be described by the wave-dynamical laws for the averaged variables with the integrodifferential equation of state containing the characteristics of the medium microstructure. A rigorous mathematical proof is given to show that finite-

amplitude long waves respond to the structure of the medium in such a way that the homogeneous medium model is insufficient for the description of the behavior of the structured medium. An important result that follows from this model is that, for

On the other hand, the system analyzed here is not expressed in the average hydrodynamical terms; hence the dynamical behavior of the medium cannot be modelled by a homogeneous medium even for long waves, if they are nonlinear.

nonlinearity that does not arise in a homogeneous medium. This effect enabled one to formulate the theoretical grounds of a new diagnostic method that determines the characteristics of a heterogeneous medium with the use of finite-amplitude long waves (inverse problem). This diagnostic method can also be employed to find the

Physics and Astronomy Classification Scheme (PACS): 43.25.Zx, 43.25.Ba,

1 Institute of Geophysics, Ukrainian Academy of Sciences, Kyiv, Ukraine

\*Address all correspondence to: vakhnenko@ukr.net

provided the original work is properly cited.

2 Instituto Politécnico Nacional, SEPI-ESIME-Zacatenco, Ciudad de México,

© 2020 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium,

\*, Dmitri Vengrovich<sup>1</sup> and Alexandre Michtchenko<sup>2</sup>

a finite-amplitude wave, the medium structure (in particular, existence of microcracks) produces nonlinear effects even if the individual components of the medium are described by a linear law. Finding the wave fields in the structured

*Mathematical Fundamentals of a Diagnostic Method by Long Nonlinear Waves…*

The heterogeneity of the medium structure always introduces additional

medium is the direct problem, on the one hand.

*DOI: http://dx.doi.org/10.5772/intechopen.91462*

mass contents of individual components.

**Additional information**

43.25.Ed, 43.25.Gf

**Author details**

México

**155**

Vyacheslav Vakhnenko<sup>1</sup>

#### **Figure 3.**

*Approximation of the diagnosed medium* <sup>V</sup>*=*c2 <sup>¼</sup> <sup>0</sup>*:*<sup>2</sup> <sup>þ</sup> <sup>0</sup>*:*8 1ð Þ � <sup>ζ</sup> <sup>2</sup> *by* <sup>N</sup>*-component medium.*

location of the structure elements inside the period. Hence, using this method, only the mass contents of the particular components can be determined.

We present, as an example, the results of the calculation to define the structure of layer media, which can properly approximate the diagnosed medium. The structure of the diagnosed medium is *<sup>V</sup>=c*<sup>2</sup> <sup>¼</sup> <sup>0</sup>*:*<sup>2</sup> <sup>þ</sup> <sup>0</sup>*:*8 1ð Þ � *<sup>ζ</sup>* <sup>2</sup> in **Figure 3**. In order to approximate the diagnosed medium by layer periodic medium, which has *N* layers within the period, it is necessary to know 2*<sup>N</sup>* � 1 values *V Vc*�<sup>2</sup> *<sup>n</sup>* for finite series (53). If we regard that the 2*<sup>N</sup>* � 1 averaged characteristics *V Vc*�<sup>2</sup> *<sup>n</sup>* coincide for the diagnosed medium and the layer medium, these averaged values at *n* ≤2*N* � 1 can be calculated from the known distributions *<sup>V</sup>=c*<sup>2</sup> <sup>¼</sup> <sup>0</sup>*:*<sup>2</sup> <sup>þ</sup> <sup>0</sup>*:*8 1ð Þ � *<sup>ζ</sup>* <sup>2</sup> . At *<sup>n</sup>*<sup>&</sup>gt; <sup>2</sup>*<sup>N</sup>* � 1 the values *V Vc*�<sup>2</sup> *<sup>n</sup>* for diagnosed medium and for approximated layer medium are different. The distributions of *<sup>V</sup>=c*<sup>2</sup>ð Þ*<sup>ζ</sup>* within the period for diagnosed medium and for approximated media with *N* components are shown in **Figure 3**. On the one hand, the calculated distributions for layered media are the best approximation for the medium we test. On the other hand, we have illustrated the accuracy of the approximation of the diagnosed medium by the finite series (53).

Thus, the new method for the diagnostics of the medium characteristics by long nonlinear waves is suggested on the basis of the asymptotic averaged model of the structured medium. The mass contents of the particular components can be denoted by the abovementioned diagnostic method.

#### **5. Conclusion**

The asymptotic averaged model is suggested for the description of the wave processes in nonequilibrium heterogeneous media. The obtained integral differential system of equations cannot be reduced to the average terms (pressure, mass velocity, specific volume) and contains the terms with characteristic sizes of individual components.

On the microstructure level of the medium, the dynamical behavior is governed only by the laws of thermodynamics. On the macrolevel, the motion of the medium *Mathematical Fundamentals of a Diagnostic Method by Long Nonlinear Waves… DOI: http://dx.doi.org/10.5772/intechopen.91462*

can be described by the wave-dynamical laws for the averaged variables with the integrodifferential equation of state containing the characteristics of the medium microstructure. A rigorous mathematical proof is given to show that finiteamplitude long waves respond to the structure of the medium in such a way that the homogeneous medium model is insufficient for the description of the behavior of the structured medium. An important result that follows from this model is that, for a finite-amplitude wave, the medium structure (in particular, existence of microcracks) produces nonlinear effects even if the individual components of the medium are described by a linear law. Finding the wave fields in the structured medium is the direct problem, on the one hand.

On the other hand, the system analyzed here is not expressed in the average hydrodynamical terms; hence the dynamical behavior of the medium cannot be modelled by a homogeneous medium even for long waves, if they are nonlinear. The heterogeneity of the medium structure always introduces additional nonlinearity that does not arise in a homogeneous medium. This effect enabled one to formulate the theoretical grounds of a new diagnostic method that determines the characteristics of a heterogeneous medium with the use of finite-amplitude long waves (inverse problem). This diagnostic method can also be employed to find the mass contents of individual components.

#### **Additional information**

Physics and Astronomy Classification Scheme (PACS): 43.25.Zx, 43.25.Ba, 43.25.Ed, 43.25.Gf

### **Author details**

Vyacheslav Vakhnenko<sup>1</sup> \*, Dmitri Vengrovich<sup>1</sup> and Alexandre Michtchenko<sup>2</sup>

1 Institute of Geophysics, Ukrainian Academy of Sciences, Kyiv, Ukraine

2 Instituto Politécnico Nacional, SEPI-ESIME-Zacatenco, Ciudad de México, México

\*Address all correspondence to: vakhnenko@ukr.net

© 2020 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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*Mathematical Fundamentals of a Diagnostic Method by Long Nonlinear Waves…*

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### *Edited by Francisco Bulnes and Olga Hachay*

The complex analysis, also known as theory of analytic functions or complex variable function theory, is the part of mathematical analysis that investigates the functions of complex numbers, their analyticity, holomorphicity, and integration of these functions on complex domains that can be complex manifolds or submanifolds. Also the extensions of these domains to the complex projective spaces and complex topological groups are study themes. The analytic continuing of complex domains where complex series representations are used and the exploring of singularities whose integration invariants obtain values as zeros of certain polynomials of the complex rings of certain vector bundles are important in the exploring of new function classes in the meromorphic context and also arithmetic context. Also important are established correspondences with complex vector spaces, or even in their real parts, using several techniques of complex geometrical analysis, Nevanlinna methods, and other techniques as the modular forms. All this is just some examples of great abundance of the problems in mathematics research that require the complex analysis application. This book covers some interesting and original research of certain topics of complex analysis. Also included are some applications for inverse and ill posed problems developed in engineering and applied research.

Published in London, UK © 2020 IntechOpen © Maxger / iStock

Advances in Complex Analysis and Applications

Advances in Complex

Analysis and Applications

*Edited by Francisco Bulnes and Olga Hachay*