**Electric Components of Acoustic Waves in the Vicinity of Nonpiezoactive Directions**

V.I. Alshits, V.N. Lyubimov and A. Radowicz

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/56067

## **1. Introduction**

The acoustic wave of displacements in piezoelectric media is usually accompanied by a quasistatic wave of the electric potential. This implies that, using acoustic waves, electric signals can be transmitted over a crystal at the velocity of sound. Such possibility opened the way to numerous applications of acoustic waves in electronic devices and even led to the formation of a special field of science called acoustoelectronics. The applied aspect provides an important stimulus for extensive investigations devoted to various features of acoustic fields in piezoelectric crystals (Royer & Dieulesaint, 2000). These investigations are also stimulated by basic interest in the study of new effects in media with electromechanical couplings (Lyubimov, 1968; Balakirev & Gilinskii, 1982; Lyamov, 1983). The acoustics of piezoelectric crystals is still an extensively developing field of solid state physics [see, e.g., the review article by Gulyaev (1998)], the more so that even purely basic investigations in this field frequently contain ideas for fruitful, however not immediately evident, applications.

It should also be noted that the anisotropy often influences the properties of piezoelectric crystals in a nontrivial way, and may sometimes lead to qualitatively new phenomena. In particular, it is very important from the practical standpoint to know the wave propagation directions **m** for which the electric field components possess maximum amplitudes (Alshits & Lyubimov, 1990) and, on the contrary, to reveal the nonpiezoactive directions (Royer & Dieulesaint, 2000; Lyamov, 1983) in which the electric signals are not transmitted. Taking into account that, irrespective of the anisotropy, the electric field in an acoustic wave is always longitudinal (**E** || **m**) and the electric induction is always transverse (**D m**), we have to distinguish (Lyamov, 1983) between the directions of longitudinal and transverse nonpiezoactivity in which **E** = 0 and **D** = 0, respectively. This paper presents the results of investigations aimed at a detailed analysis of the nonpiezoactivity of both types.

Another important aspect of this problem is related to directions **m**, in the vicinity of which the vector fields of displacements (**u**) and the acompanying electric components (**E**, **D**) exhibit singularities. According to the results obtained by Alshits, Sarychev & Shuvalov (1985), Alshits *et al* (1987), and Shuvalov (1998) this very situation takes place near the acoustic axes, where the orientational singularities in the degenerate branches of eigenwaves are observed for the **u** and **D** fields, and the amplitude singularities, for the **E**  field. This paper deals with orientational singularities of another type, which occur in the vicinity of the transverse nonpiezoactivity directions in the vector fields **D**(**m**), i.e. around the points **m**0 on the unit sphere such that 0 **D m**( 0 ) (Alshits, Lyubimov & Radowicz, 2005 a, b).

Electric Components of Acoustic Waves in the Vicinity of Nonpiezoactive Directions 5

*ij ijkl kl kij k i ikl kl ik k σ c β e E, D e β ε E ,* (2)

*ik* **m r** 0 0 { , } { , }exp{ ( )}. *vt* (3)

Div , Div 0, *σ ρ* ˆ **u D** (4)

0 0 *Fv F* (, ) [ /] (5)

*I* is the unit matrix. A necessary condition for

*<sup>α</sup> F* matrix, which is adjoint to the

ˆ det ( , ) 0. *F v* **m** (7)

<sup>ˆ</sup> **u m m c.** ( )|| <sup>0</sup> ( ) *α α <sup>F</sup>* (8)

The interrelation of these characteristics is determined by the constitutive equations (Landau

where *c*ˆ *ijkl c* is the tensor of elastic moduli, *e*ˆ *kij e* is the tensor of piezoelectric moduli,

where *ρ* is the density of medium. Here, we use the well-known quasi-static approximation valid to within the terms proportional to the ratio 2 10 ( ) 10 *v/c ~* (*c* is the velocity of light). Combining the above relations, we readily obtain a homogeneous equation for the

ˆ ˆ **mu e e u**

This is a cubic equation for the square of phase velocity *v*, which determines the three

Orientations of the corresponding mutually orthogonal polarization vectors **u**0*<sup>α</sup>* (**m**) of the

matrix ˆ ˆ () (() ) **m mm** *α αα F Fv ,* and is determined from the condition ˆˆ ˆ ˆ det *αα α FF I F* . As can be

It should be emphasized that the direct relation (8) between the polarization vector ( ) **u m** <sup>0</sup>*<sup>α</sup>*

0, (0)

ˆ ˆ **m m e mm m m** ˆ ˆ (0) <sup>2</sup> *F c ρv I, e , ε ε ,* (6)

**u u** 

polarization vector **u**0 (Landau & Lifshitz, 1984):

symbol denotes the dyadic product, and ˆ

These fields obey the usual equations of motion (Landau & Lifshitz, 1984):

the existence of nontrivial solutions of the homogeneous equation (5) is

branches of the velocity of the bulk acoustic waves *vα*(**m**) (*α* = 1, 2, 3).

readily checked, Eq. (5) for any vector **c** such that ˆ **<sup>c</sup>** <sup>0</sup> *<sup>α</sup> <sup>F</sup>* is satisfied for

and the wave normal **m** will be widely used in the subsequent analysis.

isonormal eigenwaves can be expressed in terms of the ˆ

ˆ *ik ε* is the permittivity tensor. In such a piezoelectric medium, the bulk acoustic wave with the phase velocity *v* and the wave vector **k** = *k***m** must be a superposition of mechanical

& Lifshitz, 1984):

and electrical dynamic fields:

and

where

Below we will formulate the equations determining special directions **m** for which either **E***<sup>α</sup>* (**m**) = 0 or **D***<sup>α</sup>* (**m**) = 0 for all three branches of the acoustic spectrum ( *α* = 1, 2, 3). These directions have different dimensionalities: the typical solutions appear as lines of zero electric field ( **<sup>E</sup>***<sup>α</sup>* = 0) and points of zero induction ( **<sup>D</sup>***<sup>α</sup>* = 0) on the unit sphere **<sup>m</sup>** <sup>2</sup> <sup>1</sup> . The equations obtained will be analyzed both in the general case and in application to various particular crystal symmetry classes. The two types on nonpiezoactivity are closely related to the crystal symmetry, but they can also exist in triclinic crystals possessing no elements of symmetry. The corresponding theorems of existence are proved.

The possible types of singularities in the vector field **D***<sup>α</sup>* (**m**) in the vicinity of the transverse nonpiezoactivity directions will be considered. In particular, it will be shown that, depending on the material moduli, the singularity in an isolated point **m**0 may be characterized by the Poincaré indices (topological charges) **D***n* = 0, ±1, ±2. The general analytical expressions will be obtained for the **D***n* values in triclinic crystals with arbitrary anisotropy and specified for a large series of crystals belonging to particular crystal symmetry classes. Only the solutions corresponding to singularities with **D***n* = ±1 are topologically stable, while singularities of the other types either split or disappear upon an arbitrary triclinic perturbation of the material tensors. However, the sum of indices for any splitting must be equal to the initial index **D***n .* 

The chapter is mainly based on our papers (Alshits, Lyubimov & Radowicz, 2005 a, b).

### **2. Statement of the problem and general equations**

In piezoelectric crystals, purely mechanical characteristics, the elastic displacement vector **u**(**r**, *t*), the distortion tensor ˆ *β* (**r**, *t*), and the stress tensor *σ*ˆ (**r**, *t*), are related to such electrical quantities as the potential (**r,** *t*) and the electric field strength **E**(**r**, *t*), and induction **D**(**r**, *t*). The fields of ˆ *β* (**r**, *t*) and **E**(**r**, *t*) can be expressed in terms of their own potentials as

$$
\hat{\beta}(\mathbf{r},t) \,\,\, = \nabla \mathbf{u}(\mathbf{r},t), \qquad \mathbf{E}(\mathbf{r},t) \,\, = \,\, -\nabla \phi(\mathbf{r},t). \tag{1}
$$

The interrelation of these characteristics is determined by the constitutive equations (Landau & Lifshitz, 1984):

$$
\sigma\_{ij} = \varepsilon\_{ijkl}\beta\_{kl} - e\_{kij}E\_{k\cdot\prime} \qquad D\_i = e\_{ikl}\beta\_{kl} + \varepsilon\_{ik}E\_{k\cdot\prime} \tag{2}
$$

where *c*ˆ *ijkl c* is the tensor of elastic moduli, *e*ˆ *kij e* is the tensor of piezoelectric moduli, and ˆ *ik ε* is the permittivity tensor. In such a piezoelectric medium, the bulk acoustic wave with the phase velocity *v* and the wave vector **k** = *k***m** must be a superposition of mechanical and electrical dynamic fields:

$$\{\mathbf{u},\ \ \ \ \ \phi\} = \{\mathbf{u}\_{0'} \ \ \ \ \phi\_0\} \exp\{ik(\mathbf{m}\cdot\mathbf{r}-vt)\}.\tag{3}$$

These fields obey the usual equations of motion (Landau & Lifshitz, 1984):

$$\text{Div}\,\hat{\sigma} = \rho \ddot{\mathbf{u}}\_{\prime} \qquad\qquad \text{Div}\mathbf{D} = \,\mathbf{0}\_{\prime} \tag{4}$$

where *ρ* is the density of medium. Here, we use the well-known quasi-static approximation valid to within the terms proportional to the ratio 2 10 ( ) 10 *v/c ~* (*c* is the velocity of light). Combining the above relations, we readily obtain a homogeneous equation for the polarization vector **u**0 (Landau & Lifshitz, 1984):

$$\hat{F}(\upsilon, \mathbf{m}) \mathbf{u}\_0 \equiv [\hat{F}^{(0)} + \mathbf{e} \otimes \mathbf{e} / \,\varepsilon] \mathbf{u}\_0 = \mathbf{0},\tag{5}$$

where

4 Modeling and Measurement Methods for Acoustic Waves and for Acoustic Microdevices

symmetry. The corresponding theorems of existence are proved.

**2. Statement of the problem and general equations** 

splitting must be equal to the initial index **D***n .* 

**u**(**r**, *t*), the distortion tensor ˆ

quantities as the potential

The fields of ˆ

2005 a, b).

Another important aspect of this problem is related to directions **m**, in the vicinity of which the vector fields of displacements (**u**) and the acompanying electric components (**E**, **D**) exhibit singularities. According to the results obtained by Alshits, Sarychev & Shuvalov (1985), Alshits *et al* (1987), and Shuvalov (1998) this very situation takes place near the acoustic axes, where the orientational singularities in the degenerate branches of eigenwaves are observed for the **u** and **D** fields, and the amplitude singularities, for the **E**  field. This paper deals with orientational singularities of another type, which occur in the vicinity of the transverse nonpiezoactivity directions in the vector fields **D**(**m**), i.e. around the points **m**0 on the unit sphere such that 0 **D m**( 0 ) (Alshits, Lyubimov & Radowicz,

Below we will formulate the equations determining special directions **m** for which either **E***<sup>α</sup>* (**m**) = 0 or **D***<sup>α</sup>* (**m**) = 0 for all three branches of the acoustic spectrum ( *α* = 1, 2, 3). These directions have different dimensionalities: the typical solutions appear as lines of zero electric field ( **<sup>E</sup>***<sup>α</sup>* = 0) and points of zero induction ( **<sup>D</sup>***<sup>α</sup>* = 0) on the unit sphere **<sup>m</sup>** <sup>2</sup> <sup>1</sup> . The equations obtained will be analyzed both in the general case and in application to various particular crystal symmetry classes. The two types on nonpiezoactivity are closely related to the crystal symmetry, but they can also exist in triclinic crystals possessing no elements of

The possible types of singularities in the vector field **D***<sup>α</sup>* (**m**) in the vicinity of the transverse nonpiezoactivity directions will be considered. In particular, it will be shown that, depending on the material moduli, the singularity in an isolated point **m**0 may be characterized by the Poincaré indices (topological charges) **D***n* = 0, ±1, ±2. The general analytical expressions will be obtained for the **D***n* values in triclinic crystals with arbitrary anisotropy and specified for a large series of crystals belonging to particular crystal symmetry classes. Only the solutions corresponding to singularities with **D***n* = ±1 are topologically stable, while singularities of the other types either split or disappear upon an arbitrary triclinic perturbation of the material tensors. However, the sum of indices for any

The chapter is mainly based on our papers (Alshits, Lyubimov & Radowicz, 2005 a, b).

In piezoelectric crystals, purely mechanical characteristics, the elastic displacement vector

*β* (**r**, *t*) and **E**(**r**, *t*) can be expressed in terms of their own potentials as

ˆ

*β* (**r**, *t*), and the stress tensor *σ*ˆ (**r**, *t*), are related to such electrical

(**r,** *t*) and the electric field strength **E**(**r**, *t*), and induction **D**(**r**, *t*).

 **r ur Er** ,*tt t t* , , , – , . **r** (1)

$$\hat{F}^{(0)} = \mathbf{m}c\mathbf{m} - \rho v^2 \hat{l}, \qquad \mathbf{e} = \mathbf{m}\hat{\mathbf{e}}\mathbf{m}, \qquad \varepsilon = \mathbf{m} \cdot \hat{\varepsilon}\mathbf{m}, \tag{6}$$

symbol denotes the dyadic product, and ˆ *I* is the unit matrix. A necessary condition for the existence of nontrivial solutions of the homogeneous equation (5) is

$$\det \hat{F}(\upsilon, \mathbf{m}) = 0.\tag{7}$$

This is a cubic equation for the square of phase velocity *v*, which determines the three branches of the velocity of the bulk acoustic waves *vα*(**m**) (*α* = 1, 2, 3).

Orientations of the corresponding mutually orthogonal polarization vectors **u**0*<sup>α</sup>* (**m**) of the isonormal eigenwaves can be expressed in terms of the ˆ *<sup>α</sup> F* matrix, which is adjoint to the matrix ˆ ˆ () (() ) **m mm** *α αα F Fv ,* and is determined from the condition ˆˆ ˆ ˆ det *αα α FF I F* . As can be readily checked, Eq. (5) for any vector **c** such that ˆ **<sup>c</sup>** <sup>0</sup> *<sup>α</sup> <sup>F</sup>* is satisfied for

$$|\mathfrak{u}\_{0a}(\mathbf{m})\rangle \mid \overline{\hat{F}}\_a(\mathbf{m})\mathbf{c}.\tag{8}$$

It should be emphasized that the direct relation (8) between the polarization vector ( ) **u m** <sup>0</sup>*<sup>α</sup>* and the wave normal **m** will be widely used in the subsequent analysis.

Once the field of elastic displacements for a given wave branch ( ) **u r** *<sup>α</sup> ,t* is known, we can also determine the corresponding electric components (Landau & Lifshitz, 1984). For the subsequent analysis, these components are conveniently represented [by analogy with Eqs. (3)–(8)] in a coordinate-free form as

$$
\boldsymbol{\phi}\_a = \mathbf{e} \cdot \mathbf{u}\_a / \,\boldsymbol{\varepsilon}, \qquad \mathbf{E}\_a = -ik\phi\_a \mathbf{m}, \qquad \mathbf{D}\_a = ik\hat{N}\mathbf{u}\_a. \tag{9}
$$

$$
\hat{N} = \hat{e}\mathbf{m} \cdot (\hat{\epsilon}\mathbf{m}) \otimes \mathbf{m} \cdot \hat{\epsilon}\mathbf{m} / \,\mathbf{m} \cdot \hat{\epsilon}\mathbf{m}.\tag{10}
$$

Electric Components of Acoustic Waves in the Vicinity of Nonpiezoactive Directions 7

2 2 1 3 ˆ**m** ( . 11 33 1 3 *ε ε m* , 0, ), *ε m εε ε m m* (18)

2 2 1 3 **e** 15 31 1 3 15 33 {( ) , 0, *e e mm e e m m* }, (20)

2 22

*F* matrix in (5) for classes (13) and

(21)

*F* matrix of the

(22)

*,*

(17)

11 1 44 3 1 3

*cm cm ρv dm m*

0

0

66 1 44 3

where *d* = *c*13 + *c*44. The *ε*ˆ**m** vector and, hence, the *ε* scalar in Eq. (6) are also the same for all

However, the form of the electric vector **e** according to Eq. (6) for the transversely isotropic crystals of three types is different. For the piezoelectric media belonging to classes (13) and

( **e** 14 1 3 *e mm* 0, , 0), (19)

respectively. For a medium of the symmetry class (15), the electric vector is given by a sum

symmetry classes (15), which contains no vanishing elements. In the same coordinates, the matrix for the piezoelectric media belonging to classes (13) and (14) has the following

> ( ) ˆ , ( )

*em em e e ε mm e m e m ε m*

15 3 15 1 15 31 1 1 3 15 1 33 3 1 1

*em em e e ε mm e m e m ε m*

respectively. For a medium of the symmetry classes (15), the matrix *N*ˆ is (by analogy with vector **e**) given by a sum of expressions (21) and (22). In classes (13) and (14) of higher

> 2 2 2 2 22 1 3 14 1 3

Such purely transverse waves of the *t* mode polarized orthogonally to the propagation plane are frequently called SH waves. The other two branches are polarized in the {*m*1, *m*3}

(0, 1, 0),

*||*

( /) *<sup>t</sup> t*

31 1 33 3 15 31 3 1 3 15 1 33 3 3 3

*Ne m m*

0 0 0 0 0

*ε / m*

2 3 3 14 3 1

2

2 22

2 22

.

*v m m e mm cc <sup>ε</sup>* (23)

*ε / mm*

3 13

0 ( ) 0( ) 0 0 00 0 , 0 ( ) 0( )

(14) is the same as in (17), but this conclusion is not valid for the ˆ


symmetry, one of the eigenwave branches for any direction **m** is purely transverse:

 **u**

66 44

ˆ 0 0

1 3 44 1 33 3

*dm m cm cm ρv*

2 22

symmetry classes (13)–(15) (Sirotin & Shaskolskaya, 1982):

of expressions (19) and (20). Thus, the structure of the ˆ

15 3

*N em ε*

(14), the electric vectors are expressed as

forms:

plane:

ˆ

(0) 2 22

*F cm cm ρv*

Relations (9) together with condition (8) determine the functions **E m**( ) *<sup>α</sup>* and **D m**( ) *<sup>α</sup>* necessary for the subsequent analysis.

As can be readily seen, ˆ **m***N* 0 . This identity and the third relation in (9) clearly illustrate the well-known property (see Introduction) according to which the electric field **E m**( ) *<sup>α</sup>* is purely longitudinal, whereas the induction **D m**( ) *<sup>α</sup>* is purely transverse:

$$\mathbf{E}\_a \mid \mid \mathbf{m}, \qquad \mathbf{D}\_a \perp \mathbf{m}.\tag{11}$$

On the other hand, the same identity ˆ **m***N* 0 implies one useful property of the *N*ˆ matrix:

$$\text{det }\hat{N} = 0,\tag{12}$$

which indicates that this matrix is planar and, hence, can be represented as a sum of two dyads.

#### **3. Examples of transversely isotropic piezoelectrics**

There are three groups of piezoelectrics which exhibit a transverse isotropy of their acoustic properties. They belong to the following classes of symmetry (Sirotin & Shaskolskaya, 1982):

$$\{\infty2, 622\}\tag{13}$$

$$
\omega\omega\mathfrak{m}\_{\prime}\mathfrak{G}\mathfrak{m}\mathfrak{m}\_{\prime}\tag{14}
$$

$$
\alpha \grave{,} \mathsf{6}.\tag{15}
$$

Owing to the transverse isotropy, the formulas presented below contain only the polar angle *θ* between the **m** vector and the *z* axis coinciding with the principal axis of symmetry. Without loss of generality, we may proceed with the analysis upon selecting any cross section containing the main axis. Here, it is convenient to choose

$$\mathbf{m} = (m\_{1'} \ 0 \ \prime \ m\_3) = (\sin \theta \ \prime \ 0 \ \prime \ \cos \theta). \tag{16}$$

In these coordinates, the ˆ(0) *F* matrix in Eq. (6) for all the six classes of symmetry (13)–(15) has the same quasi-diagonal form (Fedorov, 1968):

$$\hat{F}^{(0)} = \begin{pmatrix} c\_{11}m\_1^2 + c\_{44}m\_3^2 - \rho v^2 & 0 & dm\_1m\_3 \\ 0 & c\_{66}m\_1^2 + c\_{44}m\_3^2 - \rho v^2 & 0 \\ dm\_1m\_3 & 0 & c\_{44}m\_1^2 + c\_{33}m\_3^2 - \rho v^2 \end{pmatrix},\tag{17}$$

where *d* = *c*13 + *c*44. The *ε*ˆ**m** vector and, hence, the *ε* scalar in Eq. (6) are also the same for all symmetry classes (13)–(15) (Sirotin & Shaskolskaya, 1982):

6 Modeling and Measurement Methods for Acoustic Waves and for Acoustic Microdevices

purely longitudinal, whereas the induction **D m**( ) *<sup>α</sup>* is purely transverse:

**3. Examples of transversely isotropic piezoelectrics** 

section containing the main axis. Here, it is convenient to choose

has the same quasi-diagonal form (Fedorov, 1968):

(3)–(8)] in a coordinate-free form as

necessary for the subsequent analysis.

dyads.

Once the field of elastic displacements for a given wave branch ( ) **u r** *<sup>α</sup> ,t* is known, we can also determine the corresponding electric components (Landau & Lifshitz, 1984). For the subsequent analysis, these components are conveniently represented [by analogy with Eqs.

> **eu E m D u** / ,

Relations (9) together with condition (8) determine the functions **E m**( ) *<sup>α</sup>* and **D m**( ) *<sup>α</sup>*

As can be readily seen, ˆ **m***N* 0 . This identity and the third relation in (9) clearly illustrate the well-known property (see Introduction) according to which the electric field **E m**( ) *<sup>α</sup>* is

On the other hand, the same identity ˆ **m***N* 0 implies one useful property of the *N*ˆ matrix:

which indicates that this matrix is planar and, hence, can be represented as a sum of two

There are three groups of piezoelectrics which exhibit a transverse isotropy of their acoustic properties. They belong to the following classes of symmetry (Sirotin & Shaskolskaya, 1982):

Owing to the transverse isotropy, the formulas presented below contain only the polar angle *θ* between the **m** vector and the *z* axis coinciding with the principal axis of symmetry. Without loss of generality, we may proceed with the analysis upon selecting any cross

**m** ( (sin cos

In these coordinates, the ˆ(0) *F* matrix in Eq. (6) for all the six classes of symmetry (13)–(15)

1 3 *m m* , 0, ) , 0, ). (16)

, ˆ

, *α α αα α α -ik ikN* (9)

ˆ*N e* ˆˆ ˆ ˆ **m m m m m m.** -( ) / *ε e ε* (10)

**E m D m.** *α α || ,* (11)

ˆ det 0, *N* (12)

2, 622, (13)

*m mm* , 6, (14)

, 6. (15)

$$
\hat{\varepsilon}\mathbf{m} = (\varepsilon\_1 m\_1 \; \; \; 0 \; \; \; \varepsilon\_3 m\_3 \;) \; \; \; \; \varepsilon = \varepsilon\_1 m\_1^2 + \varepsilon\_3 m\_3^2. \tag{18}
$$

However, the form of the electric vector **e** according to Eq. (6) for the transversely isotropic crystals of three types is different. For the piezoelectric media belonging to classes (13) and (14), the electric vectors are expressed as

$$\mathbf{e} = e\_{14}(0, \ m\_1 m\_3, \ \ 0),\tag{19}$$

$$\mathbf{e} = \{ (e\_{15} + e\_{31})m\_1 m\_{3'} \; 0 \; \; e\_{15} m\_1^2 + e\_{33} m\_3^2 \} \; \tag{20}$$

respectively. For a medium of the symmetry class (15), the electric vector is given by a sum of expressions (19) and (20). Thus, the structure of the ˆ *F* matrix in (5) for classes (13) and (14) is the same as in (17), but this conclusion is not valid for the ˆ *F* matrix of the symmetry classes (15), which contains no vanishing elements. In the same coordinates, the matrix for the piezoelectric media belonging to classes (13) and (14) has the following forms:

$$
\hat{N} = e\_{14} \begin{pmatrix} 0 & (\varepsilon\_3 / \varepsilon) m\_3^2 & 0 \\ -m\_3 & 0 & -m\_1 \\ 0 & -(\varepsilon\_3 / \varepsilon) m\_1 m\_3^2 & 0 \end{pmatrix} \tag{21}
$$

$$
\hat{N} = \begin{pmatrix}
 e\_{15}m\_{3} & 0 & e\_{15}m\_{1} \\
 0 & e\_{15}m\_{3} & 0 \\
 e\_{31}m\_{1} & 0 & e\_{35}m\_{3}
\end{pmatrix} - e^{1} \begin{pmatrix}
 (e\_{15} + e\_{31})e\_{1}m\_{1}^{2}m\_{3} & 0 & (e\_{15}m\_{1}^{2} + e\_{33}m\_{3}^{2})e\_{1}m\_{1} \\
 0 & 0 & 0 \\
 (e\_{15} + e\_{31})e\_{3}m\_{1}m\_{3}^{2} & 0 & (e\_{15}m\_{1}^{2} + e\_{33}m\_{3}^{2})e\_{3}m\_{3}
\end{pmatrix},\tag{22}
$$

respectively. For a medium of the symmetry classes (15), the matrix *N*ˆ is (by analogy with vector **e**) given by a sum of expressions (21) and (22). In classes (13) and (14) of higher symmetry, one of the eigenwave branches for any direction **m** is purely transverse:

$$\mathbf{u}\_t \parallel \langle 0, 1, 0 \rangle\_{\prime} $$

$$\rho \sigma\_t^2 = c\_{66} m\_1^2 + c\_{44} m\_3^2 + (e\_{14}^2 / \epsilon) m\_1^2 m\_3^2 . \tag{23}$$

Such purely transverse waves of the *t* mode polarized orthogonally to the propagation plane are frequently called SH waves. The other two branches are polarized in the {*m*1, *m*3} plane:

$$\begin{aligned} \mathbf{u}\_{l,l'} &\quad \mathbb{I} \mid \{2dm\_1m\_3, \quad 0, \ -\Delta\_{14}^-m\_1^2 + \Delta\_{34}^-m\_3^2 \pm R\} \\ \rho \boldsymbol{\upsilon}\_{l,l'}^2 &= \left(\Delta\_{14}^+m\_1^2 + \Delta\_{34}^+m\_3^2 \pm R\right) / 2 \end{aligned} \tag{24}$$

Electric Components of Acoustic Waves in the Vicinity of Nonpiezoactive Directions 9

33 15 31 15 31 33 15 tan /(2 ), tan ( )/ . *l t θ e ee θ eee e* (31)

**u m um** <sup>02</sup> <sup>02</sup> ( ) ( ), (32)

**u m um u m um** <sup>01</sup> <sup>01</sup> <sup>03</sup> <sup>03</sup> ( ) ( ), ( ) ( ). (33)

in some special cases can be satisfied even on the whole **m** <sup>2</sup> 1 sphere. This takes place, in particular, in the transversely isotropic crystals belonging to the symmetry classes (13) (for the *l* and *t*' modes (26)) and (14) (for the *t* mode (27)). For all other crystals, including transversely isotropic crystals belonging to the symmetry classes (15), the geometric locus of the longitudinal nonpiezoactivity has the form of lines on the unit sphere **m** <sup>2</sup> 1 . Such lines also exist in the piezoactive branches of the aforementioned high-symmetry media belonging to symmetry classes (13) and (14). For example, the zero-field lines **E***<sup>α</sup>* = 0 in the *l*  and *t*' branches of the media of classes (14) and (15) appear at the intersection of the **m** <sup>2</sup> 1

sphere with the cones of directions defined by the polar angles *<sup>l</sup> θ* and *t θ* as

can be arranged on the **m** <sup>2</sup> 1 sphere so that one is even,

approximate formulas (31).

and two are odd,

2 2

For simplicity, these expressions are written in an approximate form corresponding to the case of a weak electromechanical interaction and a small elastic anisotropy. Nevertheless, one can readily check that the exact condition for the existence of the aforementioned nonpiezoactivity cones is the positive determinacy of the right-hand parts of the

It is possible to prove that the longitudinal nonpiezoactivity lines in fact exist practically in all (even triclinic) crystals. Let us consider a crystal with arbitrary anisotropy, which contains at least one acoustic axis of the general (conical) type. Here, it should be noted that no one real crystal without acoustic axes and no one triclinic crystal without conical axes are known so far. As was demonstrated by Alshits & Lothe (1979) and Holm (1992), the polarization fields of elastic displacements **u m** <sup>0</sup> ( ) *<sup>α</sup>* for the bulk eigenwaves in such a crystal

The nondegenerate branch **u m** <sup>03</sup>( ) is always odd and continuous on the entire sphere of wave directions. As for the degenerate branches, **u m** <sup>01</sup>( ) and **u m** <sup>02</sup> ( ) , their evenness depends on the representation and can be changed simultaneously. These branches are continuous at all points of the sphere except for some open-ended lines on which the **u m** <sup>01</sup>( ) and **u m** <sup>02</sup> ( ) functions change sign. Such "anti-sign" lines can be arbitrarily deformed on the unit sphere without changing the positions of end points (coinciding with the points of degeneracy). In fact, the representation is chosen by setting certain fixed

positions of the anti-sign lines (coinciding for both degenerate branches).

**em u m** ( ) ( ), *<sup>α</sup>* (30)

where

$$R = \sqrt{\left(\Delta\_{14}^{-} m\_1^2 - \Delta\_{34}^{-} m\_3^2\right)^2 + \left(2dm\_1m\_3\right)^2},\tag{25}$$

$$\Delta\_{ij}^{\pm} = c\_{ii} \pm c\_{jj}.\tag{25}$$

The electric components of the above wave fields can be also determined for an arbitrary direction **m**. For a medium of the symmetry class (13):

$$\begin{aligned} \phi\_t &= (e\_{14} \land \varepsilon) m\_1 m\_3 u\_{t'} \\ \mathbf{D}\_t &\quad \mid \mid e\_{14} (e\_3 \land \varepsilon) m\_3^2 (m\_3, \; 0, \; \mathbf{m}\_1) \boldsymbol{\mu}\_{t'} \\ \phi\_{t, t'} (\mathbf{m}) &\equiv 0, \\ \mathbf{D}\_{t, t'} &\mid \mid -e\_{14} (\mathbf{0}, \; \mathbf{1}\_{\prime} \; \mathbf{0}) [m\_1 (\mathbf{u}\_{t, t'})\_3 + m\_3 (\mathbf{u}\_{t, t'})\_1]. \end{aligned} \tag{26}$$

For the less simple symmetry class (14), we present only the result for the SH-branch (23):

$$
\phi\_t(\mathbf{m}) \equiv 0, \qquad \mathbf{D}\_t \mid \mid \text{ (0, } e\_{15}m\_{3\prime} \text{ 0)}\mu\_t. \tag{27}
$$

The structure of acoustic waves in media of the symmetry classes (15) is more complicated. In this case, even a purely transverse solution ( **u***<sup>t</sup>* || *y*) exists only in the *xy* basis plane.

#### **4. Lines of zero electric field on the unit sphere**

According to the second relation in (9), the electric field amplitude distribution on the unit sphere of the wave propagation directions is described by the equation

$$\mathbf{E}\_a(\mathbf{m}) = \text{const} \cdot \boldsymbol{\phi}\_a(\mathbf{m}) \mathbf{m}\_\prime \tag{28}$$

which shows that zero values of **E m**( ) *<sup>α</sup>* coincide with those of the potential / **e u** *α α* . According to condition (8), these directions are determined by the equation,

$$\mathbf{e}(\mathbf{m}) \cdot \overline{\hat{\mathbb{P}}}\_{a}(\mathbf{m}) \mathbf{c} = 0. \tag{29}$$

The acoustic waves (3) propagating in these directions contain no electrostatic components **E***<sup>α</sup>* , as in a nonpiezoelectric medium. Even a nonzero induction field *D eu i ijk k, j* in these directions does not influence the parameters of the displacement wave.

The scalar equation (29) poses only one limitation on the direction of the wave normal **m** ≡ **m**(*θ*,) as a function of two spherical angular coordinates. In other words, Eq. (29) determines a line (or several lines) of nonpiezoelectric directions (in which **E***<sup>α</sup>* = 0) on the sphere **m** <sup>2</sup> 1 . It should be noted that the condition of longitudinal nonpiezoactivity,

$$\mathbf{e}(\mathbf{m}) \perp \mathbf{u}\_a(\mathbf{m}),\tag{30}$$

in some special cases can be satisfied even on the whole **m** <sup>2</sup> 1 sphere. This takes place, in particular, in the transversely isotropic crystals belonging to the symmetry classes (13) (for the *l* and *t*' modes (26)) and (14) (for the *t* mode (27)). For all other crystals, including transversely isotropic crystals belonging to the symmetry classes (15), the geometric locus of the longitudinal nonpiezoactivity has the form of lines on the unit sphere **m** <sup>2</sup> 1 . Such lines also exist in the piezoactive branches of the aforementioned high-symmetry media belonging to symmetry classes (13) and (14). For example, the zero-field lines **E***<sup>α</sup>* = 0 in the *l* 

and *t*' branches of the media of classes (14) and (15) appear at the intersection of the **m** <sup>2</sup> 1 sphere with the cones of directions defined by the polar angles *<sup>l</sup> θ* and *t θ* as

$$
\tan^2 \theta\_l = -e\_{33} \left/ \left< 2e\_{15} + e\_{31} \right> , \quad \tan^2 \theta\_{l'} = \left( e\_{15} + e\_{31} - e\_{33} \right) / e\_{15} . \tag{31}
$$

For simplicity, these expressions are written in an approximate form corresponding to the case of a weak electromechanical interaction and a small elastic anisotropy. Nevertheless, one can readily check that the exact condition for the existence of the aforementioned nonpiezoactivity cones is the positive determinacy of the right-hand parts of the approximate formulas (31).

It is possible to prove that the longitudinal nonpiezoactivity lines in fact exist practically in all (even triclinic) crystals. Let us consider a crystal with arbitrary anisotropy, which contains at least one acoustic axis of the general (conical) type. Here, it should be noted that no one real crystal without acoustic axes and no one triclinic crystal without conical axes are known so far. As was demonstrated by Alshits & Lothe (1979) and Holm (1992), the polarization fields of elastic displacements **u m** <sup>0</sup> ( ) *<sup>α</sup>* for the bulk eigenwaves in such a crystal can be arranged on the **m** <sup>2</sup> 1 sphere so that one is even,

$$\mathbf{u}\_{02}(-\mathbf{m}) = \mathbf{u}\_{02}(\mathbf{m}),\tag{32}$$

and two are odd,

8 Modeling and Measurement Methods for Acoustic Waves and for Acoustic Microdevices

 

where

**m**(*θ*, *||*

direction **m**. For a medium of the symmetry class (13):

*l,t*

*l,t*

2 22 14 1 34 3

()

(

*||*

**4. Lines of zero electric field on the unit sphere** 

sphere of the wave propagation directions is described by the equation

According to condition (8), these directions are determined by the equation,

directions does not influence the parameters of the displacement wave.

**D**

*||*

 **u** 1 3 <sup>14</sup> (2 , 0, - Δ Δ ), (Δ Δ ) / 2,

> 

( )( -

*l,t*

*R m m dm m*

The electric components of the above wave fields can be also determined for an arbitrary

*c c*

((

14 1 33 1

) 0, (0, 1, 0)[ ) ) ].

14 1 3 2 14 3 3 3 1

*e ε mmu e ε ε mm mu*

*t t t t*

/ ,

/ , 0, ) ,

**m D** *||* 15 3 ) 0, 0, , 0) . *tt t em u* (27)

**m m,** *α α* (28)

<sup>ˆ</sup> **em mc** () () 0 . *<sup>α</sup> <sup>F</sup>* (29)

The structure of acoustic waves in media of the symmetry classes (15) is more complicated.

According to the second relation in (9), the electric field amplitude distribution on the unit

**E m**( ) const ( )

which shows that zero values of **E m**( ) *<sup>α</sup>* coincide with those of the potential /

The acoustic waves (3) propagating in these directions contain no electrostatic components **E***<sup>α</sup>* , as in a nonpiezoelectric medium. Even a nonzero induction field *D eu i ijk k, j* in these

The scalar equation (29) poses only one limitation on the direction of the wave normal **m** ≡

determines a line (or several lines) of nonpiezoelectric directions (in which **E***<sup>α</sup>* = 0) on the

sphere **m** <sup>2</sup> 1 . It should be noted that the condition of longitudinal nonpiezoactivity,

) as a function of two spherical angular coordinates. In other words, Eq. (29)

*e mm*

*l,t l,t l,t*

For the less simple symmetry class (14), we present only the result for the SH-branch (23):

( (

In this case, even a purely transverse solution ( **u***<sup>t</sup>* || *y*) exists only in the *xy* basis plane.

**m D uu**

 *R*

2 2 1 34 3

*m m*

*v mm* (24)

(25)

(26)

**e u**

*α α* .

*dm m R*

2 22 2 14 1 34 3 1 3 (Δ Δ ) (2 ) , Δ . *ij ii jj*

$$\mathbf{u}\_{01}(-\mathbf{m}) = -\mathbf{u}\_{01}(\mathbf{m}), \qquad \mathbf{u}\_{03}(-\mathbf{m}) = -\mathbf{u}\_{03}(\mathbf{m}).\tag{33}$$

The nondegenerate branch **u m** <sup>03</sup>( ) is always odd and continuous on the entire sphere of wave directions. As for the degenerate branches, **u m** <sup>01</sup>( ) and **u m** <sup>02</sup> ( ) , their evenness depends on the representation and can be changed simultaneously. These branches are continuous at all points of the sphere except for some open-ended lines on which the **u m** <sup>01</sup>( ) and **u m** <sup>02</sup> ( ) functions change sign. Such "anti-sign" lines can be arbitrarily deformed on the unit sphere without changing the positions of end points (coinciding with the points of degeneracy). In fact, the representation is chosen by setting certain fixed positions of the anti-sign lines (coinciding for both degenerate branches).

One can readily check that the aforementioned properties of the fields of elastic displacements, which were established in (Alshits & Lothe, 1979; Holm, 1992) for purely elastic media, are also valid for piezoelectrics. Taking into account that, according to relations (33), the **u m** <sup>03</sup>( ) function is odd and the **e**(**m**) function is [by definition (6)] even, we may conclude that the potential

$$\phi\_{03}(\mathbf{m}) = \mathbf{e}(\mathbf{m}) \cdot \mathbf{u}\_{03}(\mathbf{m}) / \,\varepsilon$$

is an odd function

$$
\phi\_{03}(-\mathbf{m}) = -\phi\_{03}(\mathbf{m}).\tag{34}
$$

Electric Components of Acoustic Waves in the Vicinity of Nonpiezoactive Directions 11

*Any direction in symmetry planes*  *Any direction in planes orthogonal to symmetry axes of even order* 

Let us consider, for example, a monoclinic piezoelectric crystal belonging to one of the two possible symmetry classes: *m* or 2. In the first case, the electric vector **e** of any wave propagating in a plane of symmetry *m* must, obviously, lie in the same plane being, hence, orthogonal to the polarization vector **u**0*<sup>t</sup>* of all SH waves of the *t* branch. In the second case, the **e** vector for a wave normal occurring in the plane perpendicular to the dyad (2-fold) axis of symmetry must be parallel to this axis and, hence, orthogonal to polarization vectors (belonging to said plane) of the *l* and *t*' waves. Naturally, the latter property is valid for any other symmetry axis of even order. In monograph (Royer & Dieulesaint, 2000), this rule was

Wave branches *Transverse waves SH waves In-plane polarized waves* 

In particular, the coordinate planes of the crystal system orthogonal to the tetrad and dyad axes in cubic piezoelectrics (symmetry classes 43*m* and 23) must be nonpiezoactive for the corresponding *l* and *t*' branches. At the same time, the diagonal symmetry planes {110} are nonpiezoactive for the corresponding *t* (SH) waves. One can check that, in the vicinity of the coordinate axes, the potential amplitudes for these branches can be represented in spherical

> <sup>2</sup>

<sup>2</sup>

> ( ) *l,t* and (b)

of the (0, 0, 1) direction in a cubic piezoelectric crystal. Numbers 1, 2, and 3 refer to the angles

<sup>123</sup> *θθθ* ; solid and dashed lines relate to the potentials of different signs.

sin2 , *l,t <sup>θ</sup>* (35)

cos2 . *<sup>t</sup> <sup>θ</sup>* (36)

 ( ) 

*<sup>t</sup>* at *θ* = const in the vicinity

formulated for planes orthogonal to the tetrad (4-fold) and hexad (6-fold) axes.

*Directions of acoustic and symmetry axes* 

**Table 1.** Directions of obligatory longitudinal nonpiezoactivity ( **E***<sup>α</sup>* = 0) in crystals

The above theorems are summarized in Table 1.

Directions of nonpiezoactivity

coordinates (

, 

) as (Fig. 1)

**Figure 1.** Polar diagrams of the electric potentials (a)

This result implies that, for any path connecting the opposite points **m** and –**m** on the unit sphere, there exists at least one point **m**0 such that **m** 03 0 ( )0 . In scanning the paths on the unit sphere, points **m**0 will apparently form a closed line representing a geometric locus of the directions of longitudinal nonpiezoactivity for the nondegenerate branch.

For the degenerate branches <sup>01</sup>( ) **m** and <sup>02</sup> ( ) **m** , the considerations should be somewhat modified, while being still generally analogous to those used in solving a similar problem (Alshits & Lothe, 1979) concerning the existence of the lines of solutions for exceptional bulk waves related to the same degenerate branches in semi-infinite elastic media. Not reproducing these considerations here, we only formulate the result: the longitudinal nonpiezoactivity lines exist in both degenerate branches and pass from one branch to another at the degeneratcy points. Thus, the following theorem of existence is valid:

*All three wave branches in an arbitrary crystal, which contains conical acoustic axes, must possess lines of longitudinal nonpiezoactivity directions on the unit sphere.* 

It should be also noted that, when a wave propagates along an acoustic axis **m***<sup>d</sup>* of any type, the continuum of possible orientations of the wave polarization **u** in the plane of degeneracy always contains a vector orthogonal to the ) **e m**( *<sup>d</sup>* direction. In view of the criterion (30), this ensures nonpiezoactivity of the corresponding wave. Therefore,

*acoustic axes must belong to the lines of longitudinal nonpiezoactivity.* 

The elements of crystal symmetry can become an additional factor accounting for the phenomenon of nonpiezoactivity. According to (Royer & Dieulesaint, 2000), also

*symmetry axes determine the directions of longitudinal nonpiezoactivity for purely transverse modes,* 

while

*a symmetry plane is the geometric locus of directions of longitudinal nonpiezoactivity for the related SH waves.* 

#### One can add that

*the planes orthogonal to symmetry axes of even order are the geometric locus of directions of longitudinal nonpiezoactivity for in-plane polarized waves.* 

Let us consider, for example, a monoclinic piezoelectric crystal belonging to one of the two possible symmetry classes: *m* or 2. In the first case, the electric vector **e** of any wave propagating in a plane of symmetry *m* must, obviously, lie in the same plane being, hence, orthogonal to the polarization vector **u**0*<sup>t</sup>* of all SH waves of the *t* branch. In the second case, the **e** vector for a wave normal occurring in the plane perpendicular to the dyad (2-fold) axis of symmetry must be parallel to this axis and, hence, orthogonal to polarization vectors (belonging to said plane) of the *l* and *t*' waves. Naturally, the latter property is valid for any other symmetry axis of even order. In monograph (Royer & Dieulesaint, 2000), this rule was formulated for planes orthogonal to the tetrad (4-fold) and hexad (6-fold) axes.

The above theorems are summarized in Table 1.

10 Modeling and Measurement Methods for Acoustic Waves and for Acoustic Microdevices

the directions of longitudinal nonpiezoactivity for the nondegenerate branch.

<sup>01</sup>( ) **m** and

we may conclude that the potential

For the degenerate branches

sphere, there exists at least one point **m**0 such that

*lines of longitudinal nonpiezoactivity directions on the unit sphere.* 

ensures nonpiezoactivity of the corresponding wave. Therefore,

*acoustic axes must belong to the lines of longitudinal nonpiezoactivity.* 

*longitudinal nonpiezoactivity for in-plane polarized waves.* 

is an odd function

while

*SH waves.* 

One can add that

One can readily check that the aforementioned properties of the fields of elastic displacements, which were established in (Alshits & Lothe, 1979; Holm, 1992) for purely elastic media, are also valid for piezoelectrics. Taking into account that, according to relations (33), the **u m** <sup>03</sup>( ) function is odd and the **e**(**m**) function is [by definition (6)] even,

**m em u m**

 **m m** 

This result implies that, for any path connecting the opposite points **m** and –**m** on the unit

unit sphere, points **m**0 will apparently form a closed line representing a geometric locus of

modified, while being still generally analogous to those used in solving a similar problem (Alshits & Lothe, 1979) concerning the existence of the lines of solutions for exceptional bulk waves related to the same degenerate branches in semi-infinite elastic media. Not reproducing these considerations here, we only formulate the result: the longitudinal nonpiezoactivity lines exist in both degenerate branches and pass from one branch to

*All three wave branches in an arbitrary crystal, which contains conical acoustic axes, must possess* 

It should be also noted that, when a wave propagates along an acoustic axis **m***<sup>d</sup>* of any type, the continuum of possible orientations of the wave polarization **u** in the plane of degeneracy always contains a vector orthogonal to the ) **e m**( *<sup>d</sup>* direction. In view of the criterion (30), this

The elements of crystal symmetry can become an additional factor accounting for the

*symmetry axes determine the directions of longitudinal nonpiezoactivity for purely transverse modes,* 

*a symmetry plane is the geometric locus of directions of longitudinal nonpiezoactivity for the related* 

*the planes orthogonal to symmetry axes of even order are the geometric locus of directions of* 

phenomenon of nonpiezoactivity. According to (Royer & Dieulesaint, 2000), also

another at the degeneratcy points. Thus, the following theorem of existence is valid:

<sup>03</sup> <sup>03</sup> ( ) ( ) ( )/

<sup>03</sup> <sup>03</sup> ( ) ( ). (34)

**m** 03 0 ( )0 . In scanning the paths on the

<sup>02</sup> ( ) **m** , the considerations should be somewhat


**Table 1.** Directions of obligatory longitudinal nonpiezoactivity ( **E***<sup>α</sup>* = 0) in crystals

In particular, the coordinate planes of the crystal system orthogonal to the tetrad and dyad axes in cubic piezoelectrics (symmetry classes 43*m* and 23) must be nonpiezoactive for the corresponding *l* and *t*' branches. At the same time, the diagonal symmetry planes {110} are nonpiezoactive for the corresponding *t* (SH) waves. One can check that, in the vicinity of the coordinate axes, the potential amplitudes for these branches can be represented in spherical coordinates (, ) as (Fig. 1)

$$
\phi\_{l,t'} \propto \theta^2 \sin 2\varphi\_{\prime} \tag{35}
$$

$$
\phi\_t \propto \theta^2 \cos 2\varphi.\tag{36}
$$

**Figure 1.** Polar diagrams of the electric potentials (a) ( ) *l,t* and (b) ( ) *<sup>t</sup>* at *θ* = const in the vicinity of the (0, 0, 1) direction in a cubic piezoelectric crystal. Numbers 1, 2, and 3 refer to the angles <sup>123</sup> *θθθ* ; solid and dashed lines relate to the potentials of different signs.

### **5. Zero-induction points on the unit sphere**

#### **5.1. General case of arbitrary anisotropy**

Now let us consider the conditions determining the propagation directions **m**0 in which the electric induction vector <sup>ˆ</sup> **D u** *α α ikN* defined in (9) vanishes. Taking into account identity (12) and the definition of the adjoint tensor

$$
\hat{N}\hat{N}\hat{\mathbb{N}} = \hat{I}\text{det}\hat{N},\tag{37}
$$

Electric Components of Acoustic Waves in the Vicinity of Nonpiezoactive Directions 13

. <sup>ˆ</sup> **D mm** 0 0 ( ) *<sup>l</sup> ikN* (39)

, } . **uD m** <sup>0</sup> { *t,t t,t* (40)

**m mmm**

**m m** 0 000

0 0

**D m** || <sup>0</sup> ( 2) 0. *<sup>l</sup>* (43)

*<sup>ε</sup>* (41)

(42)

**5.2. Zero-induction points related to elements of the crystal symmetry** 

Let us consider a wave propagating along the direction **m**<sup>0</sup> , which coincides with a symmetry axis of any order except for dyad axes (e.g., this can be the 3, 4, 4 , 6, or 6 -fold axis). As is known (Fedorov, 1968), any symmetry axis (including a dyad axis) is a longitudinal normal. Evidently, the electric induction **D m**<sup>0</sup> ( ) *<sup>l</sup>* accompanying the longitudinal wave **u m** *||* ) <sup>0</sup> ( *<sup>l</sup>* must be zero, otherwise the **D m**<sup>0</sup> ( ) *<sup>l</sup>* vector would possess two equivalent orientations, in contradiction with the single-valued third relation in (9):

It should be noted that this argument does not work in the case of transverse branches. For the selected symmetry directions they are always degenerate, that is, possessing equal phase

The wave propagating along a dyad axis should be treated separately (albeit with the same result). In the general case, this direction is not an acoustic axis. On the other hand, the transverse isonormal vectors **D m** *<sup>α</sup>* <sup>0</sup> ( *α t,t ,l* ) are determined to within the sign (like **u***<sup>α</sup>* ) and, hence, their symmetry rotations due to the dyad axis cannot be considered as different solutions. So, one can readily check that, for a propagation direction along the dyad axis, the transverse branches *α t,t* are again characterized by nonzero induction vectors. However, the longitudinal branch in the same direction always obeys the relation **D m** <sup>0</sup> ( )0 *<sup>l</sup>* . Indeed,

ˆ ˆ <sup>ˆ</sup> / <sup>ˆ</sup> . <sup>ˆ</sup>

( )[( ) ] () ( ) *<sup>l</sup> <sup>ε</sup> eik N e*

Let us check that the right-hand part of this expression vanishes even for a monoclinic crystal of the class 2. Selecting the *z* axis in (41) along the dyad axis *||* **m**<sup>0</sup> (2 ) , we obtain

> / .

, , 0 *<sup>l</sup> <sup>ε</sup> <sup>e</sup> <sup>ε</sup> <sup>e</sup> ik e <sup>e</sup>*

However, according to (Royer & Dieulesaint, 2000; Sirotin & Shaskolskaya, 1982), the offdiagonal components of *e*ˆ and *ε*ˆ tensors, entering into Eq. (42) for the symmetry class 2 in

33 33

*ε ε*

**D** 13 33 23 33 13 23

00 00

velocities ( ) *t t v v* and, hence, arbitrary orientations of **u***t,t* and **D***t,t* in the plane:

*5.2.1. Longitudinal waves propagating along symmetry axes* 

combining Eqs. (10) and (39) for **m m** <sup>0</sup> , we obtain

**D mm mm**

this coordinate system, are vanishing: 13 23 13 23 *e e ε ε* 0 and, hence,

one can readily check that **D***<sup>α</sup>* = 0 for the directions **m**0 such that **u***<sup>α</sup>* || *<sup>N</sup>*ˆ**<sup>d</sup>** , where **d** is any vector obeying the condition *N*ˆ**<sup>d</sup>** <sup>≠</sup> 0. For these directions **m**<sup>0</sup> , according to condition (8), we also have

$$
\overline{\hat{F}}\_a(\mathbf{m})\mathbf{c} \parallel \overline{\hat{N}}\mathbf{d} \tag{38}
$$

In the general case, this condition gives two equations with two unknowns *θ* and , which determine the positions of isolated points **m** <sup>0</sup>( ) *,* such that **D***<sup>α</sup>* = 0 on the unit sphere **m** <sup>2</sup> 1 . There is the well-known Brouwer theorem in the topology, according to which

*any continuous transform on a sphere, not mapping any point by its antipode, has at least two stationary points.* 

Now let us consider a distribution of vectors **D m**( ) *<sup>α</sup>* continuous everywhere on the unit sphere. The continuity of **D m**( ) *<sup>α</sup>* is ensured when the corresponding branch *α* is nondegenerate. According to Brouwer's theorem, this distribution of **D***α* vectors tangent to the sphere must have two stationary points for which **D***<sup>α</sup>* = 0. On the other hand, relations (9) and (10) imply that this distribution also possesses an additional property: **D m**(- ) *<sup>α</sup>* || **D m**( ) *<sup>α</sup>* . For this reason, the pair of points stipulated by Brouwer's theorem includes the inversion-equivalent stationary points **m**0 and – **m**<sup>0</sup> . Thus, the following theorem of existence of the transverse nonpiezoactivity directions is valid:

*In any crystal of unrestricted anisotropy each nondegenerate branch must contain at least one pair of inversion-equivalent zero-induction points m0 such that <sup>α</sup> <sup>0</sup> D m ( )0 on the unit sphere.* 

Therefore, the zero-induction points in a wave field **D***α*(**m**) must exist even in triclinic crystals. Of course, the positions of such points in the general case (i.e., the solutions of Eq. (38) in the general form) cannot be found analytically. However, in some more symmetric crystals, nonpiezoactive directions **m**0 can be found without cumbersome computations.

#### **5.2. Zero-induction points related to elements of the crystal symmetry**

#### *5.2.1. Longitudinal waves propagating along symmetry axes*

12 Modeling and Measurement Methods for Acoustic Waves and for Acoustic Microdevices

Now let us consider the conditions determining the propagation directions **m**0 in which the electric induction vector <sup>ˆ</sup> **D u** *α α ikN* defined in (9) vanishes. Taking into account identity

one can readily check that **D***<sup>α</sup>* = 0 for the directions **m**0 such that **u***<sup>α</sup>* || *<sup>N</sup>*ˆ**<sup>d</sup>** , where **d** is any

vector obeying the condition *N*ˆ**<sup>d</sup>** <sup>≠</sup> 0. For these directions **m**<sup>0</sup> , according to condition (8),

sphere **m** <sup>2</sup> 1 . There is the well-known Brouwer theorem in the topology, according to

*any continuous transform on a sphere, not mapping any point by its antipode, has at least two* 

Now let us consider a distribution of vectors **D m**( ) *<sup>α</sup>* continuous everywhere on the unit sphere. The continuity of **D m**( ) *<sup>α</sup>* is ensured when the corresponding branch *α* is nondegenerate. According to Brouwer's theorem, this distribution of **D***α* vectors tangent to the sphere must have two stationary points for which **D***<sup>α</sup>* = 0. On the other hand, relations (9) and (10) imply that this distribution also possesses an additional property: **D m**(- ) *<sup>α</sup>* || **D m**( ) *<sup>α</sup>* . For this reason, the pair of points stipulated by Brouwer's theorem includes the inversion-equivalent stationary points **m**0 and – **m**<sup>0</sup> . Thus, the following theorem of

*In any crystal of unrestricted anisotropy each nondegenerate branch must contain at least one pair of* 

Therefore, the zero-induction points in a wave field **D***α*(**m**) must exist even in triclinic crystals. Of course, the positions of such points in the general case (i.e., the solutions of Eq. (38) in the general form) cannot be found analytically. However, in some more symmetric crystals, nonpiezoactive directions **m**0 can be found without cumbersome

*inversion-equivalent zero-induction points m0 such that <sup>α</sup> <sup>0</sup> D m ( )0 on the unit sphere.* 

 

In the general case, this condition gives two equations with two unknowns *θ* and

ˆˆ ˆ ˆ *NN I N,* det (37)

ˆ ˆ ( ) **mc d** *|| <sup>α</sup> F N* (38)

<sup>0</sup>( ) *,* such that **D***<sup>α</sup>* = 0 on the unit

, which

**5. Zero-induction points on the unit sphere** 

**5.1. General case of arbitrary anisotropy** 

(12) and the definition of the adjoint tensor

determine the positions of isolated points **m**

existence of the transverse nonpiezoactivity directions is valid:

we also have

which

*stationary points.* 

computations.

Let us consider a wave propagating along the direction **m**<sup>0</sup> , which coincides with a symmetry axis of any order except for dyad axes (e.g., this can be the 3, 4, 4 , 6, or 6 -fold axis). As is known (Fedorov, 1968), any symmetry axis (including a dyad axis) is a longitudinal normal. Evidently, the electric induction **D m**<sup>0</sup> ( ) *<sup>l</sup>* accompanying the longitudinal wave **u m** *||* ) <sup>0</sup> ( *<sup>l</sup>* must be zero, otherwise the **D m**<sup>0</sup> ( ) *<sup>l</sup>* vector would possess two equivalent orientations, in contradiction with the single-valued third relation in (9):

$$\mathbf{D}\_{l} = ik\hat{N}(\mathbf{m}\_{0})\mathbf{m}\_{0}.\tag{39}$$

It should be noted that this argument does not work in the case of transverse branches. For the selected symmetry directions they are always degenerate, that is, possessing equal phase velocities ( ) *t t v v* and, hence, arbitrary orientations of **u***t,t* and **D***t,t* in the plane:

$$\{\mathbf{u}\_{t,t'}, \mathbf{D}\_{t,t'}\} \perp \mathbf{m}\_0. \tag{40}$$

The wave propagating along a dyad axis should be treated separately (albeit with the same result). In the general case, this direction is not an acoustic axis. On the other hand, the transverse isonormal vectors **D m** *<sup>α</sup>* <sup>0</sup> ( *α t,t ,l* ) are determined to within the sign (like **u***<sup>α</sup>* ) and, hence, their symmetry rotations due to the dyad axis cannot be considered as different solutions. So, one can readily check that, for a propagation direction along the dyad axis, the transverse branches *α t,t* are again characterized by nonzero induction vectors. However, the longitudinal branch in the same direction always obeys the relation **D m** <sup>0</sup> ( )0 *<sup>l</sup>* . Indeed, combining Eqs. (10) and (39) for **m m** <sup>0</sup> , we obtain

$$\mathbf{D}\_1 / ik = \hat{N}(\mathbf{m}\_0)\mathbf{m}\_0 = (\hat{\varepsilon}\mathbf{m}\_0)\mathbf{m}\_0 - \frac{(\hat{\varepsilon}\mathbf{m}\_0)[(\mathbf{m}\_0\hat{\varepsilon}\mathbf{m}\_0)\mathbf{m}\_0]}{\mathbf{m}\_0 \cdot \hat{\varepsilon}\mathbf{m}\_0}.\tag{41}$$

Let us check that the right-hand part of this expression vanishes even for a monoclinic crystal of the class 2. Selecting the *z* axis in (41) along the dyad axis *||* **m**<sup>0</sup> (2 ) , we obtain

$$\mathbf{D}\_1 / ik = \begin{pmatrix} \varepsilon\_{13} - \frac{\varepsilon\_{13}\varepsilon\_{33}}{\varepsilon\_{33}}, & \varepsilon\_{23} - \frac{\varepsilon\_{23}\varepsilon\_{33}}{\varepsilon\_{33}}, & 0 \end{pmatrix}. \tag{42}$$

However, according to (Royer & Dieulesaint, 2000; Sirotin & Shaskolskaya, 1982), the offdiagonal components of *e*ˆ and *ε*ˆ tensors, entering into Eq. (42) for the symmetry class 2 in this coordinate system, are vanishing: 13 23 13 23 *e e ε ε* 0 and, hence,

$$\mathbf{D}\_l(\mathbf{m}\_0 \mid l \ 2) = 0. \tag{43}$$

Evidently, Eq. (43) is valid for all crystals of various classes possessing dyad axes. Thus, the following statement is valid:

*A longitudinal wave propagating along any axis of symmetry in a piezoelectric crystal is accompanied by an electric component with zero induction.* 

For example, let us consider a piezoelectric crystal of the orthorhombic symmetry class 222. According to the above theorem, all three dyad axes in this crystal are the zero-induction directions **m**0 for the longitudinal modes. However, it can be shown that another four inversion-nonequivalent asymmetric directions **m**0 with zero induction ( **D** 0 *<sup>l</sup>* ) may exist in a quasi-longitudinal branch of this crystal:

$$(\theta\_{0'} \pm \phi\_0), \ (\theta\_{0'} \pm \phi\_0 + \pi) \,, \tag{44}$$

Electric Components of Acoustic Waves in the Vicinity of Nonpiezoactive Directions 15

*||* 1 2 ( , , 0), (0, 0, 1). **m u** *m m <sup>t</sup>* (46)

**D u** *|| ||* . 35 1 34 2 (0, 0, ) *<sup>t</sup> t t em emu* (47)

**m** - max 02 01 ( , , 0), *m m* (49)

*||* 2 3 (0, , ), (1, 0, 0), **m u** *m m <sup>t</sup>* (50)

*|| ||* , 22 2 15 3 (- , 0, 0) *<sup>t</sup> t t* **D u** *em em u* (51)

03 22

*em u* (53)

(52)

*m e*

/ / . 0 01 02 0 02 01 35 34 **m** ( , , 0), tan *m m mm ee* (48)

<sup>0</sup> (counted as in Fig. 1):

Therefore, the symmetry plane always contains a single direction **m**0 corresponding to zero

*In any symmetry plane there is always at least one direction for propagation of an SH wave with* 

The other examples below just specify orientations of the zero-induction direction in various

**Example 2: symmetry class 3***m***.** In trigonal crystals, the situation with transverse nonpiezoactive directions for the *t* waves in each of the three symmetry planes containing the triad axis is completely analogous to the above case of a monoclinic crystal. For example,

**m** 02 15

**Example 3: symmetry class** *mm***2.** For a *t* wave propagating in the *yz* symmetry plane of an

*||*

.

*|| ||*

*t t t*

(0, , ), (1, 0, 0), ( , 0, 0)

*t*

**D u**

(0, , ), tan *m e m m θ*

0 02 03 0

2 3 15 3

**m u**

*m m*

One can readily check that in such waves

which is perpendicular to **m**<sup>0</sup> .

*vanishing electric induction.* 

orthorhombic crystal, we have

Evidently, in this case

symmetry classes.

induction **D***<sup>t</sup>* , which is determined by the azimuthal angle

The maximum amplitude of **D***<sup>t</sup>* in this plane corresponds to the direction

This is the most general example. Thus, the following theorem is valid:

in the *yz* symmetry plane, relations (46)–(48) have to be replaced by

where <sup>0</sup> *θ* is the polar angle between **m**0 and the triad axis.

where the angles of spherical coordinate system are determined by approximate relations

$$\rho\_0 = \text{arccot}\sqrt{\frac{e\_{36}\varepsilon\_1\varepsilon\_2}{(e\_{14}\varepsilon\_2 + e\_{25}\varepsilon\_1)\varepsilon\_3}}, \qquad \qquad \wp\_0 = \text{arcctan}\sqrt{\frac{e\_{25}\varepsilon\_1}{e\_{14}\varepsilon\_2}}.\tag{45}$$

For simplicity, solutions (45) are written in the approximation of small piezoelectric moduli and weak elastic anisotropy. In this approximation, a necessary condition for the existence of the above series of zero induction points is that all the piezoelectric moduli entering into relations (45) must have the same sign (Fig. 2). It should be noted that cubic piezoelectric crystals (symmetry classes 43m and 23) are always described by Fig. 2b, since additional zero-induction directions (44), (45) always appear along the triad axes.

**Figure 2.** Diagrams of the directions of propagation of the transversely nonpiezoactive acoustic waves of quasi-longitudinal branch in crystals of the symmetry class 222. The stereographic projections are given for the cases when (a) the sign of the piezoelectric modulus <sup>36</sup> *e* is opposite to that of <sup>14</sup> *e* and/or <sup>25</sup> *e* and (b) all piezoelectric moduli have the same sign.

#### *5.2.2. Transverse (SH) waves propagating in symmetry planes*

**Example 1: symmetry class** *m***.** Let the *z* axis be perpendicular to the plane of symmetry of a monoclinic crystal (*z* ⊥ *m*) and consider the *t* branch of a wave propagating in this plane:

Electric Components of Acoustic Waves in the Vicinity of Nonpiezoactive Directions 15

$$\mathbf{m} = (m\_{1'} \ m\_{2'} \ \ 0), \qquad \qquad \qquad \mathfrak{u}\_t \ \ \mid \ \ \ \ \ \ \ \ \ 0, \ \ \ \ 1). \tag{46} \tag{46}$$

One can readily check that in such waves

14 Modeling and Measurement Methods for Acoustic Waves and for Acoustic Microdevices

*accompanied by an electric component with zero induction.* 

in a quasi-longitudinal branch of this crystal:

following statement is valid:

Evidently, Eq. (43) is valid for all crystals of various classes possessing dyad axes. Thus, the

*A longitudinal wave propagating along any axis of symmetry in a piezoelectric crystal is* 

For example, let us consider a piezoelectric crystal of the orthorhombic symmetry class 222. According to the above theorem, all three dyad axes in this crystal are the zero-induction directions **m**0 for the longitudinal modes. However, it can be shown that another four inversion-nonequivalent asymmetric directions **m**0 with zero induction ( **D** 0 *<sup>l</sup>* ) may exist

where the angles of spherical coordinate system are determined by approximate relations

arccot arctan

*<sup>e</sup> ε ε <sup>e</sup> <sup>ε</sup> <sup>θ</sup>*

For simplicity, solutions (45) are written in the approximation of small piezoelectric moduli and weak elastic anisotropy. In this approximation, a necessary condition for the existence of the above series of zero induction points is that all the piezoelectric moduli entering into relations (45) must have the same sign (Fig. 2). It should be noted that cubic piezoelectric crystals (symmetry classes 43m and 23) are always described by Fig. 2b, since additional

**Figure 2.** Diagrams of the directions of propagation of the transversely nonpiezoactive acoustic waves of quasi-longitudinal branch in crystals of the symmetry class 222. The stereographic projections are given for the cases when (a) the sign of the piezoelectric modulus <sup>36</sup> *e* is opposite to that of <sup>14</sup> *e* and/or

**Example 1: symmetry class** *m***.** Let the *z* axis be perpendicular to the plane of symmetry of a monoclinic crystal (*z* ⊥ *m*) and consider the *t* branch of a wave propagating in this plane:

 

 . 36 1 2 25 1

14 2 25 1 3 14 2

<sup>0000</sup> ( , ), ( , ) *θ θπ* , (44)

*<sup>e</sup> <sup>ε</sup> <sup>e</sup> ε ε <sup>e</sup> <sup>ε</sup>* (45)

0 0

( )

,

zero-induction directions (44), (45) always appear along the triad axes.

<sup>25</sup> *e* and (b) all piezoelectric moduli have the same sign.

*5.2.2. Transverse (SH) waves propagating in symmetry planes* 

$$\mathbf{D}\_t \parallel \begin{array}{c} \begin{array}{c} 0, \ \end{array} \ l \end{array} \begin{array}{c} 0, \ e\_{35}m\_1 + e\_{34}m\_2 \end{array} \mu\_t \parallel \begin{array}{c} \mathbf{u}\_t \end{array} . \tag{47}$$

Therefore, the symmetry plane always contains a single direction **m**0 corresponding to zero induction **D***<sup>t</sup>* , which is determined by the azimuthal angle <sup>0</sup> (counted as in Fig. 1):

$$\mathbf{m}\_0 = \begin{pmatrix} m\_{01'} \ m\_{02'} & 0 \end{pmatrix}, \qquad \tan \rho\_0 = m\_{02} \ / \ m\_{01} = -e\_{35} \ / \ e\_{34}. \tag{48}$$

The maximum amplitude of **D***<sup>t</sup>* in this plane corresponds to the direction

$$\mathbf{m}\_{\text{max}} = (m\_{02} \, \text{ } \text{ } \text{ } m\_{01} \, \text{ } \text{ } \text{ }), \tag{49}$$

which is perpendicular to **m**<sup>0</sup> .

This is the most general example. Thus, the following theorem is valid:

*In any symmetry plane there is always at least one direction for propagation of an SH wave with vanishing electric induction.* 

The other examples below just specify orientations of the zero-induction direction in various symmetry classes.

**Example 2: symmetry class 3***m***.** In trigonal crystals, the situation with transverse nonpiezoactive directions for the *t* waves in each of the three symmetry planes containing the triad axis is completely analogous to the above case of a monoclinic crystal. For example, in the *yz* symmetry plane, relations (46)–(48) have to be replaced by

$$\mathbf{m} = \begin{pmatrix} 0, & m\_2, & m\_3 \end{pmatrix}, \qquad \qquad \mathbf{u}\_t \mid \mid \ \begin{pmatrix} 1, & 0, & 0 \end{pmatrix} \tag{50}$$

$$\mathbf{D}\_t \parallel \begin{pmatrix} -e\_{22}m\_2 + e\_{15}m\_{3'} & 0 & 0 \end{pmatrix} \mu\_t \parallel \mathbf{u}\_t \,, \tag{51}$$

$$\tan\_0 = \begin{pmatrix} 0, & m\_{02}, & m\_{03} \end{pmatrix}, \qquad \tan \theta\_0 = \frac{m\_{02}}{m\_{03}} = \frac{e\_{15}}{e\_{22}} \tag{52}$$

where <sup>0</sup> *θ* is the polar angle between **m**0 and the triad axis.

**Example 3: symmetry class** *mm***2.** For a *t* wave propagating in the *yz* symmetry plane of an orthorhombic crystal, we have

$$\begin{aligned} \mathbf{m} &= \begin{pmatrix} 0, & m\_{2'} & m\_3 \end{pmatrix} & \mathbf{u}\_t &\parallel & \begin{pmatrix} 1, & 0, & 0 \end{pmatrix} \\ \mathbf{D}\_t &\parallel & \begin{pmatrix} e\_{15}m\_{3'} & 0, & 0 \end{pmatrix} \boldsymbol{u}\_t &\parallel & \mathbf{u}\_t. \end{aligned} \tag{53}$$

Evidently, in this case

$$\mathbf{m}\_0 = \begin{pmatrix} 0, & 1, & 0 \end{pmatrix}, \qquad \qquad \mathbf{m}\_{\text{max}} = \begin{pmatrix} 0, & 0, & 1 \end{pmatrix}. \tag{54}$$

Electric Components of Acoustic Waves in the Vicinity of Nonpiezoactive Directions 17

ˆ ˆ **D m m m c.** ( ) ( )( ) *|| α α N F* (60)

<sup>ˆ</sup> ˆ ˆ , **<sup>m</sup> m m c|m m**<sup>0</sup> ( )( ) *Q NF α α* (62)

0

*<sup>m</sup>* (61)

**6. Orientational singularities of the induction fields** 

on the directions of transverse nonpiezoactivity.

**6.1. General case of arbitrary anisotropy** 

The vector fields **D m**( ) *<sup>α</sup>* , which are orthogonal to the wave normal **m**, may exhibit orientational singularity in the vicinity of directions of the two types: zero-induction points, where rotations are topologically allowed (Alshits, Lyubimov & Radowicz, 2005a, 2005b), and along acoustic axes, where inductions of degenerate branches, as a rule, do not vanish, but rotate together with the corresponding displacement vectors **u**0*<sup>α</sup>* (Alshits *et al*, 1987).

Below we shall consider the both types of singularities concentrating our attention basically

As was mentioned above, the vector fields **D m**( ) *<sup>α</sup>* in the zero-induction points **m**0 may exibit rotations (Fig 3). Let us consider the **D m**( ) *<sup>α</sup>* function for **m** = **m**<sup>0</sup> + Δ**m**, where

ˆ ˆ ˆ *||* .

*i Q m NF*

**m m**

Δ**m m** <sup>0</sup> and |Δ**m**| << 1. Using condition (8) and the third relation in (9), we obtain

Taking into account that **D m** <sup>0</sup> ( )0 *<sup>α</sup>* , one has from (60) to a first approximation:

**Dm m m mc**

**Figure 3.** A singular vector distribution **D m**( ) *<sup>α</sup>* in the vicinity of a zero-induction point **m**<sup>0</sup> .

For the transverse **D m**( ) *<sup>α</sup>* field [see (11)], the asymmetric tensor entering into formula (61),

must be planar, that is, its spectral expansion can be represented as a sum of two dyads:

( ) Δ Δ [ ( ) ( )] *α α <sup>i</sup> <sup>α</sup>*

*6.1.1. Orientational singularities in the vicinities of zero-induction points* 

Relations (53) and (54) are also valid for tetragonal crystals of the symmetry class 4*mm*.

**Example 4: symmetry classes 42***m* **, 43***m* **, and 23.** For the *x* axis parallel to the dyad axis ( 2) *x ||* , transverse waves propagating in the diagonal plane (1, 1 , 0) obey the relations

$$\mathbf{m} = (m\_{1'} \ \ -m\_{1'} \ \ m\_3), \tag{55} \\ \tag{56} \\ \tag{57}$$

These waves exhibit electric components with the amplitude of induction

$$\mathbf{D}\_t \parallel e\_{14} m\_3(\mathbf{1}, \mathbf{1}, \mathbf{0}) \mu\_t \parallel \mathbf{u}\_t \tag{56}$$

and, hence, have the following special directions:

$$\mathbf{m}\_0 = \begin{pmatrix} 1, & -1, & 0 \end{pmatrix} / \sqrt{2}, \qquad \mathbf{m}\_{\text{max}} = \begin{pmatrix} 0, & 0, & 1 \end{pmatrix}. \tag{57}$$

The found above orientations of zero-induction direction **m**0 for a series of crystal classes are summarized in Table 2.


**Table 2.** Propagation directions of transversely nonpiezoactive SH waves in the symmetry planes of crystals of various symmetry systems. Certainly, in Table 2 for crystals more symmetric than monoclinic (*m*) and containing other equivalent symmetry planes, the directions **m**0 of zero induction found are accordingly multiplied. For instance, in crystals of the orthorhombic (*mm*2) and tetragonal (4*mm*) classes there is also the symmetry plane *m* || *xz* where the corresponding transversely non-piezoactive direction is **m**0 = (1, 0, 0).

In conclusion, let us consider the more exclusive case of hexagonal symmetry classes (14).

**Example 5: symmetry classes 6***mm* **and** *m* **(14).** Any plane containing the principal symmetry axis in such a crystal is the plane of symmetry *m*. According to relations (27), the electric induction vector of *t* waves propagating in such planes is orthogonal to *m* and proportional to *m*<sup>3</sup> . Therefore, **D***<sup>t</sup>* vanishes along the entire equator <sup>3</sup> *m* 0 :

$$\mathbf{D}\_t(m\_{1'}, m\_{2'}, \ 0) = \ \ 0. \tag{58}$$

Note in passing that at the same equator for the same symmetry classes the other transverse branch (*t*') polarized along the principal axis also forms a line of zero electric displacement:

$$\mathbf{D}\_{\mathbf{i}'} (m\_{1'} \ m\_{2'} \quad \mathbf{0}) = \begin{array}{c} \mathbf{0}. \\ \end{array} \tag{59}$$

## **6. Orientational singularities of the induction fields**

The vector fields **D m**( ) *<sup>α</sup>* , which are orthogonal to the wave normal **m**, may exhibit orientational singularity in the vicinity of directions of the two types: zero-induction points, where rotations are topologically allowed (Alshits, Lyubimov & Radowicz, 2005a, 2005b), and along acoustic axes, where inductions of degenerate branches, as a rule, do not vanish, but rotate together with the corresponding displacement vectors **u**0*<sup>α</sup>* (Alshits *et al*, 1987).

Below we shall consider the both types of singularities concentrating our attention basically on the directions of transverse nonpiezoactivity.

#### **6.1. General case of arbitrary anisotropy**

16 Modeling and Measurement Methods for Acoustic Waves and for Acoustic Microdevices

These waves exhibit electric components with the amplitude of induction

and, hence, have the following special directions:

(

*m m*

/ *e e*/

*m m*, ,0)

02 35 34

 **m**0 01 02

are summarized in Table 2.

Direction **m**<sup>0</sup> <sup>01</sup>

direction is **m**0 = (1, 0, 0).

Relations (53) and (54) are also valid for tetragonal crystals of the symmetry class 4*mm*.

**Example 4: symmetry classes 42***m* **, 43***m* **, and 23.** For the *x* axis parallel to the dyad axis ( 2) *x ||* , transverse waves propagating in the diagonal plane (1, 1 , 0) obey the relations

The found above orientations of zero-induction direction **m**0 for a series of crystal classes

Class of symmetry *m* 3*m mm*2, 4*mm* 42*m* , 43*m* , 23 Symmetry plane *m z m* || *yz m* || *yz m* || ( 110 )

(

*m m*

there is also the symmetry plane *m* || *xz* where the corresponding transversely non-piezoactive

proportional to *m*<sup>3</sup> . Therefore, **D***<sup>t</sup>* vanishes along the entire equator <sup>3</sup> *m* 0 :

In conclusion, let us consider the more exclusive case of hexagonal symmetry classes (14).

**Example 5: symmetry classes 6***mm* **and** *m* **(14).** Any plane containing the principal symmetry axis in such a crystal is the plane of symmetry *m*. According to relations (27), the electric induction vector of *t* waves propagating in such planes is orthogonal to *m* and

Note in passing that at the same equator for the same symmetry classes the other transverse branch (*t*') polarized along the principal axis also forms a line of zero electric displacement:

**Table 2.** Propagation directions of transversely nonpiezoactive SH waves in the symmetry planes of crystals of various symmetry systems. Certainly, in Table 2 for crystals more symmetric than monoclinic (*m*) and containing other equivalent symmetry planes, the directions **m**0 of zero induction found are accordingly multiplied. For instance, in crystals of the orthorhombic (*mm*2) and tetragonal (4*mm*) classes

03

/ *e e*/

0, , ) *m m*

 **m**0 02 03 02 15 22

<sup>0</sup> max **m m** (0, 1, 0), (0, 0, 1). (54)

*||* 1 13 ( , - , ), (1, 1, 0). **m u** *m mm <sup>t</sup>* (55)

**D u** *|| ||* 14 3(1, 1, 0) *<sup>t</sup> t t em u* (56)

**<sup>D</sup>** 1 2 ( , , 0) 0. *<sup>t</sup> m m* (58)

**D** 1 2 ( , , 0) 0. *<sup>t</sup> m m* (59)

**m**<sup>0</sup> = (0, 1, 0) **m**<sup>0</sup> = (1, -1, 0)/ 2


#### *6.1.1. Orientational singularities in the vicinities of zero-induction points*

As was mentioned above, the vector fields **D m**( ) *<sup>α</sup>* in the zero-induction points **m**0 may exibit rotations (Fig 3). Let us consider the **D m**( ) *<sup>α</sup>* function for **m** = **m**<sup>0</sup> + Δ**m**, where Δ**m m** <sup>0</sup> and |Δ**m**| << 1. Using condition (8) and the third relation in (9), we obtain

$$\mathbf{D}\_a(\mathbf{m}) \mid \mid \hat{N}(\mathbf{m}) \overline{\hat{F}}\_a(\mathbf{m}) \mathbf{c}. \tag{60}$$

Taking into account that **D m** <sup>0</sup> ( )0 *<sup>α</sup>* , one has from (60) to a first approximation:

$$\mathbf{D}\_{a}(\mathbf{m}) \mid \mid \Delta \mathbf{m} \hat{Q}\_{a} \equiv \Delta m\_{i} \left\{ \frac{\hat{\boldsymbol{\sigma}}}{\hat{\boldsymbol{\sigma}} m\_{i}} [\hat{\boldsymbol{N}}(\mathbf{m}) \overline{\hat{\boldsymbol{F}}}\_{a}(\mathbf{m}) \mathbf{c}] \right\}\_{\mathbf{m} = \mathbf{m}\_{0}}.\tag{61}$$

**Figure 3.** A singular vector distribution **D m**( ) *<sup>α</sup>* in the vicinity of a zero-induction point **m**<sup>0</sup> .

For the transverse **D m**( ) *<sup>α</sup>* field [see (11)], the asymmetric tensor entering into formula (61),

$$
\hat{Q}\_{\text{at}} = \nabla\_{\mathbf{m}} \otimes \hat{\mathcal{N}}(\mathbf{m}) \overline{\hat{\mathcal{F}}}\_{a}(\mathbf{m}) \mathbf{c}|\_{\mathbf{m} = \mathbf{m}\_0 \prime} \tag{62}
$$

must be planar, that is, its spectral expansion can be represented as a sum of two dyads:

$$
\hat{Q}\_a = \lambda\_{a1}\tilde{\mathbf{e}}\_{a1} \otimes \mathbf{e}\_{a1} + \lambda\_{a2}\tilde{\mathbf{e}}\_{a2} \otimes \mathbf{e}\_{a2},\tag{63}
$$

Electric Components of Acoustic Waves in the Vicinity of Nonpiezoactive Directions 19

. <sup>ˆ</sup> **<sup>D</sup>** sgndet *<sup>α</sup> n Q* (68)

ˆ *||* **D m me e** <sup>Δ</sup> 1 11 (<sup>Δ</sup> ) *α α α αα <sup>Q</sup> <sup>λ</sup>* (69)

**Figure 5.** The two main types of singularities in the vicinity of zero points of the induction vector

The above considerations fail to be valid in particular cases, when one of the eigenvalues ( *<sup>α</sup>*<sup>1</sup> *<sup>λ</sup>* or *<sup>α</sup>*<sup>2</sup> *<sup>λ</sup>* ) of matrices (63) and (65) vanishes. In such cases, det <sup>ˆ</sup> *<sup>Q</sup>α* = 0, but formula (68) is

and a zero-induction line can pass via **m**0 in the direction of Δ**m** <sup>⊥</sup> **e** *<sup>α</sup>*<sup>1</sup> , but only provided that *α*<sup>2</sup> *λ* = 0 is valid. In this situation, the very concept of the Poincaré index is inapplicable. However, if the vanishing of *α*<sup>2</sup> *λ* has a strictly local character and takes place only along **m**<sup>0</sup> , then we are dealing with a very special singularity analogous to a local-wedge degeneracy known in the theory of acoustic axes (Alshits, Sarychev & Shuvalov, 1985). It can be shown that a topological charge of the corresponding singularity in the **D m**( ) *<sup>α</sup>* field in this case can take one of three values: **D***n* = 0, 1. However, both situations (point and line) of this type with zero induction amplitude are very exclusive and never encountered in real (even symmetric) crystals. Below we will consider zero-induction lines of this kind in model crystals. However it will be demonstrated that the examples of the **D***<sup>α</sup>* = 0 lines, existing in

hexagonal crystals and described by Eqs. (58) and (59), belong to a different type.

On the other hand, the ordinary singular points (68) with indices **D***n* = 1 depicted in Fig. 5 are rather widely encountered in real crystals. For example, all directions **m**0 in Fig. 2b corresponding to orthorhombic (222) or cubic ( 43m and 23) crystals are characterized by topological charges **D***n* = 1 (in Fig. 2, filled and empty circles correspond to +1 and –1, respectively). Figure 6 is a schematic diagram of the **D m**( ) *<sup>l</sup>* distribution in the central region

distribution in the normalized directed representation.

not applicable. Indeed, let *<sup>α</sup>*<sup>2</sup> *λ* = 0 at **m**<sup>0</sup> . Then,

of the circle in Fig. 2b.

where *α<sup>j</sup> <sup>λ</sup>* , **<sup>e</sup>** *<sup>α</sup><sup>j</sup>* , and **e***α<sup>j</sup>* are the eigenvalues and eigenvectors (left and right) of the <sup>ˆ</sup> *Qα* tensor ( **e***<sup>α</sup>*1 and **e***<sup>α</sup>*2 must be orthogonal to **m**<sup>0</sup> ). Note that the **e** *<sup>α</sup>j* eigenvectors (in contrast to **e***α<sup>j</sup>* ) in the general case do not belong to a plane orthogonal to **m**<sup>0</sup> , but their components *||* **<sup>e</sup>** *<sup>α</sup><sup>j</sup>* oriented along **m**0 are insignificant for our analysis.

Let us decompose each eigenvector into two components

$$
\tilde{\mathbf{e}}\_{aj} = \tilde{\mathbf{e}}\_{aj}^{\parallel} + \tilde{\mathbf{e}}\_{aj\prime}^{\perp} \qquad \tilde{\mathbf{e}}\_{aj}^{\parallel} \mid \text{l} \, \mathbf{m}\_{0\prime} \qquad \tilde{\mathbf{e}}\_{aj}^{\perp} = (\hat{I} - \mathbf{m}\_0 \otimes \mathbf{m}\_0) \tilde{\mathbf{e}}\_{aj} \perp \mathbf{m}\_0. \tag{64}
$$

and form a more convenient matrix

$$
\hat{Q}\_{a}^{\perp} = (\hat{I} - \mathbf{m}\_{0} \otimes \mathbf{m}\_{0}) \hat{Q}\_{a} = \lambda\_{a1} \mathbf{e}\_{a1}^{\perp} \otimes \mathbf{e}\_{a1} + \lambda\_{a2} \mathbf{e}\_{a2}^{\perp} \otimes \mathbf{e}\_{a2},\tag{65}
$$

which will be used below instead of ˆ *Q<sup>α</sup>* :

$$\mathbf{D}\_a \parallel \Delta \mathbf{m} \hat{Q}\_a^\perp. \tag{66}$$

Let the orientation angle Φ of the **D m**( ) *<sup>α</sup>* vector be measured from the **e***<sup>α</sup>*1 direction, and the analogous angle for Δ**m** in the same plane, from the **e** *<sup>α</sup>*1 direction (Fig. 4). In these terms, we can write

$$
\tan \Phi = \frac{\mathbf{D}\_a \cdot \mathbf{e}\_{a2}}{\mathbf{D}\_a \cdot \mathbf{e}\_{a1}} = \frac{\lambda\_{a2}}{\lambda\_{a1}} \frac{\Delta \mathbf{m} \cdot \tilde{\mathbf{e}}\_{a2}^\perp}{\Delta \mathbf{m} \cdot \tilde{\mathbf{e}}\_{a1}^\perp} = \frac{\lambda\_{a2}}{\lambda\_{a1}} \tan \varphi. \tag{67}
$$

**Figure 4.** The angles of orientation of the **D***α* and Δ**m** vectors in the plane orthogonal to **m**<sup>0</sup> .

Thus, the complete turn of Δ**m** around **m**0 in the plane orthogonal to **m**0 implies the complete turn of **D m**( ) *<sup>α</sup>* in the same or in0 the opposite direction (depending on the sign of det <sup>ˆ</sup> *<sup>Q</sup><sup>α</sup>* <sup>=</sup> *α α* 1 2 *<sup>λ</sup> <sup>λ</sup>* ), which corresponds to the Poincaré index of the given singular point (Fig. 5)

oriented along **m**0 are insignificant for our analysis.

and form a more convenient matrix

which will be used below instead of ˆ

the analogous angle

terms, we can write

Let us decompose each eigenvector into two components

where *α<sup>j</sup> <sup>λ</sup>* , **<sup>e</sup>** *<sup>α</sup><sup>j</sup>* , and **e***α<sup>j</sup>* are the eigenvalues and eigenvectors (left and right) of the <sup>ˆ</sup>

*Q<sup>α</sup>* :

( **e***<sup>α</sup>*1 and **e***<sup>α</sup>*2 must be orthogonal to **m**<sup>0</sup> ). Note that the **e** *<sup>α</sup>j* eigenvectors (in contrast to **e***α<sup>j</sup>* )

in the general case do not belong to a plane orthogonal to **m**<sup>0</sup> , but their components *||*

Let the orientation angle Φ of the **D m**( ) *<sup>α</sup>* vector be measured from the **e***<sup>α</sup>*1 direction, and

**D e me D e m e**

**Figure 4.** The angles of orientation of the **D***α* and Δ**m** vectors in the plane orthogonal to **m**<sup>0</sup> .

Thus, the complete turn of Δ**m** around **m**0 in the plane orthogonal to **m**0 implies the complete turn of **D m**( ) *<sup>α</sup>* in the same or in0 the opposite direction (depending on the sign of det <sup>ˆ</sup> *<sup>Q</sup><sup>α</sup>* <sup>=</sup> *α α* 1 2 *<sup>λ</sup> <sup>λ</sup>* ), which corresponds to the Poincaré index of the given singular point (Fig. 5)

, <sup>ˆ</sup> **ee ee** *<sup>Q</sup>α αα α αα α* 11 1 22 2 *λ λ* (63)

, ˆ || ( ) *|| || <sup>I</sup>* **e e e e m e m me m** <sup>0</sup> 00 0 , , *<sup>α</sup><sup>j</sup> <sup>α</sup><sup>j</sup> <sup>α</sup><sup>j</sup> <sup>α</sup><sup>j</sup> <sup>α</sup><sup>j</sup> <sup>α</sup>j* (64)

, ˆ ˆ ˆ( ) *<sup>I</sup>* **mm e e e e** *Q Q <sup>α</sup>* 0 0 11 1 22 2 *α αα α αα α λ λ* (65)

for Δ**m** in the same plane, from the **e** *<sup>α</sup>*1 direction (Fig. 4). In these

 

 

22 22 1 1 1 1

*λ λ*

Δ tanΦ tan Δ *αα α α α αα α α α*

. ˆ *||* **D m** *α α* <sup>Δ</sup> *<sup>Q</sup>* (66)

.

*λ λ* (67)

*Qα* tensor

**e** *αj*

**Figure 5.** The two main types of singularities in the vicinity of zero points of the induction vector distribution in the normalized directed representation.

$$m\_{\mathbf{D}} = \text{sgn} \det \hat{Q}\_a^{\perp}. \tag{68}$$

The above considerations fail to be valid in particular cases, when one of the eigenvalues ( *<sup>α</sup>*<sup>1</sup> *<sup>λ</sup>* or *<sup>α</sup>*<sup>2</sup> *<sup>λ</sup>* ) of matrices (63) and (65) vanishes. In such cases, det <sup>ˆ</sup> *<sup>Q</sup>α* = 0, but formula (68) is not applicable. Indeed, let *<sup>α</sup>*<sup>2</sup> *λ* = 0 at **m**<sup>0</sup> . Then,

$$\mathbf{D}\_a \parallel \Delta \mathbf{m} \hat{Q}\_a^\perp = \lambda\_{a1} (\Delta \mathbf{m} \cdot \tilde{\mathbf{e}}\_{a1}^\perp) \mathbf{e}\_{a1} \tag{69}$$

and a zero-induction line can pass via **m**0 in the direction of Δ**m** <sup>⊥</sup> **e** *<sup>α</sup>*<sup>1</sup> , but only provided that *α*<sup>2</sup> *λ* = 0 is valid. In this situation, the very concept of the Poincaré index is inapplicable. However, if the vanishing of *α*<sup>2</sup> *λ* has a strictly local character and takes place only along **m**<sup>0</sup> , then we are dealing with a very special singularity analogous to a local-wedge degeneracy known in the theory of acoustic axes (Alshits, Sarychev & Shuvalov, 1985). It can be shown that a topological charge of the corresponding singularity in the **D m**( ) *<sup>α</sup>* field in this case can take one of three values: **D***n* = 0, 1. However, both situations (point and line) of this type with zero induction amplitude are very exclusive and never encountered in real (even symmetric) crystals. Below we will consider zero-induction lines of this kind in model crystals. However it will be demonstrated that the examples of the **D***<sup>α</sup>* = 0 lines, existing in hexagonal crystals and described by Eqs. (58) and (59), belong to a different type.

On the other hand, the ordinary singular points (68) with indices **D***n* = 1 depicted in Fig. 5 are rather widely encountered in real crystals. For example, all directions **m**0 in Fig. 2b corresponding to orthorhombic (222) or cubic ( 43m and 23) crystals are characterized by topological charges **D***n* = 1 (in Fig. 2, filled and empty circles correspond to +1 and –1, respectively). Figure 6 is a schematic diagram of the **D m**( ) *<sup>l</sup>* distribution in the central region of the circle in Fig. 2b.

**Figure 6.** A schematic image of the **D m**( ) *<sup>l</sup>* vector field distribution over a group of five singular points in the central region of Fig. 2b in the representation of nondirected segments.

## *6.1.2. Orientational singularities in the* **D m**( ) *<sup>α</sup> fields around acoustic axes*

Let us consider the vector polarization fields ( ) **u m** 01,2 of degenerate branches in the vicinity of the direction **m***d* of the acoustic axis. In this region the considered vector distributions should be very close to the plane orthogonal to the unit polarization vector ( ) **u m** <sup>03</sup> *<sup>d</sup>* of the non-degenerate isonormal eigenwave being singular at **m***<sup>d</sup>* . Their rotations around the acoustic axis are equal to each other being described by the Poincaré index **<sup>u</sup>** *n* which is determined by the type of the acoustic axis (Alshits *et al*, 1987). The appropriate fields of electric induction **D m** 1,2( ) due to the coupling (9) <sup>ˆ</sup> **D u** *|| α α <sup>N</sup>* have similar rotations characterized by the Poincaré index **D***n* which ordinarily may differ from **<sup>u</sup>** *n* only by a sign. In order to find this sign we should take into consideration that the *N*ˆ matrix is degenerate ( ˆ **m***N* 0 ). However, one can replace *N*ˆ by a matrix ˆ*N* such that we have ˆ ˆ *N N* **u u** for any **u u** <sup>03</sup> , but simultaneously det <sup>ˆ</sup>*N* <sup>0</sup> . These conditions are satisfied by, for example, the matrix

$$
\hat{N}' = \hat{N}(\mathbf{m}\_d) + \mathbf{m}\_d \otimes \mathbf{u}\_{03}.\tag{70}
$$

Electric Components of Acoustic Waves in the Vicinity of Nonpiezoactive Directions 21

**<sup>D</sup>** 1 1 2 2 12 *n a b a b cc* sgn{( )( ) - }. (72)

*ε* (73)

*<sup>ε</sup>* (74)

). **<sup>D</sup>** 1 2 *n bb* sgn( (75)

( ) sgn( *<sup>z</sup> n ad ad* (76)


( ) *<sup>y</sup> n* are obtained from (76) by cyclic rearrangement of

Consequently, these two vector fields are homotopic with each other. That is why they

Now let us consider some examples of crystals belonging to particular symmetry systems.

**Example 1.** For a longitudinal wave propagating along the **m**<sup>0</sup> || 2 direction in a monoclinic

 14 36 15 1 33 1 14 2 15 36

 25 36 24 2 33 2 25 1 24 36

 *i i*3 33 *dc c* . The *ε*ˆ tensor is assumed to be diagonal, which can be ensured by the appropriate choice of the *x*- and *y* axes of the crystallographic coordinate system with the *z* 

**Example 2.** For the **m**<sup>0</sup> || 2 direction in an orthorhombic crystal belonging to the symmetry

**Example 3.** For the **m**<sup>0</sup> || 2 || *z* direction in an orthorhombic crystal belonging to the

/ ). **<sup>D</sup>** 12 21

/ / 2, 1 35 2 34

and only a very large elastic anisotropy can change the signs of the ratios in formula (76). Therefore these signs for most orthorhombic crystals are determined only by the

34 35 3 34 35 , - , , *ec ed e <sup>ε</sup> e d e c ab c*

35 34 3 35 34 , - , , *e c e d <sup>e</sup> <sup>ε</sup> ed ec ab c*

correspond to the same value of the index **D***n* .

**6.2. Waves propagating along symmetry axes** 

( ) *<sup>x</sup> <sup>n</sup>* and **<sup>D</sup>**

the indices. Note that, in the isotropic limit, we obtain

( ) sgn( *<sup>z</sup> n ee* .

11 1

22 2

*6.2.1. Longitudinal waves along symmetry axes* 

crystal with dyad axis, we have

axis parallel to the dyad axis.

symmetry class 222, we have

Analogous formulas for the **<sup>D</sup>**

piezoelectric moduli: ) **<sup>D</sup>** 14 25

class *mm*2, we have

where

Indeed, in accordance with (Fedorov, 1968) we have det ˆ ˆ **u m** <sup>0</sup> *N N* <sup>03</sup> *<sup>d</sup>* . Then following to (Alshits *et al*, 1987) one obtains

$$
\hbar n\_{\mathbf{D}} = n\_{\mathbf{u}} \operatorname{sgn} \det \hat{N}'.\tag{71}
$$

This equation is valid until <sup>ˆ</sup> ( )0 **<sup>m</sup>** *<sup>N</sup> <sup>d</sup>* which holds for any known acoustic axes except of the direction **m** *||* 6 *<sup>d</sup>* (see below).

It should be noted that, in contrast to the mutually orthogonal vectors ( ) **u m** <sup>0</sup>*α* ( *α* = 1, 2, 3), the three vectors **Dm m** *<sup>α</sup>* ( ) are coplanar and generally unorthogonal in pairs. At the same time, it is clear that the vectors **D m**1( ) and **D m**<sup>2</sup> ( ) are not collinear with any **m**. Consequently, these two vector fields are homotopic with each other. That is why they correspond to the same value of the index **D***n* .

Now let us consider some examples of crystals belonging to particular symmetry systems.

#### **6.2. Waves propagating along symmetry axes**

#### *6.2.1. Longitudinal waves along symmetry axes*

**Example 1.** For a longitudinal wave propagating along the **m**<sup>0</sup> || 2 direction in a monoclinic crystal with dyad axis, we have

$$m\_{\mathbf{D}} = \text{sgn}\{ (a\_1 + b\_1)(a\_2 + b\_2) \cdot c\_1 c\_2 \}. \tag{72}$$

where

20 Modeling and Measurement Methods for Acoustic Waves and for Acoustic Microdevices

in the central region of Fig. 2b in the representation of nondirected segments.

*6.1.2. Orientational singularities in the* **D m**( ) *<sup>α</sup> fields around acoustic axes* 

Let us consider the vector polarization fields ( ) **u m** 01,2 of degenerate branches in the vicinity of the direction **m***d* of the acoustic axis. In this region the considered vector distributions should be very close to the plane orthogonal to the unit polarization vector ( ) **u m** <sup>03</sup> *<sup>d</sup>* of the non-degenerate isonormal eigenwave being singular at **m***<sup>d</sup>* . Their rotations around the acoustic axis are equal to each other being described by the Poincaré index **<sup>u</sup>** *n* which is determined by the type of the acoustic axis (Alshits *et al*, 1987). The appropriate fields of electric induction **D m** 1,2( ) due to the coupling (9) <sup>ˆ</sup> **D u** *|| α α <sup>N</sup>* have similar rotations characterized by the Poincaré index **D***n* which ordinarily may differ from **<sup>u</sup>** *n* only by a sign. In order to find this sign we should take into consideration that the *N*ˆ matrix is degenerate ( ˆ **m***N* 0 ). However, one can replace *N*ˆ by a matrix ˆ*N* such that we have ˆ ˆ *N N* **u u** for any **u u** <sup>03</sup> , but simultaneously det <sup>ˆ</sup>*N* <sup>0</sup> . These conditions are satisfied by, for example,

Indeed, in accordance with (Fedorov, 1968) we have det ˆ ˆ **u m** <sup>0</sup> *N N* <sup>03</sup> *<sup>d</sup>* . Then following

This equation is valid until <sup>ˆ</sup> ( )0 **<sup>m</sup>** *<sup>N</sup> <sup>d</sup>* which holds for any known acoustic axes except of

It should be noted that, in contrast to the mutually orthogonal vectors ( ) **u m** <sup>0</sup>*α* ( *α* = 1, 2, 3), the three vectors **Dm m** *<sup>α</sup>* ( ) are coplanar and generally unorthogonal in pairs. At the same time, it is clear that the vectors **D m**1( ) and **D m**<sup>2</sup> ( ) are not collinear with any **m**.

*<sup>l</sup>* vector field distribution over a group of five singular points

. ˆ ˆ ( ) **m mu** *N N d d* <sup>03</sup> (70)

<sup>ˆ</sup> sgndet **D u** *n n N.* (71)

**Figure 6.** A schematic image of the **D m**( )

the matrix

to (Alshits *et al*, 1987) one obtains

the direction **m** *||* 6 *<sup>d</sup>* (see below).

$$a\_1 = \frac{e\_{14}c\_{36}}{\Lambda\_{34}^-}, \quad b\_1 = \frac{e\_{15}d\_1}{\Lambda\_{35}^-} \cdot \frac{e\_{33}c\_1}{e\_3}, \ c\_1 = \frac{e\_{14}d\_2}{\Lambda\_{34}^-} + \frac{e\_{15}c\_{36}}{\Lambda\_{35}^-},\tag{73}$$

$$a\_2 = \frac{e\_{25}c\_{36}}{\Lambda\_{35}^-}, \quad b\_2 = \frac{e\_{24}d\_2}{\Lambda\_{34}^-} - \frac{e\_{33}e\_2}{e\_3}, \; c\_2 = \frac{e\_{25}d\_1}{\Lambda\_{35}^-} + \frac{e\_{24}c\_{36}}{\Lambda\_{34}^-},\tag{74}$$

 *i i*3 33 *dc c* . The *ε*ˆ tensor is assumed to be diagonal, which can be ensured by the appropriate choice of the *x*- and *y* axes of the crystallographic coordinate system with the *z*  axis parallel to the dyad axis.

**Example 2.** For the **m**<sup>0</sup> || 2 direction in an orthorhombic crystal belonging to the symmetry class *mm*2, we have

$$m\_{\mathbf{D}} = \text{sgn}(b\_1 b\_2). \tag{75}$$

**Example 3.** For the **m**<sup>0</sup> || 2 || *z* direction in an orthorhombic crystal belonging to the symmetry class 222, we have

$$m\_{\mathbf{D}}^{(z)} = \text{sgn}(a\_1 d\_2 \; / \; a\_2 d\_1). \tag{76}$$

Analogous formulas for the **<sup>D</sup>** ( ) *<sup>x</sup> <sup>n</sup>* and **<sup>D</sup>** ( ) *<sup>y</sup> n* are obtained from (76) by cyclic rearrangement of the indices. Note that, in the isotropic limit, we obtain

$$d\_1 \mid \Lambda\_{35}^{\cdot} = d\_2 \mid \Lambda\_{34}^{\cdot} \to \mathcal{2} \tag{77}$$

and only a very large elastic anisotropy can change the signs of the ratios in formula (76). Therefore these signs for most orthorhombic crystals are determined only by the piezoelectric moduli: ) **<sup>D</sup>** 14 25 ( ) sgn( *<sup>z</sup> n ee* .

**Example 4.** As can be readily checked, for the principal symmetry axes in crystals of the symmetry classes 422, 622, ∞22, 4*mm*, 6*mm*, ∞*mm*, 4, 6, ∞, 32, 3*m*, and 3 we have

$$m\_{\mathbf{D}} = \mathbf{1}\_{\prime} \tag{78}$$

Electric Components of Acoustic Waves in the Vicinity of Nonpiezoactive Directions 23

 

of the vector **m**. The induction vector **D***l* of the non-

. The direction of **D***l* in the accepted main order coincides with that of

(82)

In these terms, the vector fields **D m**( ) *<sup>α</sup>* , in the main order, have the following form

*μ e e μ e e*

 

*μ de e*

**D ρ ρ m D ρ ρ m**

depend on the orientation

does not depend on

direction of **m***<sup>d</sup>* )**.**

**D**

*l t t* 2

**D ρ ρ m**

22 11 0 11 22 0

{ ( 2 ) [ ( 2 ) ]}, { ( 2 ) [ ( 2 ) ]}.

Here the notation is introduced: 13 44 34 33 44 *dc c c c* , Δ . We note that in the symmetry classes 6 2 *m* (*m X*1) and 62*m* (2 || *X*1) one can put in (82), respectively, *e*11 = 0 and *e*22 = 0.

It is easily seen that the induction vectors **D***t* and **D***t* of the degenerate branches are mutually orthogonal, while their absolute value is proportional to |**m**| and does not

degenerate quasi-longitudinal branch has much smaller length, **|D | m** <sup>2</sup> <sup>|</sup><sup>Δ</sup> <sup>|</sup> *<sup>l</sup>* , which also

**D***t* . Certainly, in the next approximation this coincidence disappears. It is clear from (82) that during the full rotation of the vector **m** around the axis **m** *||* 6 *<sup>d</sup>* each of three vectors

**m** *||* 6 *<sup>d</sup>* in all three vector fields is characterized by the Poincaré index **<sup>D</sup>***n* 2 (Table 3). The corresponding singular configuration for one of these vector fields is shown in Fig. 7.

**Figure 7.** Vector induction field **D***<sup>α</sup>* , *α t,t l* or **,** near the acoustic axis **m** *||* 6 *<sup>d</sup>* related to the

**6.3. Transverse (SH) waves propagating in symmetry planes** 

Poincaré index **<sup>D</sup>***n* 2 (top view of the plane orthogonal to **m***<sup>d</sup>* **;** the central point corresponds to the

The directions of transverse nonpiezoactivity in symmetry planes (Table 2) are also characterized by rotations **D***n* in the induction vector fields of appropriate SH wave branches. We will not write lengthy expressions determining the choice between **D***n* = 1

twice circumvents the same axis in opposite direction. This means that the point

 

( /Δ ){ ( 2 ) [ ( 2 ) ]},

34 11 22 0

 

and in crystals of the symmetry classes 42*m* , 4 , 43*m* , and 23,

$$m\_{\mathbf{D}} = -1.\tag{79}$$

#### *6.2.2. Degenerate transverse waves along symmetry axes*

In accordance with Eq. (71), the transverse waves propagating along directions near acoustic axes, which coincide with symmetry axes in piezoelectric crystals, are characterized by rotations of both polarization fields ( ) **u m** 01,2 and accompanied induction fields **D m** 1,2 ( ) . The direct analysis for various types of symmetry axes gives the related Poincaré indices **<sup>u</sup>** *n* and **D***n* shown in Table 3.


**Table 3.** The Poincaré indices of vector fields of polarizations and inductions along acoustic axes coinciding with symmetry axes *N* for transverse and longitudinal wave branches. The sign of **<sup>u</sup>** *n* <sup>=</sup><sup>1</sup> when it is not universal may be found from the equations given in (Alshits, Sarychev & Shuvalov, 1985; Alshits & Shuvalov, 1987; Shuvalov, 1998). The indices **D***n* for the direction of a 6 -fold symmetry axis are found in the next sub-section.

#### *6.2.3. The both types of induction singularities along a* **6** *-fold symmetry axis*

An interesting configuration of the vector fields **D m**( ) *<sup>α</sup>* arises near an acoustic axis **m** *||* 6 *<sup>d</sup>* . In this case we have an exclusive situation <sup>ˆ</sup> ( )0 **<sup>m</sup>** *<sup>N</sup> <sup>d</sup>* . So, by Eq. <sup>3</sup> (9) , along the direction **m***d* the induction components vanish, **D m**( )0 *<sup>α</sup> <sup>d</sup>* , in all branches, both degenerate ( *α* 1,2 *t,t* ) and nondegenerate ( *α* 3 *l* ). Accordingly, in the vicinity of the 6 -fold symmetry axis these fields should be small. Let us consider the direction

$$\mathbf{m} = \mathbf{m}\_d + \Delta \mathbf{m} \tag{80}$$

where

$$\begin{aligned} \mathbf{m}\_d &= \langle 0, \ 0, \ 1 \rangle \parallel \overline{6}, & \Delta \mathbf{m} &= \mu \mathfrak{p}(\boldsymbol{\varphi}), \\ \mathfrak{p}(\boldsymbol{\varphi}) &= \langle \cos \boldsymbol{\varphi}, \ \sin \boldsymbol{\varphi}, \ 0 \rangle, & 0 \le \mu << 1. \end{aligned} \tag{81}$$

In these terms, the vector fields **D m**( ) *<sup>α</sup>* , in the main order, have the following form

22 Modeling and Measurement Methods for Acoustic Waves and for Acoustic Microdevices

and in crystals of the symmetry classes 42*m* , 4 , 43*m* , and 23,

*6.2.2. Degenerate transverse waves along symmetry axes* 

and **D***n* shown in Table 3.

are found in the next sub-section.

where

symmetry classes 422, 622, ∞22, 4*mm*, 6*mm*, ∞*mm*, 4, 6, ∞, 32, 3*m*, and 3 we have

**Example 4.** As can be readily checked, for the principal symmetry axes in crystals of the

In accordance with Eq. (71), the transverse waves propagating along directions near acoustic axes, which coincide with symmetry axes in piezoelectric crystals, are characterized by rotations of both polarization fields ( ) **u m** 01,2 and accompanied induction fields **D m** 1,2 ( ) . The direct analysis for various types of symmetry axes gives the related Poincaré indices **<sup>u</sup>** *n*

N , 6 6 4 4 and 2 23 3 branch trans long trans long trans long trans long trans long **<sup>u</sup>** *n* 1 0 1 0 1 0 1 0 -1/2 0 **<sup>D</sup>***n* 1 1 -2 -2 **<sup>u</sup>** *n* 1 *-* **<sup>u</sup>** *n* -1 -1/2 1

**Table 3.** The Poincaré indices of vector fields of polarizations and inductions along acoustic axes coinciding with symmetry axes *N* for transverse and longitudinal wave branches. The sign of **<sup>u</sup>** *n* <sup>=</sup><sup>1</sup> when it is not universal may be found from the equations given in (Alshits, Sarychev & Shuvalov, 1985; Alshits & Shuvalov, 1987; Shuvalov, 1998). The indices **D***n* for the direction of a 6 -fold symmetry axis

*6.2.3. The both types of induction singularities along a* **6** *-fold symmetry axis* 

symmetry axis these fields should be small. Let us consider the direction

**ρ**

 *||*

 **m m ρ**

(0, 0, 1) 6, Δ ( ), ( ) (cos , sin , 0), 0 1. *<sup>d</sup> μ*

 

An interesting configuration of the vector fields **D m**( ) *<sup>α</sup>* arises near an acoustic axis **m** *||* 6 *<sup>d</sup>* . In this case we have an exclusive situation <sup>ˆ</sup> ( )0 **<sup>m</sup>** *<sup>N</sup> <sup>d</sup>* . So, by Eq. <sup>3</sup> (9) , along the direction **m***d* the induction components vanish, **D m**( )0 *<sup>α</sup> <sup>d</sup>* , in all branches, both degenerate ( *α* 1,2 *t,t* ) and nondegenerate ( *α* 3 *l* ). Accordingly, in the vicinity of the 6 -fold

**<sup>D</sup>***n* 1, (78)

**<sup>D</sup>***n* 1. (79)

**mm m** *<sup>d</sup>* Δ (80)

(81)

*μ*

$$\begin{aligned} \mathbf{D}\_{l} &= \mu^{2} (d \;/ \, \Delta\_{34}) \{ e\_{11} \mathbf{p} (-2\boldsymbol{\varphi}) + e\_{22} [\mathbf{p} (-2\boldsymbol{\varphi}) \times \mathbf{m}\_{0}] \}, \\ \mathbf{D}\_{l} &= \mu \{ e\_{22} \mathbf{p} (-2\boldsymbol{\varphi}) - e\_{11} [\mathbf{p} (-2\boldsymbol{\varphi}) \times \mathbf{m}\_{0}] \}, \\ \mathbf{D}\_{l'} &= \mu \{ e\_{11} \mathbf{p} (-2\boldsymbol{\varphi}) + e\_{22} [\mathbf{p} (-2\boldsymbol{\varphi}) \times \mathbf{m}\_{0}] \}. \end{aligned} \tag{82}$$

Here the notation is introduced: 13 44 34 33 44 *dc c c c* , Δ . We note that in the symmetry classes 6 2 *m* (*m X*1) and 62*m* (2 || *X*1) one can put in (82), respectively, *e*11 = 0 and *e*22 = 0.

It is easily seen that the induction vectors **D***t* and **D***t* of the degenerate branches are mutually orthogonal, while their absolute value is proportional to |**m**| and does not depend on the orientation of the vector **m**. The induction vector **D***l* of the nondegenerate quasi-longitudinal branch has much smaller length, **|D | m** <sup>2</sup> <sup>|</sup><sup>Δ</sup> <sup>|</sup> *<sup>l</sup>* , which also does not depend on . The direction of **D***l* in the accepted main order coincides with that of **D***t* . Certainly, in the next approximation this coincidence disappears. It is clear from (82) that during the full rotation of the vector **m** around the axis **m** *||* 6 *<sup>d</sup>* each of three vectors **D** twice circumvents the same axis in opposite direction. This means that the point **m** *||* 6 *<sup>d</sup>* in all three vector fields is characterized by the Poincaré index **<sup>D</sup>***n* 2 (Table 3). The corresponding singular configuration for one of these vector fields is shown in Fig. 7.

**Figure 7.** Vector induction field **D***<sup>α</sup>* , *α t,t l* or **,** near the acoustic axis **m** *||* 6 *<sup>d</sup>* related to the Poincaré index **<sup>D</sup>***n* 2 (top view of the plane orthogonal to **m***<sup>d</sup>* **;** the central point corresponds to the direction of **m***<sup>d</sup>* )**.**

#### **6.3. Transverse (SH) waves propagating in symmetry planes**

The directions of transverse nonpiezoactivity in symmetry planes (Table 2) are also characterized by rotations **D***n* in the induction vector fields of appropriate SH wave branches. We will not write lengthy expressions determining the choice between **D***n* = 1 indices for the waves along **m**0 directions in monoclinic and trigonal crystals (see relations (49) and (52), respectively) and instead start our analysis from orthorhombic crystals.

**Example 1.** For the transverse acoustic waves (53) and (54) propagating in the vicinity of the zero-induction direction **m**<sup>0</sup> = (0, 1, 0) in an orthorhombic crystal belonging to the symmetry class *mm*2, the singular induction field is characterized by the Poincaré index

$$m\_{\mathbf{D}} = \text{sgn}\left[\frac{e\_{32}(c\_{12} + c\_{66}) / \Delta\_{26}^{-} - e\_{31}}{e\_{15}}\right] \tag{83}$$

Electric Components of Acoustic Waves in the Vicinity of Nonpiezoactive Directions 25

*Qα* matrix vanishing is offered by a crystal with hexad axis 6 .

31 15 24 32 33 | | *e eeee* | |,| |,| |,| |. (88)

*Qα* tensor (62) is

*Q<sup>α</sup>* ≠ 0. However, in some very exclusive

**6.4. Special types of singularities** 

known alternative example of ˆ

**<sup>m</sup>**<sup>0</sup> , where ˆ **Dm m** <sup>0</sup> 0 0 () () *α α <sup>Q</sup>* .

axis parallel to the dyad axis, in which

*6.4.1. A model crystal of the symmetry class mm2* 

**<sup>D</sup>***<sup>n</sup>* <sup>2</sup> (Figs. 7 and 8c).

nonzero. As was shown above, usual systems have ˆ

The general analysis in Sec. 6.1 is exhaustive only provided that the ˆ

cases, this tensor may vanish in some special directions because of high symmetry or as a result of vanishing of certain combinations of the material tensor components. In such cases, the general expressions are very lengthy and we only present here some final results. For <sup>ˆ</sup> **<sup>m</sup>** <sup>0</sup> ( )0 *<sup>Q</sup><sup>α</sup>* , the distribution of the induction vector field in the vicinity of **m**0 has four additional variants depicted in Fig. 8. The first three of them correspond to isolated singular points with the Poincaré indices **D***n* = 0, 2 (Figs. 8a–8c), while the fourth variant corresponds to the existence of a **D** = 0 line passing via the **m**0 point (Fig. 8d). This very situation is observed on the equator <sup>3</sup> *m* 0 for the transverse tangentially polarized *t* waves (58) in all transverse-isotropic media (13)–(15) and for the transverse *t*' waves (59) polarized along the principal symmetry axis in the media of symmetry classes 6*mm* and ∞*m*. The only

In this case, all three wave branches have along the direction 6 the identical singularities with

**Figure 8.** Four possible types of the **D m**( ) *<sup>α</sup>* vector field distribution around a zero-induction point

Let us assume that one piezoelectric modulus in the crystal under consideration is much smaller than the other moduli. In particular, we consider a conventional crystallographic coordinate system with the *x* and *y* axes perpendicular to the symmetry planes and the *z* 

One can readily check that, in a zero-order approximation with <sup>31</sup> *e* = 0, a quasi-longitudinal nondegenerate wave branch along the **m**<sup>0</sup> = (1, 0, 0) direction features the above special

**Example 2.** For the same direction in a tetragonal crystal belonging to the symmetry class 4*mm*, we have

$$n\_{\mathbf{D}} = \text{sgn}\left[\frac{e\_{31}(c\_{12} - c\_{11} + 2c\_{66})}{e\_{15}\Delta\_{16}^{-}}\right] \tag{84}$$

**Example 3.** For the direction **m** <sup>0</sup> (1, - 1, 0) / 2 in a crystal belonging to the other tetragonal symmetry class 42*m* , we obtain

$$m\_{\mathbf{D}} = -\text{sgn}(e\_{14} \mid e\_{36}).\tag{85}$$

**Example 4.** For the same direction in a cubic crystals of the symmetry class 43*m* or 23 in the diagonal symmetry plane, as well as for the symmetry-equivalent direction **m** <sup>0</sup> (-1, 1, 0) / 2 , we have (for any combinations of the moduli)

$$m\_{\mathbf{D}} = -1\tag{86}$$

The results presented in this subsection are summarized in Table 4.


**Table 4.** Topological charges of **D** fields for SH acoustic waves propagating in symmetry planes of various crystals. In this table the notation is introduced:

$$\kappa\_p = \frac{c\_{12} + c\_{66}}{c\_{pp} - c\_{66}} \tag{87}$$

#### **6.4. Special types of singularities**

24 Modeling and Measurement Methods for Acoustic Waves and for Acoustic Microdevices

indices for the waves along **m**0 directions in monoclinic and trigonal crystals (see relations

**Example 1.** For the transverse acoustic waves (53) and (54) propagating in the vicinity of the zero-induction direction **m**<sup>0</sup> = (0, 1, 0) in an orthorhombic crystal belonging to the symmetry

> **<sup>D</sup>** 32 12 66 26 31 15

**Example 2.** For the same direction in a tetragonal crystal belonging to the symmetry class

 **<sup>D</sup>** 31 12 11 66 15 16 ( ) sgn <sup>Δ</sup> *ec c c*

**Example 3.** For the direction **m** <sup>0</sup> (1, - 1, 0) / 2 in a crystal belonging to the other

**Example 4.** For the same direction in a cubic crystals of the symmetry class 43*m* or 23 in the diagonal symmetry plane, as well as for the symmetry-equivalent direction

Classes of symmetry *mm*2 4*mm* 42*m* 43*m* **,** 23

Symmetry plane *m || yz* (110)

32 31 2 15 15

*p*

*κ*

**Table 4.** Topological charges of **D** fields for SH acoustic waves propagating in symmetry planes of

 12 66 66

*pp c c*

*c c*

 

*e e e e* 

Direction of propagation **m** <sup>0</sup> (0, 1, 0) **m** <sup>0</sup> (1, 1, 0) / 2

*e*

<sup>2</sup>

( )/Δ

*ec c e*

(83)

(84)

*e*

/ ). **<sup>D</sup>** 14 36 *n ee* sgn( (85)

**<sup>D</sup>***n* 1 (86)

 14 36

*e*

*<sup>e</sup>* <sup>1</sup>

(87)

sgn

 31 1 15 sgn ( 1) *<sup>e</sup> κ e*

(49) and (52), respectively) and instead start our analysis from orthorhombic crystals.

class *mm*2, the singular induction field is characterized by the Poincaré index

sgn

tetragonal symmetry class 42*m* , we obtain

Poincare index *n*

4*mm*, we have

*n*

*n*

**m** <sup>0</sup> (-1, 1, 0) / 2 , we have (for any combinations of the moduli)

The results presented in this subsection are summarized in Table 4.

sgn

various crystals. In this table the notation is introduced:

The general analysis in Sec. 6.1 is exhaustive only provided that the ˆ *Qα* tensor (62) is nonzero. As was shown above, usual systems have ˆ*Q<sup>α</sup>* <sup>≠</sup> 0. However, in some very exclusive cases, this tensor may vanish in some special directions because of high symmetry or as a result of vanishing of certain combinations of the material tensor components. In such cases, the general expressions are very lengthy and we only present here some final results. For <sup>ˆ</sup> **<sup>m</sup>** <sup>0</sup> ( )0 *<sup>Q</sup><sup>α</sup>* , the distribution of the induction vector field in the vicinity of **m**0 has four additional variants depicted in Fig. 8. The first three of them correspond to isolated singular points with the Poincaré indices **D***n* = 0, 2 (Figs. 8a–8c), while the fourth variant corresponds to the existence of a **D** = 0 line passing via the **m**0 point (Fig. 8d). This very situation is observed on the equator <sup>3</sup> *m* 0 for the transverse tangentially polarized *t* waves (58) in all transverse-isotropic media (13)–(15) and for the transverse *t*' waves (59) polarized along the principal symmetry axis in the media of symmetry classes 6*mm* and ∞*m*. The only known alternative example of ˆ *Qα* matrix vanishing is offered by a crystal with hexad axis 6 . In this case, all three wave branches have along the direction 6 the identical singularities with **<sup>D</sup>***<sup>n</sup>* <sup>2</sup> (Figs. 7 and 8c).

**Figure 8.** Four possible types of the **D m**( ) *<sup>α</sup>* vector field distribution around a zero-induction point **<sup>m</sup>**<sup>0</sup> , where ˆ **Dm m** <sup>0</sup> 0 0 () () *α α <sup>Q</sup>* .

#### *6.4.1. A model crystal of the symmetry class mm2*

Let us assume that one piezoelectric modulus in the crystal under consideration is much smaller than the other moduli. In particular, we consider a conventional crystallographic coordinate system with the *x* and *y* axes perpendicular to the symmetry planes and the *z*  axis parallel to the dyad axis, in which

$$\left| \left| e\_{31} \right| << \left| \left| e\_{15} \right| \right| \left| \left| e\_{24} \right| \right| \left| e\_{32} \right| \left| \left| e\_{33} \right| \right| \left| \left| e\_{33} \right| \right|. \tag{88}$$

One can readily check that, in a zero-order approximation with <sup>31</sup> *e* = 0, a quasi-longitudinal nondegenerate wave branch along the **m**<sup>0</sup> = (1, 0, 0) direction features the above special

 

situation, whereby simultaneously **D m** <sup>0</sup> ( )0 *<sup>l</sup>* and <sup>ˆ</sup> **<sup>m</sup>** <sup>0</sup> ( )0 *Ql* . In this case, the **D m**( ) *<sup>l</sup>* vector field distribution in the *yz* plane in the vicinity of **m**0 is described by the expression

$$\mathbf{D}\_{1} \parallel \{ \mathbf{0}, \ \mathbf{g}\_{2} \sin 2\phi, \ \mathbf{g}\_{1} + \varepsilon\_{1} \mathbf{e}\_{32} \mathbf{y}\_{2} - (\mathbf{g}\_{1} - \varepsilon\_{1} \mathbf{e}\_{32} \mathbf{y}\_{2}) \cos 2\phi \}\tag{89}$$

where *ϕ* is a polar angle of the **m** = **m** – **m**<sup>0</sup> direction measured from the *y* axis in the *yz*  plane and

$$\begin{aligned} g\_1 &= \chi\_1(\varepsilon\_1 e\_{33} - \varepsilon\_3 e\_{15}), \\ g\_2 &= (\chi\_1 + \chi\_2)\varepsilon\_1 e\_{24} - \varepsilon\_2 e\_{15}, \\ \chi\_1 &= \overline{d}\_5 / \overline{\Delta}\_{15'} \ \chi\_2 = d\_6 / \Delta\_{16'}, \\ \overline{d}\_5 &= c\_{13} + \overline{c}\_{55'} \ \overline{\Delta}\_{15} = c\_{11} - \overline{c}\_{55'} \\ \overline{c}\_{55} &= c\_{55} + c\_{15}^2 / \, c\_{1'} \ \, d\_6 = c\_{12} + c\_{66}. \end{aligned} \tag{90}$$

Electric Components of Acoustic Waves in the Vicinity of Nonpiezoactive Directions 27

*<sup>g</sup> <sup>ε</sup> <sup>e</sup> <sup>γ</sup>* (93)

Finally, when <sup>2</sup> *g* 0 and 1 32 2 *g e γ* 0 , the system features an oblique cross of zero-

*6.4.2. Behavior of point singularities in response to perturbations in the material moduli* 

and disappear either completely or leaving a certain number of isolated zero points.

the direction **m**<sup>0</sup> = (1, 0, 0) exhibits splitting so as to form two or four singular points:

1 0 1 0

0 2

the vicinity of the given direction for / 31 1 *e g* 1 ) as depicted in Fig. 10b.

**m m**

10a).

As is seen from Eq.(91) with 1 32 2 *g e γ* 0 , the zero-order approximation along **m**<sup>0</sup> corresponds to a singularity with 2 **<sup>D</sup>***n* or –2. The introduction of a small <sup>31</sup> *e* modulus leads to a symmetric splitting of this singularity into a pair of zero-induction points with equal indices 1 **<sup>D</sup>***n* or –1 along the *y* or *z* axis, depending on the sign of / 31 1 *e g* (Fig.

For 1 32 2 *g e γ* 0 , when the initial topological charge in the zero-order approximation is zero, the perturbed pattern comprises either four singularities with a zero total index **D***n* (for / 31 1 *e g* 1 ) or none of them (which corresponds to the absence of zero-induction points in

**Example 2.** It should be noted that the discussed splitting of unstable singularities is by no means reduced to abstract mathematical games. Perturbations in the material moduli of real crystals are frequently caused by various external factors such as electric fields, mechanical stresses, or temperature fluctuations arising in the vicinity of phase transitions. For example, the phase transition from a crystal of the symmetry class 62*m* , 6 2 *m* , or 6 to a trigonal

*, e /e <sup>γ</sup> <sup>δ</sup>*

 

( , ), , ( , ), .

2 2 2 31 32 2

*, e ε / g*

3 3 31 1 1

(94)

 

**Example 1.** The above general properties can be illustrated by a particular example using a model crystal of the symmetry class *mm*2 with a small modulus <sup>31</sup> *e* described above. It should be recalled that relations (89)–(93) were obtained in the zero-order approximation for <sup>31</sup> *e* = 0. In the next order with respect to the small parameter <sup>31</sup> *e* , the initial singularity along

The point singularities of various types in vector fields **D m**( ) *<sup>α</sup>* behave differently in response to perturbations in the material moduli: they shift, split, or disappear. An analysis of this situation, analogous to that carried out by Alshits, Sarychev & Shuvalov (1985), showed that singularities with **<sup>D</sup>***n* 1 (Fig. 5) are topologically stable and can only be displaced by such perturbations. The singular points of other types (Figs. 8a–8c) are unstable and either split (in accordance with the law of topological charge conservation) or disappear (provided only that 0 **<sup>D</sup>***n* ). The zero-induction lines (Fig. 8d and Fig. 9) are also unstable

cos2 . *<sup>g</sup> <sup>ε</sup> <sup>e</sup> <sup>γ</sup> <sup>φ</sup>*

1 1 32 2 1 1 32 2

induction lines (Fig. 9d) with the mutual orientation determined by the equation

Expression (89) shows that, depending on the material constants, the **D m**( ) *<sup>l</sup>* field always corresponds to one of the possible variants depicted in Fig. 8. The Poincaré indices for the point singularities corresponding to Figs. 8a–8c are as follows:

$$m\_l = \begin{cases} 0, & g\_1 e\_{32} \chi\_2 > 0, \\ 2 \text{sgn} \{ (g\_1 - e\_1 e\_{32} \chi\_2) g\_2 \} & g\_1 e\_{32} \chi\_2 < 0. \end{cases} \tag{91}$$

A condition for the existence of zero-induction lines in the **D m**( ) *<sup>l</sup>* field for Fig. 8d is

$$
\varrho\_1 e\_{32} \mathbb{1}\_2 = 0 \quad \text{or} \quad \varrho\_2 = 0, \quad \varrho\_1 e\_{32} \mathbb{1}\_2 < 0. \tag{92}
$$

According to expression (89), a zero-induction line for <sup>1</sup> *g* 0 passes via the vector **m**<sup>0</sup> along the *z* axis (Fig. 9a). For 32 2 *e γ* 0 , a similar zero-induction line is directed along the *y*  axis (Fig. 9b). If <sup>1</sup> *g* 0 simultaneously with 32 2 *e γ* 0 , the two lines coexist (Fig. 9c).

**Figure 9.** Four possible types of the **D m**()0 *<sup>l</sup>* lines in a model orthorhombic crystal.

Finally, when <sup>2</sup> *g* 0 and 1 32 2 *g e γ* 0 , the system features an oblique cross of zeroinduction lines (Fig. 9d) with the mutual orientation determined by the equation

26 Modeling and Measurement Methods for Acoustic Waves and for Acoustic Microdevices

 

point singularities corresponding to Figs. 8a–8c are as follows:

A condition for the existence of zero-induction lines in the **D m**( )

*n*

**D** *||*

plane and

situation, whereby simultaneously **D m** <sup>0</sup> ( )0 *<sup>l</sup>* and <sup>ˆ</sup> **<sup>m</sup>** <sup>0</sup> ( )0 *Ql* . In this case, the **D m**( )

vector field distribution in the *yz* plane in the vicinity of **m**0 is described by the expression

where *ϕ* is a polar angle of the **m** = **m** – **m**<sup>0</sup> direction measured from the *y* axis in the *yz* 

( ), () , /Δ , /Δ , , Δ , / , .

corresponds to one of the possible variants depicted in Fig. 8. The Poincaré indices for the

 

According to expression (89), a zero-induction line for <sup>1</sup> *g* 0 passes via the vector **m**<sup>0</sup> along the *z* axis (Fig. 9a). For 32 2 *e γ* 0 , a similar zero-induction line is directed along the *y* 

axis (Fig. 9b). If <sup>1</sup> *g* 0 simultaneously with 32 2 *e γ* 0 , the two lines coexist (Fig. 9c).

**Figure 9.** Four possible types of the **D m**()0 *<sup>l</sup>* lines in a model orthorhombic crystal.

1 1 32 2 2 1 32 2

0, 0, 2sgn[( ε ) ], 0. *<sup>l</sup>*

 

Expression (89) shows that, depending on the material constants, the **D m**( )

*g γ ε e ε e g γ γε e ε e γ d γ d dc c c c cce ε dc c*

1 1 1 33 3 15 2 1 2 1 24 2 15 1 5 15 2 6 16 5 13 55 15 11 55 2 55 55 15 1 6 12 66

<sup>2</sup> 1 1 32 2 1 1 32 2 {0, sin2 , ( )cos2 } *<sup>l</sup> g g ε e γ g ε e γ* (89)

1 32 2

or 1 32 2 2 1 32 2 0 0, 0. *g e γ g ge γ* (92)

*g e <sup>γ</sup> g ge <sup>γ</sup>* (91)

*<sup>l</sup>* field for Fig. 8d is

*g e γ*

*l*

(90)

*<sup>l</sup>* field always

$$\cos 2\phi = \frac{g\_1 + \varepsilon\_1 e\_{32} \mathcal{V}\_2}{g\_1 - \varepsilon\_1 e\_{32} \mathcal{V}\_2}. \tag{93}$$

#### *6.4.2. Behavior of point singularities in response to perturbations in the material moduli*

The point singularities of various types in vector fields **D m**( ) *<sup>α</sup>* behave differently in response to perturbations in the material moduli: they shift, split, or disappear. An analysis of this situation, analogous to that carried out by Alshits, Sarychev & Shuvalov (1985), showed that singularities with **<sup>D</sup>***n* 1 (Fig. 5) are topologically stable and can only be displaced by such perturbations. The singular points of other types (Figs. 8a–8c) are unstable and either split (in accordance with the law of topological charge conservation) or disappear (provided only that 0 **<sup>D</sup>***n* ). The zero-induction lines (Fig. 8d and Fig. 9) are also unstable and disappear either completely or leaving a certain number of isolated zero points.

**Example 1.** The above general properties can be illustrated by a particular example using a model crystal of the symmetry class *mm*2 with a small modulus <sup>31</sup> *e* described above. It should be recalled that relations (89)–(93) were obtained in the zero-order approximation for <sup>31</sup> *e* = 0. In the next order with respect to the small parameter <sup>31</sup> *e* , the initial singularity along the direction **m**<sup>0</sup> = (1, 0, 0) exhibits splitting so as to form two or four singular points:

$$
\delta \mathbf{m}\_0 + \delta \mathbf{m} = \begin{cases}
\langle \mathbf{1}, \ \pm \boldsymbol{\mu}\_2, \ \mathbf{0} \rangle, & \boldsymbol{\mu}\_2^2 = -e\_{31} \ \boldsymbol{\mu}\_{32} \boldsymbol{\chi}\_{2'} \\
\langle \mathbf{1}, \ \mathbf{0}, \pm \boldsymbol{\mu}\_3 \rangle, & \boldsymbol{\mu}\_3^2 = -e\_{31} e\_1 \ \boldsymbol{\mu}\_1.
\end{cases}
\tag{94}$$

As is seen from Eq.(91) with 1 32 2 *g e γ* 0 , the zero-order approximation along **m**<sup>0</sup> corresponds to a singularity with 2 **<sup>D</sup>***n* or –2. The introduction of a small <sup>31</sup> *e* modulus leads to a symmetric splitting of this singularity into a pair of zero-induction points with equal indices 1 **<sup>D</sup>***n* or –1 along the *y* or *z* axis, depending on the sign of / 31 1 *e g* (Fig. 10a).

For 1 32 2 *g e γ* 0 , when the initial topological charge in the zero-order approximation is zero, the perturbed pattern comprises either four singularities with a zero total index **D***n* (for / 31 1 *e g* 1 ) or none of them (which corresponds to the absence of zero-induction points in the vicinity of the given direction for / 31 1 *e g* 1 ) as depicted in Fig. 10b.

**Example 2.** It should be noted that the discussed splitting of unstable singularities is by no means reduced to abstract mathematical games. Perturbations in the material moduli of real crystals are frequently caused by various external factors such as electric fields, mechanical stresses, or temperature fluctuations arising in the vicinity of phase transitions. For example, the phase transition from a crystal of the symmetry class 62*m* , 6 2 *m* , or 6 to a trigonal

crystal of the symmetry class 32, 3*m*, or 3, respectively, leads to replacement of the hexad axis by a triad axis. Simultaneously, in accordance with Table 3, the polarization singularity **<sup>u</sup>** *n* 1 in the vector fields **u**0*<sup>t</sup>* and **u**0*t* of degenerate branches along this direction radically changes into the point with the index **<sup>u</sup>** *n* -1 / 2 . And the induction vector fields **D***t* and **D***t* of the same branches transform the singular pattern of the index 2 **<sup>D</sup>***n* into that with the index **<sup>D</sup>***n* -1 / 2 . The requirement of the conservation of the topological charge (Poincaré index) is realized in the appearance of three additional conical acoustic axes with indices **<sup>u</sup>** *n* 1/2 (Alshits, Sarychev & Shuvalov, 1985) and **<sup>D</sup>***n* -1 / 2 (Fig. 11a, b).

Electric Components of Acoustic Waves in the Vicinity of Nonpiezoactive Directions 29

( ) *t,t* , **D m**

( ) *t,t* and **D m**( ) *<sup>l</sup>* near the principal symmetry axis (Fig.11).

( ) *t,t* and **D m**( ) *<sup>l</sup>*

**Figure 11.** The three topological transformations in the vector fields **u m** <sup>0</sup>

In the nondegenerate quasi-longitudinal branch the index 2 **<sup>D</sup>***n* is replaced after the transition by 1 **<sup>D</sup>***n* . In accordance with the same conservation law of the Poincaré index and with the final crystal symmetry, three additional zero-induction points 0 **D** *l* with the indices 1 **<sup>D</sup>***n* must be created along with the central zero-point (Fig. 11c). Thus the considered phase transition ( 6 3 ) causes the three different topological transformations

Two electric components, the electric field **E** and the electric induction **D**, accompanying a bulk acoustic wave which propagates in a piezoelectric medium, exhibit significantly different properties. The electric field is always purely longitudinal, whereas the electric induction vector is, in contrast, always purely transverse. On the unit sphere ( **m** <sup>2</sup> 1 ) of wave propagation directions, the directions of zero electric field (**E** = 0) form lines, while the zero-induction directions (**D** = 0) are usually isolated and appear as singular points of the tangential vector field **D**(**m**) orientations. The nonpiezoactive directions of both types exist practically in all (even triclinic) crystals, although the presence of crystal symmetry

The topological singularities of the **D m**( ) *<sup>α</sup>* vector fields in the vicinity of zero-induction points in most crystals are characterized by the Poincaré indices **<sup>D</sup>***n* 1 , where the sign coincides with that of the determinant of the matrix (62) [see also Eq. (68)]. However, in

elements is the additional factor determining the appearance of such directions.

(in the non-directed representation) after the phase transition 6 3 .

( ) *t,t* , **D m**

in the vector fields **u m** <sup>0</sup>

**7. Conclusions** 

**Figure 10.** Diagrams illustrating the splitting of singular points with **D***n <sup>=</sup>*2 (*a*) and **D***n* = 0 (*b*) in a model crystal of the symmetry class *mm*2.

**Figure 11.** The three topological transformations in the vector fields **u m** <sup>0</sup> ( ) *t,t* , **D m** ( ) *t,t* and **D m**( ) *<sup>l</sup>* (in the non-directed representation) after the phase transition 6 3 .

In the nondegenerate quasi-longitudinal branch the index 2 **<sup>D</sup>***n* is replaced after the transition by 1 **<sup>D</sup>***n* . In accordance with the same conservation law of the Poincaré index and with the final crystal symmetry, three additional zero-induction points 0 **D** *l* with the indices 1 **<sup>D</sup>***n* must be created along with the central zero-point (Fig. 11c). Thus the considered phase transition ( 6 3 ) causes the three different topological transformations in the vector fields **u m** <sup>0</sup> ( ) *t,t* , **D m** ( ) *t,t* and **D m**( ) *<sup>l</sup>* near the principal symmetry axis (Fig.11).

## **7. Conclusions**

28 Modeling and Measurement Methods for Acoustic Waves and for Acoustic Microdevices

1985) and **<sup>D</sup>***n* -1 / 2 (Fig. 11a, b).

model crystal of the symmetry class *mm*2.

crystal of the symmetry class 32, 3*m*, or 3, respectively, leads to replacement of the hexad axis by a triad axis. Simultaneously, in accordance with Table 3, the polarization singularity **<sup>u</sup>** *n* 1 in the vector fields **u**0*<sup>t</sup>* and **u**0*t* of degenerate branches along this direction radically changes into the point with the index **<sup>u</sup>** *n* -1 / 2 . And the induction vector fields **D***t* and **D***t* of the same branches transform the singular pattern of the index 2 **<sup>D</sup>***n* into that with the index **<sup>D</sup>***n* -1 / 2 . The requirement of the conservation of the topological charge (Poincaré index) is realized in the appearance of three additional conical acoustic axes with indices **<sup>u</sup>** *n* 1/2 (Alshits, Sarychev & Shuvalov,

**Figure 10.** Diagrams illustrating the splitting of singular points with **D***n <sup>=</sup>*2 (*a*) and **D***n* = 0 (*b*) in a

Two electric components, the electric field **E** and the electric induction **D**, accompanying a bulk acoustic wave which propagates in a piezoelectric medium, exhibit significantly different properties. The electric field is always purely longitudinal, whereas the electric induction vector is, in contrast, always purely transverse. On the unit sphere ( **m** <sup>2</sup> 1 ) of wave propagation directions, the directions of zero electric field (**E** = 0) form lines, while the zero-induction directions (**D** = 0) are usually isolated and appear as singular points of the tangential vector field **D**(**m**) orientations. The nonpiezoactive directions of both types exist practically in all (even triclinic) crystals, although the presence of crystal symmetry elements is the additional factor determining the appearance of such directions.

The topological singularities of the **D m**( ) *<sup>α</sup>* vector fields in the vicinity of zero-induction points in most crystals are characterized by the Poincaré indices **<sup>D</sup>***n* 1 , where the sign coincides with that of the determinant of the matrix (62) [see also Eq. (68)]. However, in

some specific cases, this tensor may vanish ( ˆ *Q<sup>α</sup>* = 0) in some special directions because of a high symmetry or as a result of vanishing of certain combinations of the material tensor components. In this case, the system has either an isolated zero-induction point **m**0 (and has the Poincaré indices **<sup>D</sup>***n* 0, 2 ) or a zero-induction line. Such special orientations are topologically unstable and, in response to any change in the anisotropy, either split into stable points with **<sup>D</sup>***n* 1 or disappear.

Electric Components of Acoustic Waves in the Vicinity of Nonpiezoactive Directions 31

Alshits, V.I. & Lyubimov, V.N. (1990). Acoustic waves with extremal electro- (magneto-) mechanical coupling in piezocrystals. *Kristallografiya*, Vol. 35, No. 6 (Dec. 1990) 1325- 1327, ISSN 0023-4761 [*Sov. Phys. Crystallography*, Vol. 35, No. 6 (1990) 780-782, ISSN

Alshits, V.I.; Lyubimov, V.N. & Radowicz, A. (2005a). Special features of the electric components of acoustic waves in the vicinity of nonpiezoactive directions in crystals. *Zh. Eksp. Teor. Fiz*., Vol. 128, No. 1 (Jan. 2005) 125-138, ISSN 0044-4510 [*JETP*, Vol. 101,

Alshits, V.I.; Lyubimov, V.N. & Radowicz, A. (2005b). Non-piezoactivity in piezoelectrics: basic properties and topological features. *Arch. Appl. Mech.,* Vol. 74, No. 11-12 (Dec.

Alshits, V.I.; Lyubimov, V.N.; Sarychev, A.V. & Shuvalov, A.L. (1987). Topological characteristics of singular points of the electric field accompanying sound propagation in piezoelectrics. *Zh. Eksp. Teor. Fiz*., Vol. 93, No. 2 (8) (Aug. 1987) 723-732, ISSN 0044-

Alshits, V.I.; Sarychev, A.V. & Shuvalov, A.L. (1985). Classification of degeneracies and analysis of their stability in the theory of elastic waves in crystals. *Zh. Eksp. Teor. Fiz*., Vol. 89, No. 3(9) (Sept. 1985) 922-938, ISSN 0044-4510 [*Sov. Phys. JETP*, Vol. 62, No. 3

Alshits, V.I. & Shuvalov, A.L. (1988). On acoustic axes in piezoelectric crystals. *Kristallografiya*, Vol. 33, No. 1 (Jan. 1988) 7-12, ISSN 0023-4761 [*Sov. Phys. Crystallography*,

Balakirev, M.K. & Gilinskii, I.A. (1982). *Waves in Piezoelectric Crystals*, Nauka, ISBN,

Gulyaev, Yu.V. (1998). Review of shear surface acoustic waves in solids. *IEEE Trans. Ultrason. Ferroel. Freq. Control*, Vol. 45, No. 4 (April 1998) 935-938, ISSN 0885-3010 Holm, P. (1992). Generic elastic media. *Phys. Scr.,* Vol. T44 (1992) 122-127, ISSN 0031-

Landau, L.D. & Lifshitz, E.M. (1984). *Electrodynamics of Continuous Media*, Pergamon Press,

Lyamov, V.E. (1983). *Polarization Effects and Interaction Anisotropy of Acoustic Waves in* 

Lyubimov, V.N. (1969). Сonsideration of the piezoelectric effect in the theory of elastic waves for crystals of different symmetries. *Doklady AN SSSR*, Vol. 186, No. 5 (May 1969) 1055-1058, ISSN 0869-5652 [Sov. Phys. Doklady, Vol. 14, No. 5 (1969) 567-570, ISSN

Royer, D. & Dieulesaint, E. (2000). Elastic *Waves in Solids*. Springer, ISBN 3-540-65932-3,

*Crystals*, Moscow State University, ISBN, Moscow [in Russian]

Fedorov, F.I. (1968). *Theory of Elastic Waves in Crystals*, Plenum Press, ISBN, New York

4510 [*Sov. Phys. JETP*, Vol. 66, No. 2 (8) (1987) 408-413, ISSN 1063-7761]

1063-7745]

No. 1 (2005) 107-119, ISSN 1063-7761]

2005) 739-745, ISSN 0939-1533

(1985) 531-539, ISSN 1063-7761]

Novosibirsk [in Russian]

ISBN 0080302750, New York

8949

1085-1992]

Berlin

Vol. 33, No. 1 (1988) 1-4, ISSN 1063-7745]

It is interesting to note that singularities of the induction vector field **D m**( ) *<sup>α</sup>* in the vicinity of the zero-induction points substantially differ from analogous singularities near the acoustic axes (see Eq. (71) and Table 3). According to (Alshits *et al*, 1987), stable singularities in the latter case are characterized by the Poincaré indices **<sup>D</sup>***n* 1/2 , while the unstable ones have **<sup>D</sup>***n* 0, 1 . The only exception to this rule is the acoustic axis along the hexad axis 6 , for which the all branches, both degenerate *α t,t* and non-degenerate *α l* , are characterized by ˆ *Q<sup>α</sup>* = 0, **D***<sup>α</sup>* = 0 and **<sup>D</sup>***n* 2 . However, as we have seen, in the latter case the transformation of the same singularity **<sup>D</sup>***n* 2 due to the phase transition 6 3 has radically different topology [see Fig. 11 (*b*) and (*c*)].

## **Author details**

V.I. Alshits *A.V. Shubnikov Institute of Crystallography, Russian Academy of Sciences, Moscow, Russia Polish-Japanese Institute of Information Technology, Warsaw, Poland* 

V.N. Lyubimov *A.V. Shubnikov Institute of Crystallography, Russian Academy of Sciences, Moscow, Russia* 

A. Radowicz *Kielce University of Technology, Kielce, Poland* 

## **Acknowledgement**

This study was performed within the framework of the Agreement on Cooperation between the Shubnikov Institute of Crystallography (Russia) and the Kielce University of Technology (Poland). V.I.A. and V.N.L. are grateful to the Kielce University of Technology for a hospitality and support.

## **8. References**

Alshits, V.I. & Lothe, J. (1979). Elastic waves in triclinic crystals I, II, and III. *Kristallografiya*, Vol. 24, No. 4, 6 (Aug., Dec. 1979) 972-993, 1122-1130, ISSN 0023- 4761 [*Sov. Phys. Crystallography*, Vol. 24, No. 4, 6 (1979) 387-398, 644-648, ISSN 1063- 7745]

Alshits, V.I. & Lyubimov, V.N. (1990). Acoustic waves with extremal electro- (magneto-) mechanical coupling in piezocrystals. *Kristallografiya*, Vol. 35, No. 6 (Dec. 1990) 1325- 1327, ISSN 0023-4761 [*Sov. Phys. Crystallography*, Vol. 35, No. 6 (1990) 780-782, ISSN 1063-7745]

30 Modeling and Measurement Methods for Acoustic Waves and for Acoustic Microdevices

high symmetry or as a result of vanishing of certain combinations of the material tensor components. In this case, the system has either an isolated zero-induction point **m**0 (and has the Poincaré indices **<sup>D</sup>***n* 0, 2 ) or a zero-induction line. Such special orientations are topologically unstable and, in response to any change in the anisotropy, either split into

It is interesting to note that singularities of the induction vector field **D m**( ) *<sup>α</sup>* in the vicinity of the zero-induction points substantially differ from analogous singularities near the acoustic axes (see Eq. (71) and Table 3). According to (Alshits *et al*, 1987), stable singularities in the latter case are characterized by the Poincaré indices **<sup>D</sup>***n* 1/2 , while the unstable ones have **<sup>D</sup>***n* 0, 1 . The only exception to this rule is the acoustic axis along the hexad axis 6 , for which the all branches, both degenerate *α t,t* and non-degenerate *α l* , are characterized by ˆ*Q<sup>α</sup>* = 0, **D***<sup>α</sup>* = 0 and **<sup>D</sup>***<sup>n</sup>* <sup>2</sup> . However, as we have seen, in the latter case the transformation of the same singularity **<sup>D</sup>***n* 2 due to the phase transition 6 3 has

*A.V. Shubnikov Institute of Crystallography, Russian Academy of Sciences, Moscow, Russia* 

*A.V. Shubnikov Institute of Crystallography, Russian Academy of Sciences, Moscow, Russia* 

This study was performed within the framework of the Agreement on Cooperation between the Shubnikov Institute of Crystallography (Russia) and the Kielce University of Technology (Poland). V.I.A. and V.N.L. are grateful to the Kielce University of Technology for a

Alshits, V.I. & Lothe, J. (1979). Elastic waves in triclinic crystals I, II, and III. *Kristallografiya*, Vol. 24, No. 4, 6 (Aug., Dec. 1979) 972-993, 1122-1130, ISSN 0023- 4761 [*Sov. Phys. Crystallography*, Vol. 24, No. 4, 6 (1979) 387-398, 644-648, ISSN 1063-

*Q<sup>α</sup>* = 0) in some special directions because of a

some specific cases, this tensor may vanish ( ˆ

stable points with **<sup>D</sup>***n* 1 or disappear.

radically different topology [see Fig. 11 (*b*) and (*c*)].

*Kielce University of Technology, Kielce, Poland* 

*Polish-Japanese Institute of Information Technology, Warsaw, Poland* 

**Author details** 

V.I. Alshits

V.N. Lyubimov

A. Radowicz

**Acknowledgement** 

hospitality and support.

**8. References** 

7745]

	- Shuvalov, A.L. (1998). Topological features of polarization fields of plane acoustic waves in anisotropic media. *Proc. R. Soc. Lond*. *A*, Vol. 454, (Nov. 1998) 2911-2947, ISSN 1471- 2946
	- Sirotin Yu.I. & Shaskolskaya, M.P. (1979). *Fundamentals of Crystal Physics* (in Russian), Nauka, Moscow [(1982) translation into English, Mir, ISBN, Moscow]

**1. Introduction**

http://dx.doi.org/10.5772/55876

Kuo-Ming Lee

deal with here.

Crack detecting is an important issue in many areas of science and engineering. Often it is dangerous or it is only laborious to find the cracks directly, e.g., the detection of cracks within the nuclear power plants or of the cracks buried beneath the earth. One possible way out is to make use of the phenomenon of wave scattering. Sending an incident wave into the area in which we are interested, the information about the crack is hidden in the scattered wave that can be measured at some convenient place distance away. Thus, crack detection is equivalent to the extraction of the hidden information from the measured data. This is what we want to

**Chapter 2**

**Crack Detecting via Newton's Method** 

Additional information is available at the end of the chapter

To understand the extraction of information, one has to realize the constitution of the data first. This leads to the concept of *inverse problems*. Following Keller [1], two problems are inverse to each other if the formulation of each of them requires all or partial knowledge of the other. Often, for historical reasons, one of the two problems has been studied extensively for some time, while the other is newer and not so well understood. In such cases, the former

The direct problem in our case is the computation of the scattered wave from a prescribed crack. From the aspect of wave scattering, the space setting for our task is the unbounded domain outside the crack. This is certainly not a good place to begin with. We therefore employ the method of boundary integral equations which has the advantage of transforming a physical problem in the unbounded domain into one at the boundary which is just the crack itself. Thus, the boundary integral equation approach is advantageous both from a theoretical and a numerical point of view. On one hand, the boundary integral equations method allows an elegant and concise analysis of the unique solvability and the stability of the solution based on the powerful Riesz theory from functional analysis. On the other hand, this approach reduces the computational cost by decreasing the dimension. Having sucessfully solved the direct problem by the boundary integral equations method, it is natural for us to choose the same scheme to solve the inverse problem, i.e., the extraction of the information

> ©2013 Lee, licensee InTech. This is an open access chapter 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.

© 2013 The Author(s). Licensee InTech. 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,

©2013 Lee, licensee InTech. This is a paper 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.

and reproduction in any medium, provided the original work is properly cited.

problem is called the direct problem, while the latter is called the inverse problem.

## **Crack Detecting via Newton's Method**

## Kuo-Ming Lee

32 Modeling and Measurement Methods for Acoustic Waves and for Acoustic Microdevices

2946

Shuvalov, A.L. (1998). Topological features of polarization fields of plane acoustic waves in anisotropic media. *Proc. R. Soc. Lond*. *A*, Vol. 454, (Nov. 1998) 2911-2947, ISSN 1471-

Sirotin Yu.I. & Shaskolskaya, M.P. (1979). *Fundamentals of Crystal Physics* (in Russian),

Nauka, Moscow [(1982) translation into English, Mir, ISBN, Moscow]

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/55876

## **1. Introduction**

Crack detecting is an important issue in many areas of science and engineering. Often it is dangerous or it is only laborious to find the cracks directly, e.g., the detection of cracks within the nuclear power plants or of the cracks buried beneath the earth. One possible way out is to make use of the phenomenon of wave scattering. Sending an incident wave into the area in which we are interested, the information about the crack is hidden in the scattered wave that can be measured at some convenient place distance away. Thus, crack detection is equivalent to the extraction of the hidden information from the measured data. This is what we want to deal with here.

To understand the extraction of information, one has to realize the constitution of the data first. This leads to the concept of *inverse problems*. Following Keller [1], two problems are inverse to each other if the formulation of each of them requires all or partial knowledge of the other. Often, for historical reasons, one of the two problems has been studied extensively for some time, while the other is newer and not so well understood. In such cases, the former problem is called the direct problem, while the latter is called the inverse problem.

The direct problem in our case is the computation of the scattered wave from a prescribed crack. From the aspect of wave scattering, the space setting for our task is the unbounded domain outside the crack. This is certainly not a good place to begin with. We therefore employ the method of boundary integral equations which has the advantage of transforming a physical problem in the unbounded domain into one at the boundary which is just the crack itself. Thus, the boundary integral equation approach is advantageous both from a theoretical and a numerical point of view. On one hand, the boundary integral equations method allows an elegant and concise analysis of the unique solvability and the stability of the solution based on the powerful Riesz theory from functional analysis. On the other hand, this approach reduces the computational cost by decreasing the dimension. Having sucessfully solved the direct problem by the boundary integral equations method, it is natural for us to choose the same scheme to solve the inverse problem, i.e., the extraction of the information

about the crack from the measured data. Later we will see that both for the direct and the inverse problems, the same equations are used almost all the time. This is beneficial both in the apprehension and in the computation.

## **2. Direct scattering problem**

The scattering problem is mathematically modelled by an exterior boundary value problem governed by Helmholtz equation with some prescribed boundary conditions. The two dimensional problem arises when the obstacle is an infintely long cylinder. Thus, a planar crack can be seen as the intersction of a plane with an infintely long thin cylinder which runs in the direction normal to the plane. Mathematically, a planar crack can be given by a regular non-intersecting *<sup>C</sup>*3- smooth open arc <sup>Γ</sup> <sup>⊂</sup> **<sup>R</sup>**<sup>2</sup> which can be described as

$$\Gamma = \{ z(s) : s \in [-1, 1], z \in \mathbb{C}^3[-1, 1] \text{ and } |z'(s)| \neq 0, \forall s \in [-1, 1] \}.$$

The two end points of the crack are denoted by *z*∗ −1, *<sup>z</sup>*<sup>∗</sup> <sup>1</sup> respectively. The left hand side and the right hand side of the crack are written by Γ<sup>+</sup> and Γ<sup>−</sup> respectively. The unit normal to Γ<sup>+</sup> is denoted by *ν*. Further we set Γ<sup>0</sup> := Γ \ {*z*<sup>∗</sup> <sup>−</sup>1, *<sup>z</sup>*<sup>∗</sup> <sup>1</sup> }. The direct scattering problem that we are considering is as follows:

#### **Problem 1. (DP)**

*Given an incident planar wave u<sup>i</sup>* (*x*, *d*) := e*ik<x*,*d<sup>&</sup>gt; with a wave number k >* 0 *and a unit vector d giving the direction of propagation, find a solution u<sup>s</sup>* <sup>∈</sup> *<sup>C</sup>*2(**R**<sup>2</sup> \ <sup>Γ</sup>) <sup>∩</sup> *<sup>C</sup>*(**R**<sup>2</sup> \ <sup>Γ</sup>0) *to the Helmholtz equation*

$$
\Delta u^s + k^2 u^s = 0, \quad \text{in } \mathbb{R}^2 \backslash \Gamma \tag{1}
$$

example [2] and [3]. Indeed, in terms of the fundamental solution to the Helmholtz equation

we have the following theorem which ensures the unique solvability of the direct scattering

*∂ν*(*y*) *<sup>ϕ</sup>*(*y*)*ds*(*y*) = <sup>−</sup> *<sup>∂</sup>u<sup>i</sup>*

<sup>1</sup> ) = 0, *<sup>d</sup>ϕ*(*z*(*s*))

Applying the Green's integral theorem, the uniqueness is ensured by the Rellich's lemma and the radiation condition. Using the potential ansatz, the solvability of the boundary value problem is then converted to the solvability of the induced boundary integral equation (5)

At this place we want to point out that in the scattering problem, one is particularly interested

uniformly for all directions *<sup>x</sup>*<sup>ˆ</sup> <sup>∈</sup> <sup>Ω</sup> :<sup>=</sup> {*<sup>x</sup>* <sup>∈</sup> **<sup>R</sup>**2||*x*<sup>|</sup> <sup>=</sup> <sup>1</sup>}. The one-to-one correspondence between radiating waves and their far field patterns is established by the Rellich's lemma. In

For further treatment, we would like to transform our integral equation (5) into operator form.

Φ(*x*, *y*)*ϕ*(*y*)*ds*(*y*)

 Γ  1 |*x*| *u*∞(*x*ˆ) + *O*

<sup>0</sup> (*k*|*x* − *y*|), *x* �= *y*

*∂ν*(*y*) *<sup>ϕ</sup>*(*y*)*ds*(*y*) *<sup>x</sup>* <sup>∈</sup> **<sup>R</sup>**<sup>2</sup> \ <sup>Γ</sup>. (4)

(*x*)

*ds* <sup>=</sup> *<sup>ϕ</sup>*˜(arccos *<sup>s</sup>*) √

*∂ν*(*x*) (5)

Crack Detecting via Newton's Method 35

<sup>1</sup> <sup>−</sup> *<sup>s</sup>*<sup>2</sup> , *<sup>ϕ</sup>*˜ <sup>∈</sup> *<sup>C</sup>*0,*α*[0, *<sup>π</sup>*]

. The far-field pattern describes the behavior


*< ν*(*y*), *x*ˆ *>* e−ik*<*x,yˆ *<sup>&</sup>gt;*'(y)ds(y). (6)

 *<sup>k</sup>* <sup>8</sup>*<sup>π</sup>* <sup>e</sup>−<sup>i</sup> <sup>ß</sup> 4 .

<sup>Φ</sup>(*x*, *<sup>y</sup>*) :<sup>=</sup> <sup>i</sup>

**Theorem 1.** *The direct Neumann problem 1 has a unique solution given by*

Γ

*where <sup>ϕ</sup>* <sup>∈</sup> *<sup>C</sup>*1,*α*,∗(Γ) *is the (unique) solution to the following integral equation*

 Γ

<sup>−</sup>1) = *<sup>ϕ</sup>*(*z*<sup>∗</sup>

which can be determined by the Riesz theory. For details we refer to [4].

*us*

(*x*) =

*∂ ∂ν*(*x*)

*and for* 0 *< α <* 1*, the function space C*1,*α*,∗(Γ) *is defined by*

*ϕ*| *ϕ*(*z*<sup>∗</sup>

in the far-field pattern *u*<sup>∞</sup> of the scattered field *u<sup>s</sup>*

*us*

(*x*) = <sup>e</sup>ik|x<sup>|</sup> <sup>|</sup>*x*<sup>|</sup>

*u*∞(*x*ˆ) = *ρ*

the case of a sound-hard crack, the far-field pattern is declared via

 Γ

with the density function *ϕ* given by theorem 1 and the constant *ρ* =

For this purpose, the following integral operators may be defined:

(*Sϕ*)(*x*) :=

*C*1,*α*,∗(Γ) :=

of the scattered wave at infinity

4 *H*(1)

*∂*Φ(*x*, *y*)

*∂*Φ(*x*, *y*)

in **R**<sup>2</sup>

problem.

*which satisfies the Neumann boundary conditions*

$$\frac{\partial u^s\_{\pm}}{\partial \nu} = -\frac{\partial u^i}{\partial \nu} \qquad on \,\Gamma\_0 \tag{2}$$

*on both sides of the crack and the Sommerfeld radiation condition*

$$\lim\_{r \to \infty} \sqrt{r} \left( \frac{\partial u^s}{\partial r} - iku^s \right) = 0, \qquad r := |\mathbf{x}| \tag{3}$$

*uniformly for all directions x*ˆ := *<sup>x</sup>* |*x*| *.*

In (2), the limits

$$\frac{\partial u^s\_\pm(\mathbf{x})}{\partial \nu} := \lim\_{h \to 0} < \nu(\mathbf{x}),\\\operatorname{grad} u^s(\mathbf{x} \pm h\nu(\mathbf{x})) > , \quad \mathbf{x} \in \Gamma\_0.$$

are required to exist in the sense of locally uniform convergence.

Note that the boundary conditions (2) can be reformulated as homogeneous Neumann conditions for the total field *u* := *u<sup>i</sup>* + *u<sup>s</sup>* , i.e., *<sup>∂</sup>u*<sup>±</sup> *∂ν* = 0. This means that the normal component of the velocity of the total wave vainishes on the crack, i.e., the crack is sound-hard. Using boundary integral equations, this direct problem can be solved via the layer approach, see for

example [2] and [3]. Indeed, in terms of the fundamental solution to the Helmholtz equation in **R**<sup>2</sup>

$$\Phi(\mathfrak{x}, y) := \frac{\mathfrak{i}}{4} H\_0^{(1)}(k|\mathfrak{x} - y|), \qquad \mathfrak{x} \neq y.$$

we have the following theorem which ensures the unique solvability of the direct scattering problem.

**Theorem 1.** *The direct Neumann problem 1 has a unique solution given by*

$$u^s(\mathbf{x}) = \int\_{\Gamma} \frac{\partial \Phi(\mathbf{x}, y)}{\partial v(y)} \varphi(y) ds(y) \quad \mathbf{x} \in \mathbb{R}^2 \; \vert \; \Gamma. \tag{4}$$

*where <sup>ϕ</sup>* <sup>∈</sup> *<sup>C</sup>*1,*α*,∗(Γ) *is the (unique) solution to the following integral equation*

$$\frac{\partial}{\partial \nu(\mathbf{x})} \int\_{\Gamma} \frac{\partial \Phi(\mathbf{x}, y)}{\partial \nu(y)} \varphi(y) ds(y) = -\frac{\partial u^i(\mathbf{x})}{\partial \nu(\mathbf{x})} \tag{5}$$

*and for* 0 *< α <* 1*, the function space C*1,*α*,∗(Γ) *is defined by*

2 Will-be-set-by-IN-TECH

about the crack from the measured data. Later we will see that both for the direct and the inverse problems, the same equations are used almost all the time. This is beneficial both in

The scattering problem is mathematically modelled by an exterior boundary value problem governed by Helmholtz equation with some prescribed boundary conditions. The two dimensional problem arises when the obstacle is an infintely long cylinder. Thus, a planar crack can be seen as the intersction of a plane with an infintely long thin cylinder which runs in the direction normal to the plane. Mathematically, a planar crack can be given by a regular

the right hand side of the crack are written by Γ<sup>+</sup> and Γ<sup>−</sup> respectively. The unit normal to Γ<sup>+</sup>

<sup>−</sup>1, *<sup>z</sup>*<sup>∗</sup>

*giving the direction of propagation, find a solution u<sup>s</sup>* <sup>∈</sup> *<sup>C</sup>*2(**R**<sup>2</sup> \ <sup>Γ</sup>) <sup>∩</sup> *<sup>C</sup>*(**R**<sup>2</sup> \ <sup>Γ</sup>0) *to the Helmholtz*

−1, *<sup>z</sup>*<sup>∗</sup>

(*s*)| �= 0, ∀*s* ∈ [−1, 1]}.

(*x*, *d*) := e*ik<x*,*d<sup>&</sup>gt; with a wave number k >* 0 *and a unit vector d*

<sup>Δ</sup>*u<sup>s</sup>* <sup>+</sup> *<sup>k</sup>*2*u<sup>s</sup>* <sup>=</sup> 0, *in* **<sup>R</sup>**<sup>2</sup> \ <sup>Γ</sup> (1)

(*x* ± *hν*(*x*)) *>*, *x* ∈ Γ<sup>0</sup>

<sup>1</sup> respectively. The left hand side and

<sup>1</sup> }. The direct scattering problem that we are

*∂ν on* <sup>Γ</sup><sup>0</sup> (2)

= 0, *r* := |*x*| (3)

*∂ν* = 0. This means that the normal component

non-intersecting *<sup>C</sup>*3- smooth open arc <sup>Γ</sup> <sup>⊂</sup> **<sup>R</sup>**<sup>2</sup> which can be described as

<sup>Γ</sup> <sup>=</sup> {*z*(*s*) : *<sup>s</sup>* <sup>∈</sup> [−1, 1], *<sup>z</sup>* <sup>∈</sup> *<sup>C</sup>*3[−1, 1] and <sup>|</sup>*z*�

*∂u<sup>s</sup>* ± *∂ν* <sup>=</sup> <sup>−</sup> *<sup>∂</sup>u<sup>i</sup>*

*<sup>∂</sup><sup>r</sup>* <sup>−</sup> *iku<sup>s</sup>*

*< ν*(*x*), grad*u<sup>s</sup>*

Note that the boundary conditions (2) can be reformulated as homogeneous Neumann

of the velocity of the total wave vainishes on the crack, i.e., the crack is sound-hard. Using boundary integral equations, this direct problem can be solved via the layer approach, see for

, i.e., *<sup>∂</sup>u*<sup>±</sup>

*on both sides of the crack and the Sommerfeld radiation condition*

lim*r*→<sup>∞</sup> √*r ∂u<sup>s</sup>*

> |*x*| *.*

are required to exist in the sense of locally uniform convergence.

The two end points of the crack are denoted by *z*∗

is denoted by *ν*. Further we set Γ<sup>0</sup> := Γ \ {*z*<sup>∗</sup>

*which satisfies the Neumann boundary conditions*

considering is as follows:

*Given an incident planar wave u<sup>i</sup>*

*uniformly for all directions x*ˆ := *<sup>x</sup>*

*∂u<sup>s</sup>* <sup>±</sup>(*x*) *∂ν* :<sup>=</sup> lim *h*→0

conditions for the total field *u* := *u<sup>i</sup>* + *u<sup>s</sup>*

In (2), the limits

**Problem 1. (DP)**

*equation*

the apprehension and in the computation.

**2. Direct scattering problem**

$$\mathcal{C}^{1,\mathfrak{a},\*}(\Gamma) := \left\{ \mathfrak{q} \,|\, \mathfrak{q}(z\_{-1}^\*) = \mathfrak{q}(z\_1^\*) = 0, \frac{d\mathfrak{q}(z(s))}{ds} = \frac{\mathfrak{q}(\arccos s)}{\sqrt{1 - s^2}}, \mathfrak{q} \in \mathbb{C}^{0,\mathfrak{a}}[0, \pi] \right\}$$

Applying the Green's integral theorem, the uniqueness is ensured by the Rellich's lemma and the radiation condition. Using the potential ansatz, the solvability of the boundary value problem is then converted to the solvability of the induced boundary integral equation (5) which can be determined by the Riesz theory. For details we refer to [4].

At this place we want to point out that in the scattering problem, one is particularly interested in the far-field pattern *u*<sup>∞</sup> of the scattered field *u<sup>s</sup>* . The far-field pattern describes the behavior of the scattered wave at infinity

$$u^s(\mathbf{x}) = \frac{\mathbf{e}^{i\mathbf{k}|\mathbf{x}|}}{\sqrt{|\mathbf{x}|}} \left\{ u\_\infty(\mathbf{x}) + O\left(\frac{1}{|\mathbf{x}| }\right) \right\} \qquad |\mathbf{x}| \to \infty$$

uniformly for all directions *<sup>x</sup>*<sup>ˆ</sup> <sup>∈</sup> <sup>Ω</sup> :<sup>=</sup> {*<sup>x</sup>* <sup>∈</sup> **<sup>R</sup>**2||*x*<sup>|</sup> <sup>=</sup> <sup>1</sup>}. The one-to-one correspondence between radiating waves and their far field patterns is established by the Rellich's lemma. In the case of a sound-hard crack, the far-field pattern is declared via

$$
\mu\_{\infty}(\mathfrak{x}) = \rho \int\_{\Gamma} <\nu(y), \mathfrak{x} > \mathrm{e}^{-\mathrm{ik}\, < \mathfrak{k}, \mathrm{y} > \prime}(\mathfrak{y}) \mathrm{ds}(\mathfrak{y}).\tag{6}
$$

with the density function *ϕ* given by theorem 1 and the constant *ρ* = *<sup>k</sup>* <sup>8</sup>*<sup>π</sup>* <sup>e</sup>−<sup>i</sup> <sup>ß</sup> 4 .

For further treatment, we would like to transform our integral equation (5) into operator form. For this purpose, the following integral operators may be defined:

$$(S\varphi)(x) := \int\_{\Gamma} \Phi(x, y)\rho(y)ds(y)$$

#### 4 Will-be-set-by-IN-TECH 36 Modeling and Measurement Methods for Acoustic Waves and for Acoustic Microdevices Crack Detecting Via Newton's Method <sup>5</sup>

$$(T\_0\varphi)(\mathfrak{x}) := \frac{\mathfrak{d}}{\partial \nu(\mathfrak{x})} \int\_{\Gamma} \frac{\partial \Phi(\mathfrak{x}, y)}{\partial \nu(y)} \varphi(y) ds(y).$$

To reduce the hypersingularity of the operator *T*0, we use the Maue's identity to split it into two milder parts (for a proof of this splitting, see for example Theorem 7.29 in [3])

$$T\_0 \varphi = \frac{\partial}{\partial \theta} S \frac{\partial \varphi}{\partial \theta} + k^2 < \nu\_\prime S \varphi \nu > \nu\_\prime$$

where *ϑ* is the unit tangent vector. The integral equation (5) now becomes

$$k\frac{\partial}{\partial\theta}S\frac{\partial\varphi}{\partial\theta} + k^2 < \nu,\\ S\varphi\nu > = \text{g} \tag{7}$$

Therefore, certain regularization techniques must be introduced at this place. For the sake of completeness, we briefly discuss regularization for linear compact operators in the next section. We note that the operator *F* will be differently interpreted for different methods in the

**Definition 3** (Regularization)**.** *Assume that X*, *Y are normed spaces. Let the operator A* : *X* → *Y be linear, bounded and injective. A family of bounded linear operators R<sup>α</sup>* : *Y* → *X*, *α >* 0 *is called a*

*RαAϕ* = *ϕ*, *for all ϕ* ∈ *X*

From the above definition, it is easy to see that if the operator *A* has a bounded inverse, then we can simply choose *<sup>R</sup><sup>α</sup>* <sup>=</sup> *<sup>A</sup>*−1, <sup>∀</sup>*<sup>α</sup> <sup>&</sup>gt;* 0. In the case where the inverse of the injective operator *A* is not bounded, for example when *A* is compact and the dimension of *X* is infinite, then difficulty arises since we are trying to approximate an unbounded operator with a sequence of bounded operators. The task of regularization is to find a stable approximate solution. The following theorem, which is proved in [3] (theorem 15.6), shows the limitations

**Theorem 3.** *Let X and Y be normed spaces with* dim*X* = ∞ *and let A* : *X* → *Y be linear and compact. Then for a regularization scheme the operators R<sup>α</sup> cannot be uniformly bounded with respect*

Because of the lack of uniform convergence, the choice of the regularization parameter *α* is apparently crucial. To have a vivid impression of this, let's examine the aproximation error in a more pratical setting where measurement errors or noises are present. Let's denote the contaminated data by *<sup>f</sup> <sup>δ</sup>* assuming an error level *<sup>δ</sup>*, i.e., � *<sup>f</sup>* <sup>−</sup> *<sup>f</sup> <sup>δ</sup>*� ≤ *<sup>δ</sup>*. For *<sup>α</sup> <sup>&</sup>gt;* 0, we wirte the

*<sup>α</sup>* := *<sup>R</sup><sup>α</sup> <sup>f</sup> <sup>δ</sup>*

*<sup>α</sup>* <sup>−</sup> *<sup>ϕ</sup>* <sup>=</sup> *<sup>R</sup><sup>α</sup> <sup>f</sup> <sup>δ</sup>* <sup>−</sup> *<sup>R</sup><sup>α</sup> <sup>f</sup>* <sup>+</sup> *<sup>R</sup>αA<sup>ϕ</sup>* <sup>−</sup> *<sup>ϕ</sup>*

The right-hand side of (10) reveals the difficulty of the approximation. As *α* tends to 0, the second term of the right-hand side becomes small because of the regularization. At the same time, the error *δ* will be amplified by the factor �*Rα*� which tends to infinity. Therefore, the

*<sup>α</sup>* − *ϕ*� ≤ *δ*�*Rα*� + �*RαAϕ* − *ϕ*� (10)

*ϕδ*

The total approximation error of the regularization scheme can be written as

*Aϕ* = *f* (9)

Crack Detecting via Newton's Method 37

following sectons. But the right hand side of (8) will remain the same.

lim *α*→0

*In this case, the parameter α is called the* regularization parameter*.*

*to α, and the operators RαA cannot be norm convergent as α* → 0*.*

*ϕδ*

�*ϕδ*

From this, we have the following error estimate

**4. Regularization**

regularization scheme *for*

of the regularization.

regularized approximation as

*if it satisfies the following pointwise convergence*

where we set *<sup>g</sup>* <sup>=</sup> <sup>−</sup> *<sup>∂</sup>u<sup>i</sup> ∂ν* .

## **3. Inverse scattering problem**

After introducing the direct scattering problem in the last section, we consider the following inverse problem:

#### **Problem 2. (IP)**

*Determine the crack* Γ *if the far-field pattern u*∞ *is known for one incident wave.*

About this inverse problem, the first thing to ask is the uniqueness, that is, the identifiability of the crack. Unfortunately, there exists no theoretical result for our setting of inverse problem. However, if all the far-field patterns from all possible incident directions are measured, the crack can be uniquely identified. We quote the following result from [5].

**Theorem 2.** *Assume that* Γ<sup>1</sup> *and* Γ<sup>2</sup> *are two sound-hard cracks with the property that for a fixed wave number k >* 0*, the far-field patterns u*1,<sup>∞</sup> *and u*2,<sup>∞</sup> *coincide for all incident directions d. Then we have* Γ<sup>1</sup> = Γ2*.*

Our aim is to find and to reconstruct the crack from the measured far-field data resulted from just one single incident wave. Despite of the lack of a theoretical proof, our setting of inverse problem arises naturally from the viewpoint of the real applications. It is always desirable to find the crack with less effort.

For the detection of the crack, the only available information is the measured far-field data. Equation (6) relates the far-field pattern *u*∞ with the crack Γ and is therefore suitable as the starting point for the reconstruction. For future reference, we rewrite it in the more compact form

$$F(\Gamma) = \mathfrak{u}\_{\infty} \tag{8}$$

and call it the far-field equation as in the literature. The inverse problem is then equivalent to the solving of this equation. However, the sloving of (8) is not as straight forward as it appears. Mathematically this equation is not even solvable because of the nature of the operator *F*. Being a compact operator in infinite dimensional function space, *F* cannot have a bounded inverse. This means that (8) cannot be solved in any reasonable normed space. Therefore, certain regularization techniques must be introduced at this place. For the sake of completeness, we briefly discuss regularization for linear compact operators in the next section. We note that the operator *F* will be differently interpreted for different methods in the following sectons. But the right hand side of (8) will remain the same.

## **4. Regularization**

4 Will-be-set-by-IN-TECH

 Γ

To reduce the hypersingularity of the operator *T*0, we use the Maue's identity to split it into

After introducing the direct scattering problem in the last section, we consider the following

About this inverse problem, the first thing to ask is the uniqueness, that is, the identifiability of the crack. Unfortunately, there exists no theoretical result for our setting of inverse problem. However, if all the far-field patterns from all possible incident directions are measured, the

**Theorem 2.** *Assume that* Γ<sup>1</sup> *and* Γ<sup>2</sup> *are two sound-hard cracks with the property that for a fixed wave number k >* 0*, the far-field patterns u*1,<sup>∞</sup> *and u*2,<sup>∞</sup> *coincide for all incident directions d. Then we have*

Our aim is to find and to reconstruct the crack from the measured far-field data resulted from just one single incident wave. Despite of the lack of a theoretical proof, our setting of inverse problem arises naturally from the viewpoint of the real applications. It is always desirable to

For the detection of the crack, the only available information is the measured far-field data. Equation (6) relates the far-field pattern *u*∞ with the crack Γ and is therefore suitable as the starting point for the reconstruction. For future reference, we rewrite it in the more compact

and call it the far-field equation as in the literature. The inverse problem is then equivalent to the solving of this equation. However, the sloving of (8) is not as straight forward as it appears. Mathematically this equation is not even solvable because of the nature of the operator *F*. Being a compact operator in infinite dimensional function space, *F* cannot have a bounded inverse. This means that (8) cannot be solved in any reasonable normed space.

*∂*Φ(*x*, *y*)

*∂ϑ* <sup>+</sup> *<sup>k</sup>*<sup>2</sup> *<sup>&</sup>lt; <sup>ν</sup>*, *<sup>S</sup>ϕν <sup>&</sup>gt;*,

*∂ν*(*y*) *<sup>ϕ</sup>*(*y*)*ds*(*y*)

*∂ϑ* <sup>+</sup> *<sup>k</sup>*<sup>2</sup> *<sup>&</sup>lt; <sup>ν</sup>*, *<sup>S</sup>ϕν <sup>&</sup>gt;*<sup>=</sup> *<sup>g</sup>* (7)

*F*(Γ) = *u*<sup>∞</sup> (8)

*∂ν*(*x*)

two milder parts (for a proof of this splitting, see for example Theorem 7.29 in [3])

*∂ϑ <sup>S</sup> ∂ϕ*

(*T*0*ϕ*)(*x*) :<sup>=</sup> *<sup>∂</sup>*

*<sup>T</sup>*0*<sup>ϕ</sup>* <sup>=</sup> *<sup>∂</sup>*

*∂ ∂ϑ <sup>S</sup> ∂ϕ*

where we set *<sup>g</sup>* <sup>=</sup> <sup>−</sup> *<sup>∂</sup>u<sup>i</sup>*

inverse problem: **Problem 2. (IP)**

Γ<sup>1</sup> = Γ2*.*

form

find the crack with less effort.

*∂ν* .

**3. Inverse scattering problem**

where *ϑ* is the unit tangent vector. The integral equation (5) now becomes

*Determine the crack* Γ *if the far-field pattern u*∞ *is known for one incident wave.*

crack can be uniquely identified. We quote the following result from [5].

**Definition 3** (Regularization)**.** *Assume that X*, *Y are normed spaces. Let the operator A* : *X* → *Y be linear, bounded and injective. A family of bounded linear operators R<sup>α</sup>* : *Y* → *X*, *α >* 0 *is called a* regularization scheme *for*

$$A\varphi = f \tag{9}$$

*if it satisfies the following pointwise convergence*

$$\lim\_{\alpha \to 0} R\_{\mathfrak{A}} A \,\varphi = \varphi \,\, for \, all \,\, \varphi \in X$$

*In this case, the parameter α is called the* regularization parameter*.*

From the above definition, it is easy to see that if the operator *A* has a bounded inverse, then we can simply choose *<sup>R</sup><sup>α</sup>* <sup>=</sup> *<sup>A</sup>*−1, <sup>∀</sup>*<sup>α</sup> <sup>&</sup>gt;* 0. In the case where the inverse of the injective operator *A* is not bounded, for example when *A* is compact and the dimension of *X* is infinite, then difficulty arises since we are trying to approximate an unbounded operator with a sequence of bounded operators. The task of regularization is to find a stable approximate solution. The following theorem, which is proved in [3] (theorem 15.6), shows the limitations of the regularization.

**Theorem 3.** *Let X and Y be normed spaces with* dim*X* = ∞ *and let A* : *X* → *Y be linear and compact. Then for a regularization scheme the operators R<sup>α</sup> cannot be uniformly bounded with respect to α, and the operators RαA cannot be norm convergent as α* → 0*.*

Because of the lack of uniform convergence, the choice of the regularization parameter *α* is apparently crucial. To have a vivid impression of this, let's examine the aproximation error in a more pratical setting where measurement errors or noises are present. Let's denote the contaminated data by *<sup>f</sup> <sup>δ</sup>* assuming an error level *<sup>δ</sup>*, i.e., � *<sup>f</sup>* <sup>−</sup> *<sup>f</sup> <sup>δ</sup>*� ≤ *<sup>δ</sup>*. For *<sup>α</sup> <sup>&</sup>gt;* 0, we wirte the regularized approximation as

$$
\!\!\!\!\!\!\!Q\_a^{\delta} := \mathcal{R}\_a f^{\delta}
$$

The total approximation error of the regularization scheme can be written as

$$
\varphi\_\alpha^\delta - \varphi = \mathcal{R}\_\mathfrak{a} f^\delta - \mathcal{R}\_\mathfrak{a} f + \mathcal{R}\_\mathfrak{a} A \varphi - \varphi
$$

From this, we have the following error estimate

$$||\varrho\_{\mathfrak{a}}^{\delta} - \varrho|| \le \delta ||\mathcal{R}\_{\mathfrak{a}}|| + ||\mathcal{R}\_{\mathfrak{a}} A \varphi - \varrho||\tag{10}$$

The right-hand side of (10) reveals the difficulty of the approximation. As *α* tends to 0, the second term of the right-hand side becomes small because of the regularization. At the same time, the error *δ* will be amplified by the factor �*Rα*� which tends to infinity. Therefore, the

#### 6 Will-be-set-by-IN-TECH 38 Modeling and Measurement Methods for Acoustic Waves and for Acoustic Microdevices Crack Detecting Via Newton's Method <sup>7</sup>

total error is dominated by the first term and is much larger than the datat error. For larger *α*, the first term on the right-hand side is smaller but the second term will be larger. Hence certain strategy for choosing *α* must be developed to balance the effects of the two factors. The discrepancy principle proposed by Morozov ([6], [7]) which based on the consideration that a problem cannot be solved more accurate than the data error, is a natural *a posteriori* strategy for determining the parameter *α*. More precisely, one looks for an *α >* 0 such that

$$\|AR\_{\mathfrak{a}}f^{\delta} - f^{\delta}\| = \gamma \delta$$

for some prescribed *γ* ≥ 1. It can be shown that there always exists a smallest *α >* 0 for which the approxiamte solution *f <sup>δ</sup>* satisfies the above condition. (see [3]) Although the existence for a "best" *α* is theoretically assured, there is no obvious way to select it in practice. In most cases, the only way to choose a good parameter is by trial and error. For our purpose, it is sufficient to consider regularization in the Hilbert space setting. We denote by *A*∗ the adjoint operator of *A*. Without going into details, we adapt the following regularization scheme.

**Theorem 4** (Tikhonov Regularization)**.** *Let x and Y be Hilbert spaces. The operator A* : *X* → *Y is assumed to be compact and linear. Then for every α >* 0*, the operator*

$$\alpha I + A^\*A : X \to X$$

*is bijective and has a bounded inverse. Furthermore, if the operator A is injective, then*

$$R\_{\mathfrak{a}} := \left(\mathfrak{a}I + A^\*A\right)^{-1}A^\*, \qquad \mathfrak{a} > 0.$$

*describes a regularization scheme with* �*Rα*� ≤ <sup>1</sup> 2 <sup>√</sup>*<sup>α</sup> .*

Applying the Tikhonov Regularization scheme to find an approximate solution to (9), we actually solve the follwing equation

$$(\alpha I + A^\*A)\varphi\_\mathfrak{A} = A^\*f \tag{11}$$

technique from the last section. Thus, the following equation has to be solved

Numerically, starting from an initial guess *γ* of the unknown crack, (13) finds an update *q*. The same equation is then solved iteratively with *γ* replaced by *γ* + *q* until some convergence

This scheme is conceptually very simple, yet the computation of the Fréchet derivative *F*� is

characterized as the solution of that problem (see [5]). Thus additional computational cost

At this place, we want to discuss the uniqueness of the equation (12). As pointed out in [8],

This reflects the fact that different parametrizations of the arc leading to the same set of points which in turn give the same far-field pattern. We can avoid this ambiguity by limiting our solution space to the set of arcs representable as the graph of a function as suggested in [8]. Once this restriction is made, the operator *F*� is then an injective linear compact operator. The

Based on the reciprocity gap principle, Kress and Rundell [9] proposed a method using nonlinear integral equations for solving an inverse conducting problem avoiding the computation of a forward problem at each iteration step. The idea behind this approach is quite simple: in a doubly connected bounded domain to which the second Green's theorem applies, if one knows the Cauchy data of a harmonic function on the outer boundary and the type of the boundary condition on the interior boundary, then one can determine the Cauchy data on the interior boundary and also the interior boundary itself. This method is extended to other boundary conditions and cracks in [10] and to obstacle scattering in [11]. Based on a different system of two nonlinear integral equations, a method with far-field data is proposed

To understand this method, let's first reformulate our equations for the direct problem. In terms of <sup>Γ</sup> and *<sup>ϕ</sup>*, we can define the operators *<sup>B</sup>* : *<sup>C</sup>*3[−1, 1] <sup>×</sup> *<sup>C</sup>*1,*α*,∗(Γ) <sup>→</sup> *<sup>C</sup>*0,*α*(Γ) and *<sup>F</sup>* :

*∂*Φ(*x*, *y*)

*∂ν*(*y*) *<sup>ϕ</sup>*(*y*)*ds*(*y*), *<sup>x</sup>* <sup>∈</sup> <sup>Γ</sup> (14)

*<sup>&</sup>lt; <sup>ν</sup>*(*y*), *<sup>x</sup>*<sup>ˆ</sup> *<sup>&</sup>gt; <sup>e</sup>*−*ik<x*ˆ,*y>ϕ*(*y*)*ds*(*y*), *<sup>x</sup>*<sup>ˆ</sup> <sup>∈</sup> <sup>Ω</sup> (15)

) = {*q* : *ν* · *q* = 0}

the Fréchet derivative *F*� is not injective. The null space of this operator is given by

*N*(*F*�

same restriction will aslo be made in the next two sections.

in [12] without the need of an auxiliary curve around the obstacle.

*<sup>B</sup>*(Γ, *<sup>ϕ</sup>*)(*x*) :<sup>=</sup> *<sup>∂</sup>*

 Γ

*F*(Γ, *ϕ*)(*x*ˆ) := *ρ*

*∂ν*(*x*)

 Γ

**6. Nonlinear integral equations method**

*<sup>C</sup>*3[−1, 1] <sup>×</sup> *<sup>C</sup>*1,*α*,∗(Γ) <sup>→</sup> *<sup>C</sup>*0,*α*(Ω) via

and

(*γ*))*q* = *F*�∗(*γ*)(*u*<sup>∞</sup> − *F*(*γ*)) (13)

Crack Detecting via Newton's Method 39

, one must first solve another direct problem since *F*� is

(*αI* + *F*�∗(*γ*)*F*�

criterion is met.

arises.

not straightforward. To calculate *F*�

for *ϕα*.

#### **5. Classical Newton's method**

As noted earlier, (8) connects the far-field pattern with the crack and is therefore suitable for the inverse task of finding the crack. It is therefore natural to solve it for the unknown crack when one has the measured far-field data at hand. For this nonlinear problem, Newton's method provides a simple way to handle it. Utilizing the first Frechét derivative which existence is proved in [5], Newton's method approximates the nonlinear equation (8) with a linear one. This means that instead of solving (8), we endeavor to solve the linear equation

$$F'(\gamma)q = \mathfrak{u}\_{\infty} - F(\gamma) \tag{12}$$

for an update *q*. We remark here that the derivative *F*� is still a compact operator, this means that the above equation can't be solved directly. However, we can apply the regularization technique from the last section. Thus, the following equation has to be solved

$$(aI + F^{\prime \*} (\gamma) F^{\prime} (\gamma)) q = F^{\prime \*} (\gamma) (u\_{\infty} - F(\gamma)) \tag{13}$$

Numerically, starting from an initial guess *γ* of the unknown crack, (13) finds an update *q*. The same equation is then solved iteratively with *γ* replaced by *γ* + *q* until some convergence criterion is met.

This scheme is conceptually very simple, yet the computation of the Fréchet derivative *F*� is not straightforward. To calculate *F*� , one must first solve another direct problem since *F*� is characterized as the solution of that problem (see [5]). Thus additional computational cost arises.

At this place, we want to discuss the uniqueness of the equation (12). As pointed out in [8], the Fréchet derivative *F*� is not injective. The null space of this operator is given by

$$N(F') = \{q : \boldsymbol{\nu} \cdot q = 0\}$$

This reflects the fact that different parametrizations of the arc leading to the same set of points which in turn give the same far-field pattern. We can avoid this ambiguity by limiting our solution space to the set of arcs representable as the graph of a function as suggested in [8]. Once this restriction is made, the operator *F*� is then an injective linear compact operator. The same restriction will aslo be made in the next two sections.

### **6. Nonlinear integral equations method**

Based on the reciprocity gap principle, Kress and Rundell [9] proposed a method using nonlinear integral equations for solving an inverse conducting problem avoiding the computation of a forward problem at each iteration step. The idea behind this approach is quite simple: in a doubly connected bounded domain to which the second Green's theorem applies, if one knows the Cauchy data of a harmonic function on the outer boundary and the type of the boundary condition on the interior boundary, then one can determine the Cauchy data on the interior boundary and also the interior boundary itself. This method is extended to other boundary conditions and cracks in [10] and to obstacle scattering in [11]. Based on a different system of two nonlinear integral equations, a method with far-field data is proposed in [12] without the need of an auxiliary curve around the obstacle.

To understand this method, let's first reformulate our equations for the direct problem. In terms of <sup>Γ</sup> and *<sup>ϕ</sup>*, we can define the operators *<sup>B</sup>* : *<sup>C</sup>*3[−1, 1] <sup>×</sup> *<sup>C</sup>*1,*α*,∗(Γ) <sup>→</sup> *<sup>C</sup>*0,*α*(Γ) and *<sup>F</sup>* : *<sup>C</sup>*3[−1, 1] <sup>×</sup> *<sup>C</sup>*1,*α*,∗(Γ) <sup>→</sup> *<sup>C</sup>*0,*α*(Ω) via

$$B(\Gamma, \varphi)(\mathbf{x}) := \frac{\partial}{\partial \nu(\mathbf{x})} \int\_{\Gamma} \frac{\partial \Phi(\mathbf{x}, y)}{\partial \nu(y)} \varphi(y) ds(y), \quad \mathbf{x} \in \Gamma \tag{14}$$

and

6 Will-be-set-by-IN-TECH

total error is dominated by the first term and is much larger than the datat error. For larger *α*, the first term on the right-hand side is smaller but the second term will be larger. Hence certain strategy for choosing *α* must be developed to balance the effects of the two factors. The discrepancy principle proposed by Morozov ([6], [7]) which based on the consideration that a problem cannot be solved more accurate than the data error, is a natural *a posteriori* strategy

�*AR<sup>α</sup> <sup>f</sup> <sup>δ</sup>* <sup>−</sup> *<sup>f</sup> <sup>δ</sup>*� <sup>=</sup> *γδ*

for some prescribed *γ* ≥ 1. It can be shown that there always exists a smallest *α >* 0 for which the approxiamte solution *f <sup>δ</sup>* satisfies the above condition. (see [3]) Although the existence for a "best" *α* is theoretically assured, there is no obvious way to select it in practice. In most cases, the only way to choose a good parameter is by trial and error. For our purpose, it is sufficient to consider regularization in the Hilbert space setting. We denote by *A*∗ the adjoint operator

**Theorem 4** (Tikhonov Regularization)**.** *Let x and Y be Hilbert spaces. The operator A* : *X* → *Y is*

*αI* + *A*∗*A* : *X* → *X*

2 <sup>√</sup>*<sup>α</sup> .*

Applying the Tikhonov Regularization scheme to find an approximate solution to (9), we

As noted earlier, (8) connects the far-field pattern with the crack and is therefore suitable for the inverse task of finding the crack. It is therefore natural to solve it for the unknown crack when one has the measured far-field data at hand. For this nonlinear problem, Newton's method provides a simple way to handle it. Utilizing the first Frechét derivative which existence is proved in [5], Newton's method approximates the nonlinear equation (8) with a linear one. This means that instead of solving (8), we endeavor to solve the linear equation

for an update *q*. We remark here that the derivative *F*� is still a compact operator, this means that the above equation can't be solved directly. However, we can apply the regularization

<sup>−</sup><sup>1</sup> *A*∗, *α >* 0

(*αI* + *A*∗*A*)*ϕα* = *A*<sup>∗</sup> *f* (11)

(*γ*)*q* = *u*<sup>∞</sup> − *F*(*γ*) (12)

for determining the parameter *α*. More precisely, one looks for an *α >* 0 such that

of *A*. Without going into details, we adapt the following regularization scheme.

*is bijective and has a bounded inverse. Furthermore, if the operator A is injective, then*

*R<sup>α</sup>* := (*αI* + *A*∗*A*)

*F*�

*assumed to be compact and linear. Then for every α >* 0*, the operator*

*describes a regularization scheme with* �*Rα*� ≤ <sup>1</sup>

actually solve the follwing equation

**5. Classical Newton's method**

for *ϕα*.

$$F(\Gamma, \varphi)(\pounds) := \rho \int\_{\Gamma} <\nu(y), \pounds > e^{-i\mathbf{k} < \mathbf{\hat{x}}, y>} \varphi(y) ds(y), \quad \pounds \in \Omega \tag{15}$$

#### 8 Will-be-set-by-IN-TECH 40 Modeling and Measurement Methods for Acoustic Waves and for Acoustic Microdevices Crack Detecting Via Newton's Method <sup>9</sup>

respectively. Now we can consider the following system of operator equations

$$\begin{cases} B(\Gamma, \boldsymbol{\varrho}) = -\frac{\partial \boldsymbol{u}^{\boldsymbol{\ell}}}{\partial \boldsymbol{v}} | \boldsymbol{\Gamma} \\ F(\Gamma, \boldsymbol{\varrho}) = \boldsymbol{u}\_{\infty} \end{cases} \tag{16}$$

with the kernels

(*a*1, *a*2)*<sup>t</sup>*

*a*�

*H*(1) <sup>1</sup> (*k*|Δ*γ*|)

*K*�

*K*�

where

*<sup>K</sup>*0(*τ*, *<sup>σ</sup>*) :<sup>=</sup> *ik*

*<sup>K</sup>*1(*τ*, *<sup>σ</sup>*) :<sup>=</sup> *ik*<sup>2</sup>

but helpful in the theoretical treatment [4].

*A*� <sup>0</sup>(*γ*, *ψ*�

*A*�

*A*�

*K*�

*<* Δ*q*, *γ*�

<sup>1</sup>(*τ*, *<sup>σ</sup>*, *<sup>γ</sup>*; *<sup>q</sup>*) = *ik*<sup>2</sup>

<sup>∞</sup>(*γ*, *<sup>ψ</sup>*; *<sup>q</sup>*) =

(*γ*; *<sup>q</sup>*) = <sup>−</sup>2*ik*

rules of the Hankel functions *H*�

<sup>4</sup> *<sup>H</sup>*(1)�

<sup>4</sup> *<sup>&</sup>lt; <sup>γ</sup>*�

*K*∞(*x*ˆ, *σ*) :=*< n*(*σ*), *x*ˆ *> e*

<sup>0</sup> (*k*|*γ*(*τ*) − *γ*(*σ*)|)

(*σ*) *> <sup>H</sup>*(1)

−*ik<γ*(*σ*),*x*ˆ*>*

make an odd extension of our system (16) to the closed interval [0, 2*π*] which is not necessary

It is noted that all the operators are nonlinear with respect to *γ*. Since our numerical method will be based on the Newton's method, we need also the Fréchet derivatives of the operators

by the Fréchet derivatives of their kernels (see [14]). For brevity, we set **<sup>a</sup>**<sup>⊥</sup> = (*a*2, <sup>−</sup>*a*1)*<sup>t</sup>* if **<sup>a</sup>** <sup>=</sup>

. Also we write Δ*γ* = *γ*(*τ*) − *γ*(*σ*) and Δ*q* = *q*(*τ*) − *q*(*σ*) . Using the differentiation

<sup>0</sup>(*τ*, *σ*, *γ*; *q*)*ψ*�

(*τ*)⊥, *d >* +*ik < q*(*τ*), *d >< n*(*τ*), *d >*

(*τ*) *>*

(*σ*) *>* + *< γ*�

(*σ*) *> <sup>H</sup>*(1)

<sup>|</sup>Δ*γ*<sup>|</sup> <sup>−</sup> <sup>2</sup>

(*τ*), *γ*�

(*σ*)⊥, *x*ˆ *>* −*ik < q*(*σ*), *x*ˆ *>< n*(*σ*), *x*ˆ *>*

*<* Δ*γ*,Δ*q ><* Δ*γ*, *γ*�

<sup>0</sup>(*z*) = −*H*1(*z*) and (*zH*1(*z*))� = *zH*0(*z*), we have

<sup>1</sup>(*τ*, *σ*, *γ*; *q*)*ψ*(*σ*) sin *τ* sin *σdσ*

<sup>∞</sup>(*γ*, *ψ*; *q*)*ψ*(*σ*) sin *σdσ*

(*σ*) sin *τdσ*

 *ui*

(*τ*) *>* <sup>|</sup>Δ*γ*|<sup>2</sup> *kH*(1)

*<* Δ*γ*,Δ*q ><* Δ*γ*, *γ*�

(*τ*), *q*�

<sup>1</sup> (*k*|Δ*γ*|)


(*σ*) *>*)*H*(1)

 *e* (*γ*(*τ*), *d*) sin *τ*

<sup>0</sup> (*k*|Δ*γ*|) +

(*τ*) *>*

<sup>0</sup> (*k*|Δ*γ*|)

<sup>−</sup>*ik<γ*(*σ*),*x*ˆ*<sup>&</sup>gt;* sin *τ*

*<* Δ*γ*,Δ*q >* |Δ*γ*|

(*τ*), *γ*�

and *γ*(*τ*) := (*z* ◦ cos)(*τ*), *ψ*(*τ*) = sign(*π* − *τ*)((*ϕ* ◦ cos)(*τ*)), *n* = (*ν* ◦ *γ*) · |*γ*�

w.r.t. the boundary *γ*. The Fréchet derivatives of the integral operators *A*�

; *<sup>q</sup>*) = <sup>2</sup>*<sup>π</sup>* 0 *K*�

> 0 *K*�

 2*π* 0 *K*�

4

(*τ*), *γ*�

−*k < γ*�

(*τ*) *>* + *<* Δ*γ*, *q*�

<sup>1</sup>(*γ*, *<sup>ψ</sup>*; *<sup>q</sup>*) = <sup>2</sup>*<sup>π</sup>*

<sup>∞</sup>(*γ*, *ψ*; *q*) = *ρ*

<sup>0</sup>(*τ*, *<sup>σ</sup>*, *<sup>γ</sup>*; *<sup>q</sup>*) = <sup>−</sup>*ik*

*< q*�

4 (*< q*�

*< q*�

*< γ*(*τ*) − *γ*(*σ*), *γ*�

<sup>0</sup> (*k*|*γ*(*τ*) − *γ*(*σ*)|)


(*τ*) *>*

Crack Detecting via Newton's Method 41


*s* are simply given

Note that the operator *B* is just the boundary operator which maps the unknowns to the Neumann boundary data. The operator *F* calculates the far-field pattern. From the viewpoint of the direct problem, this system is just the same as the equations system (5) and (6) if Γ is given in advance.

To solve the corresponding inverse problem, the idea of [12] is to use this same system (16). The reasoning is very simple. If Γ solves the inverse scattering problem, then it follows directly from the solution theory of the direct problem that system (16) is satisfied. Conversely, if the pair (Γ, *ϕ*) solves (16), the first equation of (16) ensures that the total field *u* defined in theorem 1 satisfies the homogeneous Neumann boundary conditions on Γ. The second equation in (16) then ascertains the correct far field pattern for the scattered field *u<sup>s</sup>* . From the uniqueness theorem it follows that Γ is the solution of the inverse problem. We have thus the following main theorem (see [12]).

**Theorem 5.** Γ *is the solution of the inverse problem if and only if* Γ, *ϕ solve the system of nonlinear integral equations* (16)*.*

We note that, mathematically, the system (16) illustrates the spirit of "inverse problems". Indeed, giving the type of the boundary conditions, (16) is just our system of equations for the direct problem which solves the far field pattern of the scattered field if one knows the crack. On the other hand, this very same system is used to find the crack from the knowledge of the far field pattern which is just the solution of the corresponding direct problem. This aspect also demonstrate one of the important features of our method. No new equations are created. What we use to solve the inverse problem is just what we have from the direct problem.

According to theorem 5, our task now is to solve the system (16). After parametrising the above system with the cosine substitution *t* = cos *τ* (see [13]) incorporated to take care of the square root singularities of the solution *u<sup>s</sup>* at the crack tips, the system (16) can be written in the following form

$$A\_0(\gamma, \psi') - A\_1(\gamma, \psi) = a(\gamma) \tag{17a}$$

$$A\_{\infty}(\gamma, \psi) = u\_{\infty} \tag{17b}$$

where

$$\begin{aligned} A\_0(\gamma, \psi') &:= \int\_0^{2\pi} \mathcal{K}\_0(\tau, \sigma) \psi'(\sigma) \sin \tau d\sigma \\ A\_1(\gamma, \psi) &:= \int\_0^{2\pi} \mathcal{K}\_1(\tau, \sigma) \psi(\sigma) \sin \tau \sin \sigma d\sigma \\ A\_\infty(\gamma, \psi) &:= \rho \int\_0^{2\pi} \mathcal{K}\_\infty(\hat{x}, \sigma) \psi(\sigma) \sin \sigma d\sigma \\ a(\gamma) &:= -2ik < n(\tau), d > u^i(\gamma(\tau), d) \sin \tau \end{aligned}$$

with the kernels

8 Will-be-set-by-IN-TECH

*<sup>B</sup>*(Γ, *<sup>ϕ</sup>*) = <sup>−</sup> *<sup>∂</sup>u<sup>i</sup>*

Note that the operator *B* is just the boundary operator which maps the unknowns to the Neumann boundary data. The operator *F* calculates the far-field pattern. From the viewpoint of the direct problem, this system is just the same as the equations system (5) and (6) if Γ is

To solve the corresponding inverse problem, the idea of [12] is to use this same system (16). The reasoning is very simple. If Γ solves the inverse scattering problem, then it follows directly from the solution theory of the direct problem that system (16) is satisfied. Conversely, if the pair (Γ, *ϕ*) solves (16), the first equation of (16) ensures that the total field *u* defined in theorem 1 satisfies the homogeneous Neumann boundary conditions on Γ. The second

uniqueness theorem it follows that Γ is the solution of the inverse problem. We have thus the

**Theorem 5.** Γ *is the solution of the inverse problem if and only if* Γ, *ϕ solve the system of nonlinear*

We note that, mathematically, the system (16) illustrates the spirit of "inverse problems". Indeed, giving the type of the boundary conditions, (16) is just our system of equations for the direct problem which solves the far field pattern of the scattered field if one knows the crack. On the other hand, this very same system is used to find the crack from the knowledge of the far field pattern which is just the solution of the corresponding direct problem. This aspect also demonstrate one of the important features of our method. No new equations are created. What we use to solve the inverse problem is just what we have from the direct problem.

According to theorem 5, our task now is to solve the system (16). After parametrising the above system with the cosine substitution *t* = cos *τ* (see [13]) incorporated to take care of the square root singularities of the solution *u<sup>s</sup>* at the crack tips, the system (16) can be written in

*K*0(*τ*, *σ*)*ψ*�

(*σ*) sin *τdσ*

(*γ*(*τ*), *d*) sin *τ*

*K*1(*τ*, *σ*)*ψ*(*σ*) sin *τ* sin *σdσ*

*K*∞(*x*ˆ, *σ*)*ψ*(*σ*) sin *σdσ*

) − *A*1(*γ*, *ψ*) = *a*(*γ*) (17a) *A*∞(*γ*, *ψ*) = *u*<sup>∞</sup> (17b)

*A*0(*γ*, *ψ*�

 2*π* 0

 2*π* 0

> 2*π* 0

*<sup>a</sup>*(*γ*) :<sup>=</sup> <sup>−</sup>2*ik <sup>&</sup>lt; <sup>n</sup>*(*τ*), *<sup>d</sup> <sup>&</sup>gt; <sup>u</sup><sup>i</sup>*

) :=

*A*0(*γ*, *ψ*�

*A*1(*γ*, *ψ*) :=

*A*∞(*γ*, *ψ*) := *ρ*

equation in (16) then ascertains the correct far field pattern for the scattered field *u<sup>s</sup>*

*F*(Γ, *ϕ*) = *u*<sup>∞</sup>

*∂ν* |Γ

(16)

. From the

respectively. Now we can consider the following system of operator equations

given in advance.

following main theorem (see [12]).

*integral equations* (16)*.*

the following form

where

$$\begin{aligned} K\_0(\tau, \sigma) &:= \frac{ik}{4} H\_0^{(1)'} (k |\gamma(\tau) - \gamma(\sigma)|) \frac{<\gamma(\tau) - \gamma(\sigma), \gamma'(\tau)>}{|\gamma(\tau) - \gamma(\sigma)|} \\\\ K\_1(\tau, \sigma) &:= \frac{ik^2}{4} < \gamma'(\tau), \gamma'(\sigma) > H\_0^{(1)} (k |\gamma(\tau) - \gamma(\sigma)|) \\\\ K\_\infty(\pounds, \sigma) &:= < \eta(\sigma), \pounds > e^{-ik < \gamma(\sigma), \pounds\flat} \end{aligned}$$

and *γ*(*τ*) := (*z* ◦ cos)(*τ*), *ψ*(*τ*) = sign(*π* − *τ*)((*ϕ* ◦ cos)(*τ*)), *n* = (*ν* ◦ *γ*) · |*γ*� |. Note that we make an odd extension of our system (16) to the closed interval [0, 2*π*] which is not necessary but helpful in the theoretical treatment [4].

It is noted that all the operators are nonlinear with respect to *γ*. Since our numerical method will be based on the Newton's method, we need also the Fréchet derivatives of the operators w.r.t. the boundary *γ*. The Fréchet derivatives of the integral operators *A*� *s* are simply given by the Fréchet derivatives of their kernels (see [14]). For brevity, we set **<sup>a</sup>**<sup>⊥</sup> = (*a*2, <sup>−</sup>*a*1)*<sup>t</sup>* if **<sup>a</sup>** <sup>=</sup> (*a*1, *a*2)*<sup>t</sup>* . Also we write Δ*γ* = *γ*(*τ*) − *γ*(*σ*) and Δ*q* = *q*(*τ*) − *q*(*σ*) . Using the differentiation rules of the Hankel functions *H*� <sup>0</sup>(*z*) = −*H*1(*z*) and (*zH*1(*z*))� = *zH*0(*z*), we have

$$A\_0'(\gamma, \psi'; q) = \int\_0^{2\pi} K\_0'(\tau, \sigma, \gamma; q) \psi'(\sigma) \sin \tau d\sigma$$

$$A\_1'(\gamma, \psi; q) = \int\_0^{2\pi} K\_1'(\tau, \sigma, \gamma; q) \psi(\sigma) \sin \tau \sin \sigma d\sigma$$

$$A\_\infty'(\gamma, \psi; q) = \rho \int\_0^{2\pi} K\_\infty'(\gamma, \psi; q) \psi(\sigma) \sin \sigma d\sigma$$

$$A(\gamma; q) = -2ik \left\{ < q'(\tau)^\perp, d> + ik < q(\tau), d> < n(\tau), d> \right\} u^i(\gamma(\tau), d) \sin \tau$$

where

*a*�

$$K\_0'(\mathbf{r}, \sigma, \gamma; q) = -\frac{i\mathbf{k}}{4} \left\{ \frac{<\Delta\gamma, \Delta q > <\Delta\gamma, \gamma'(\tau)>}{|\Delta\gamma|^2} k H\_0^{(1)}(\mathbf{k}|\Delta\gamma|) + \frac{1}{2} \frac{\gamma}{|\Delta\gamma|^2} \left( \frac{1}{2} \frac{\gamma}{|\Delta\gamma'|} + \frac{1}{2} \frac{\gamma}{|\Delta\gamma'|} \right) \right\}$$

$$H\_1^{(1)}(\mathbf{k}|\Delta\gamma) \left\{ \frac{<\Delta q, \gamma'(\tau)>+<\Delta\gamma, q'(\tau)>-2}{|\Delta\gamma|} - \frac{<\Delta\gamma, \Delta q > <\Delta\gamma, \gamma'(\tau)>}{|\Delta\gamma|^3} \right\}$$

$$K\_1'(\mathbf{r}, \sigma, \gamma; q) = \frac{i\mathbf{k}^2}{4} \left\{ (+<\gamma'(\tau), q'(\sigma)>) H\_0^{(1)}(\mathbf{k}|\Delta\gamma) \right\}$$

$$-k < \gamma'(\tau), \gamma'(\sigma)>H\_1^{(1)}(\mathbf{k}|\Delta\gamma) \right\} \xrightarrow{\Delta\gamma, \Delta q > \Delta q} \left\{ \frac{\Delta\gamma, \Delta q >}{|\Delta\gamma|} \right\}$$

$$K\_{\infty}'(\gamma, \eta; q) = \left\{  -i\mathbf{k} < q(\sigma), \hat{\mathbf{x}} > \eta(\sigma), \hat{\mathbf{x}} > \big{) \right\} e^{-i\mathbf{k} < \gamma(\sigma), \Delta\gamma} \text{ sinc } \mathbf{\hat{x}}$$

#### 10 Will-be-set-by-IN-TECH 42 Modeling and Measurement Methods for Acoustic Waves and for Acoustic Microdevices Crack Detecting Via Newton's Method <sup>11</sup>

Therefore, to solve the parametrized system (17), Newton's method suggests the solving of the following system of linear approximations

$$\begin{split} A\_0(\gamma, \psi') + A\_0(\gamma, \chi') + A\_0'(\gamma, \psi'; q) - A\_1(\gamma, \psi) - A\_1(\gamma, \chi) - A\_1'(\gamma, \psi; q) \\ = a(\gamma) + a'(\gamma; q) \end{split} \tag{18a}$$

$$A\_{\infty}(\gamma,\psi) + A\_{\infty}(\gamma,\chi) + A\_{\infty}'(\gamma,\psi;q) = \mathfrak{u}\_{\infty} \tag{18b}$$

Numerically this system is solved iteratively: with a current approximation (*ψ*, *γ*), the system (18) has to be solved for the pair (*χ*, *q*). The updated data are then given by *ψ* + *χ* for the density function of the integral operators and *γ* + *q* for the unknown crack. For brevity, we rewrite the system (18) in the operator form:

$$\mathcal{A}(\chi, q) = \Upsilon \tag{19}$$

of the crack. Note that this step is nonlinear and ill-posed. With slightly abused notations, we

Note that this system is formally the same as system (17) except that the first variable is fixed in each single equation. That is, *γ* is fixed in (21a) while *ψ* is fixed in (21b). In the actual numerical computation, we have to solve a regularized version of (21b). We therefore solve

<sup>∞</sup>(*ψ*; *γ*))*q* = *A*�∗

) − *A*1(*γ*; *ψ*) = *a*(*γ*) (21a) *A*∞(*ψ*; *γ*) = *u*<sup>∞</sup> (21b)

<sup>∞</sup>(*ψ*; *γ*)(*u*<sup>∞</sup> − *A*∞(*ψ*; *γ*)) (21b�

Crack Detecting via Newton's Method 43

) − *A*1(*γn*; *ψ*) = *a*(*γn*) (22)

*γn*+<sup>1</sup> = *γ<sup>n</sup>* + *qn* (24)

<sup>∞</sup>(*ψ*; *γn*)(*u*<sup>∞</sup> − *A*∞(*ψ*; *γn*)) (23)

)

have thus splitted the system (16) into the following two separate steps:

*A*0(*γ*; *ψ*�

(*αI* + *A*�∗

1. Given an initial guess *γ*<sup>0</sup> for the unknown crack.

(*αI* + *A*�∗

3. Stopping criterion: Discrepancy principle

smaller parts and thus makes the computation cheaper.

<sup>∞</sup>(*ψ*; *γn*)*A*�

2. iterative steps, for *n* = 0, 1, 2, . . .

for the update *qn* of *γn*. (c) Update in each step

(a) Step 1: solve

**8. Numerical results**

for *ψ*. (b) Step 2: solve

instead of (21b).

<sup>∞</sup>(*ψ*; *γ*)*A*�

The algorithm for the two-steps method can be summarized as follows:

*A*0(*γn*; *ψ*�

<sup>∞</sup>(*ψ*; *γn*))*qn* = *A*�∗

At this place we comment that from the numerical point of view this scheme is very attractive. As a variant of the nonlinear integral equations method, the Fréchet derivative of the integral operator is computed directly by solving the integral equation (23). Besides, the derivative is very simple as compared to those in the last section, since only the far-field operator which has a smooth kernel is to be differentiated. Numerically it can be easily solved via the rectangular rule, for example. Another advantage of this method is that it splits the problem into two

In this section we will demonstrate the applicability of the two-steps method via some examples. One can compare the results with those from the other methods in [5] and [12]. Using just one incident wave, we reconstruct the unknown crack from the measured far-field pattern at 16 points which are evenly distributed on the unit circle. For the direct problem, the forward solver is applied with 63 collocation points. To avoid committing an inverse crime,

Since this linear equation of the first kind is still ill-posed, we have to incorporate some regularization scheme. Instead of solving (19) directly, we solve the following regularized equation

$$
\mathcal{A}\left(\begin{bmatrix} \alpha I & 0\\ 0 & \beta I \end{bmatrix} + \mathcal{A}^\* \mathcal{A}\right)\begin{bmatrix} \chi\\ q \end{bmatrix} = \mathcal{A}^\* \mathcal{Y} \tag{20}
$$

with two to be determined regularization parameters *α* and *β*.

The major advantage of this method as compared to the classical Newton's method is that the Fréchet derivatives of the integral operators can be easily computed. They are obtained simply by solving the above system which is nothing more than the Gaussian elimination.

#### **7. Two-steps method**

The idea of the nonlinear integral equations method is based on the observation that the inverse problem can be formulated in an equivalent two-by-two system (16) of nonlinear equations. This system is then treated as a coupled system of two variables (*γ*, *ψ*). In the regularized version (20), two parameters (*α*, *β*) are therefore needed at the same time. Since these parameters are in general selected by trial and error, it is rather complicated. However, the system (16) can be solved in another way. Inspired from the solution theory of the direct problem, This system can be seen as the whole procedure for the calculation of the far-field pattern of the scattered field from a given crack resulted from an incident planar wave in two steps. Indeed, the first equation of (16) solves the scattered field from the crack and the second one calculates the far-field pattern from the solution of the first equation. This motivates our inverse scheme: the inverse solver should be *inverse* to the direct solver. There are several possibilities to solve (16) in two steps separately. One can solve the first equation first and the second one next, or the other way round. One has further choice concerning which quantity is to be solved at each individual step. Since our method is based on Newton's iteration method, an initial guess for *γ* or for *ψ* or for both is unavoidable. It is therefore natural to start with a initial guess for the crack and solve the first equation for *ψ*. This is beneficial since this is just the direct problem defined by problem 1. Even better is, theorem 1 ensures the unique solvability of the equation for any crack. Next we can solve the second equation for an update of the crack. Note that this step is nonlinear and ill-posed. With slightly abused notations, we have thus splitted the system (16) into the following two separate steps:

$$A\_0(\gamma; \psi') - A\_1(\gamma; \psi) = a(\gamma) \tag{21a}$$

$$A\_{\infty}(\psi;\gamma) = u\_{\infty} \tag{21b}$$

Note that this system is formally the same as system (17) except that the first variable is fixed in each single equation. That is, *γ* is fixed in (21a) while *ψ* is fixed in (21b). In the actual numerical computation, we have to solve a regularized version of (21b). We therefore solve

$$(aI + A\_{\infty}^{\prime \*} (\psi; \gamma) A\_{\infty}^{\prime} (\psi; \gamma)) q = A\_{\infty}^{\prime \*} (\psi; \gamma) (u\_{\infty} - A\_{\infty} (\psi; \gamma)) \tag{21b'}$$

instead of (21b).

10 Will-be-set-by-IN-TECH

Therefore, to solve the parametrized system (17), Newton's method suggests the solving of

Numerically this system is solved iteratively: with a current approximation (*ψ*, *γ*), the system (18) has to be solved for the pair (*χ*, *q*). The updated data are then given by *ψ* + *χ* for the density function of the integral operators and *γ* + *q* for the unknown crack. For brevity, we

Since this linear equation of the first kind is still ill-posed, we have to incorporate some regularization scheme. Instead of solving (19) directly, we solve the following regularized

> *χ q*

+ A∗A

The major advantage of this method as compared to the classical Newton's method is that the Fréchet derivatives of the integral operators can be easily computed. They are obtained simply by solving the above system which is nothing more than the Gaussian elimination.

The idea of the nonlinear integral equations method is based on the observation that the inverse problem can be formulated in an equivalent two-by-two system (16) of nonlinear equations. This system is then treated as a coupled system of two variables (*γ*, *ψ*). In the regularized version (20), two parameters (*α*, *β*) are therefore needed at the same time. Since these parameters are in general selected by trial and error, it is rather complicated. However, the system (16) can be solved in another way. Inspired from the solution theory of the direct problem, This system can be seen as the whole procedure for the calculation of the far-field pattern of the scattered field from a given crack resulted from an incident planar wave in two steps. Indeed, the first equation of (16) solves the scattered field from the crack and the second one calculates the far-field pattern from the solution of the first equation. This motivates our inverse scheme: the inverse solver should be *inverse* to the direct solver. There are several possibilities to solve (16) in two steps separately. One can solve the first equation first and the second one next, or the other way round. One has further choice concerning which quantity is to be solved at each individual step. Since our method is based on Newton's iteration method, an initial guess for *γ* or for *ψ* or for both is unavoidable. It is therefore natural to start with a initial guess for the crack and solve the first equation for *ψ*. This is beneficial since this is just the direct problem defined by problem 1. Even better is, theorem 1 ensures the unique solvability of the equation for any crack. Next we can solve the second equation for an update

; *q*) − *A*1(*γ*, *ψ*) − *A*1(*γ*, *χ*) − *A*�

(*γ*; *q*) (18a)

<sup>∞</sup>(*γ*, *ψ*; *q*) = *u*<sup>∞</sup> (18b)

A(*χ*, *q*) = Υ (19)

= A∗Υ (20)

<sup>1</sup>(*γ*, *ψ*; *q*)

the following system of linear approximations

= *a*(*γ*) + *a*�

rewrite the system (18) in the operator form:

) + *A*0(*γ*, *χ*�

*A*∞(*γ*, *ψ*) + *A*∞(*γ*, *χ*) + *A*�

) + *A*�

 *αI* 0 0 *βI*

with two to be determined regularization parameters *α* and *β*.

<sup>0</sup>(*γ*, *ψ*�

*A*0(*γ*, *ψ*�

equation

**7. Two-steps method**

The algorithm for the two-steps method can be summarized as follows:

	- (a) Step 1: solve

$$A\_0(\gamma\_n; \psi') - A\_1(\gamma\_n; \psi) = a(\gamma\_n) \tag{22}$$

for *ψ*.

(b) Step 2: solve

$$(\mathfrak{a}I + A\_{\infty}^{\prime \*} (\psi; \gamma\_n) A\_{\infty}^{\prime} (\psi; \gamma\_n)) q\_{\mathfrak{n}} = A\_{\infty}^{\prime \*} (\psi; \gamma\_n) (\mathfrak{u}\_{\infty} - A\_{\infty} (\psi; \gamma\_n)) \tag{23}$$

for the update *qn* of *γn*.

(c) Update in each step

$$
\gamma\_{n+1} = \gamma\_n + q\_n \tag{24}
$$

3. Stopping criterion: Discrepancy principle

At this place we comment that from the numerical point of view this scheme is very attractive. As a variant of the nonlinear integral equations method, the Fréchet derivative of the integral operator is computed directly by solving the integral equation (23). Besides, the derivative is very simple as compared to those in the last section, since only the far-field operator which has a smooth kernel is to be differentiated. Numerically it can be easily solved via the rectangular rule, for example. Another advantage of this method is that it splits the problem into two smaller parts and thus makes the computation cheaper.

#### **8. Numerical results**

In this section we will demonstrate the applicability of the two-steps method via some examples. One can compare the results with those from the other methods in [5] and [12]. Using just one incident wave, we reconstruct the unknown crack from the measured far-field pattern at 16 points which are evenly distributed on the unit circle. For the direct problem, the forward solver is applied with 63 collocation points. To avoid committing an inverse crime,

#### 12 Will-be-set-by-IN-TECH 44 Modeling and Measurement Methods for Acoustic Waves and for Acoustic Microdevices Crack Detecting Via Newton's Method <sup>13</sup>

the number of collocation points used in the inverse problem is chosen to be different from that of the forward solver. We choose 31 collocation points in the inverse algorithm. In all our examples, we take the incident plane wave coming from the direction *d* = √ 1 2 (1, 1).

For the reconstruction, we take the k-th Chebyshev monomials *Tk*(*x*) = cos(*k* cos−<sup>1</sup> *x*), *k* = 1, . . . , *m* as our basis functions. The selection of the Chebyshev polynomials is based on the fact that they can take care of the square root singularity in their own right. The updates for the crack in each iterative step can be written in the form

$$q(\boldsymbol{x}) = \left(\sum\_{k=0}^{m-1} b\_k^1 T\_k(\boldsymbol{x}), \sum\_{k=0}^{m-1} b\_k^2 T\_k(\boldsymbol{x})\right)$$

−1 −0.8 −0.6 −0.4 −0.2 <sup>0</sup> 0.2 0.4 0.6 0.8 <sup>1</sup> −1

belong to our solution space. To this end, we take the arc

*π*

−1 −0.8 −0.6 −0.4 −0.2 <sup>0</sup> 0.2 0.4 0.6 0.8 <sup>1</sup> −1

Crack Detecting via Newton's Method 45

(b) 3% error, *N* = 9

<sup>2</sup> *<sup>x</sup>*)), *<sup>x</sup>* <sup>∈</sup> [−1, 1]

−1 −0.8 −0.6 −0.4 −0.2 <sup>0</sup> 0.2 0.4 0.6 0.8 <sup>1</sup> −1

(b) 3% error, *N* = 4

−0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1

From the numerical results, we see that the reconstructions for exact data are very good. In

**Example 2.** To demonstrate the benefit of our method, we choose a curve which does not

<sup>2</sup> *<sup>x</sup>*) <sup>−</sup> 0.1 cos(

−0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1

We see from the figures 3, 4 that the reconstructions are very good, even in the case of

3*π*

*π*

the case where random errors are present, the reconstructions are not bad at all.

<sup>2</sup> *<sup>x</sup>*) + 0.2 sin(

(a) exact data, *N* = 9

Γ = (*x*, 0.5 cos(

−1 −0.8 −0.6 −0.4 −0.2 <sup>0</sup> 0.2 0.4 0.6 0.8 <sup>1</sup> −1

(a) exact data, *N* = 6

−0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1

**Figure 2.** *k* = 3

.

−0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1

**Figure 3.** *k* = 1

erroneous data.

with to be determined coefficients *b*<sup>1</sup> *<sup>k</sup>* , *<sup>b</sup>*<sup>2</sup> *<sup>k</sup>*, *k* = 0, . . . , *m* − 1. We test our scheme for two different wave numbers, *k* = 1, 3. The dimension of the test space is taken to be 5, i.e., *m* = 5. The starting curve, that is, the initial guess for the regularized Newton's method, is taken to be the straight line *y* = 0 in all examples. According to the discrepancy principle, the stopping criterion for the iterative scheme is given by the relative error

$$\frac{||u\_{\infty} - u\_{\infty, n}||\_2}{||u\_{\infty}||\_2} \le \epsilon^\*$$

which is taken to be 0.001 in case of exact data and 0.03 in the case where 3% data error are present. In all our figures below, the dotted line (blue) represents the initial guess. We denote by the dashed line (red) the true solution and by the solid line (black) the reconstruction.

**Example 1.** For the first example, we take the arc

$$\Gamma = (\mathfrak{x}, 0.4\mathfrak{x}^3 + 0.2\mathfrak{x}^2 - 0.4\mathfrak{x} - 0.2), \quad \mathfrak{x} \in [-1, 1]$$

which is a polynomial. The numerical results are given in the figures 1, 2 for *k* = 1 and *k* = 3, respectively.

**Figure 1.** *k* = 1

**Figure 2.** *k* = 3

.

12 Will-be-set-by-IN-TECH

the number of collocation points used in the inverse problem is chosen to be different from that of the forward solver. We choose 31 collocation points in the inverse algorithm. In all our

For the reconstruction, we take the k-th Chebyshev monomials *Tk*(*x*) = cos(*k* cos−<sup>1</sup> *x*), *k* = 1, . . . , *m* as our basis functions. The selection of the Chebyshev polynomials is based on the fact that they can take care of the square root singularity in their own right. The updates for

wave numbers, *k* = 1, 3. The dimension of the test space is taken to be 5, i.e., *m* = 5. The starting curve, that is, the initial guess for the regularized Newton's method, is taken to be the straight line *y* = 0 in all examples. According to the discrepancy principle, the stopping

> �*u*<sup>∞</sup> − *u*∞,*n*�<sup>2</sup> �*u*∞�<sup>2</sup>

which is taken to be 0.001 in case of exact data and 0.03 in the case where 3% data error are present. In all our figures below, the dotted line (blue) represents the initial guess. We denote by the dashed line (red) the true solution and by the solid line (black) the reconstruction.

<sup>Γ</sup> = (*x*, 0.4*x*<sup>3</sup> <sup>+</sup> 0.2*x*<sup>2</sup> <sup>−</sup> 0.4*<sup>x</sup>* <sup>−</sup> 0.2), *<sup>x</sup>* <sup>∈</sup> [−1, 1]

which is a polynomial. The numerical results are given in the figures 1, 2 for *k* = 1 and *k* = 3,

*m*−1 ∑ *k*=0 *b*2 *<sup>k</sup>Tk*(*x*))

≤

−0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1

*<sup>k</sup>*, *k* = 0, . . . , *m* − 1. We test our scheme for two different

−1 −0.8 −0.6 −0.4 −0.2 <sup>0</sup> 0.2 0.4 0.6 0.8 <sup>1</sup> −1

(b) 3% error, *N* = 10

1 2 (1, 1).

examples, we take the incident plane wave coming from the direction *d* = √

*m*−1 ∑ *k*=0 *b*1 *<sup>k</sup>Tk*(*x*),

*<sup>k</sup>* , *<sup>b</sup>*<sup>2</sup>

the crack in each iterative step can be written in the form

with to be determined coefficients *b*<sup>1</sup>

*q*(*x*)=(

criterion for the iterative scheme is given by the relative error

**Example 1.** For the first example, we take the arc

−1 −0.8 −0.6 −0.4 −0.2 <sup>0</sup> 0.2 0.4 0.6 0.8 <sup>1</sup> −1

(a) exact data, *N* = 11

respectively.

−0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1

**Figure 1.** *k* = 1

From the numerical results, we see that the reconstructions for exact data are very good. In the case where random errors are present, the reconstructions are not bad at all.

**Example 2.** To demonstrate the benefit of our method, we choose a curve which does not belong to our solution space. To this end, we take the arc

$$\Gamma = (\mathfrak{x}, 0.5\cos(\frac{\pi}{2}\mathfrak{x}) + 0.2\sin(\frac{\pi}{2}\mathfrak{x}) - 0.1\cos(\frac{3\pi}{2}\mathfrak{x})), \quad \mathfrak{x} \in [-1, 1]$$

**Figure 3.** *k* = 1

We see from the figures 3, 4 that the reconstructions are very good, even in the case of erroneous data.

−1.5 −1 −0.5 <sup>0</sup> 0.5 <sup>1</sup> 1.5 <sup>2</sup> 2.5 −1.5

−1.5 −1 −0.5 <sup>0</sup> 0.5 <sup>1</sup> 1.5 <sup>2</sup> 2.5 −1.5

Crack Detecting via Newton's Method 47

(b) 3% error, *N* = 21

−1

In this article, we try to detect a planar crack via iterative Newton's method with the knowledge of the measured far-field pattern. By the one to one correspondence of the scattered field and its far-field, the measurements can be taken anywhere outside the crack. The choice of boundary integral equations method reduces the dimension of the problem by one which makes the computation cheaper. The conceptual simplicity and its numerical accuracy make the Newton's method attractive. Besides, the two variants of the Newton's method (nonlinear integral equations method and two-steps method) on one hand simplifiy the calculation of the Fréchet derivatives and on the other hand help clearifying the idea of

−0.5

0

0.5

1

1.5

(a) exact data, *N* = 26

*Department of Mathematics, National Cheng Kung University, Taiwan*

This work is partially supported by the NSC grant NSC-100-2115-M-006-003-MY2.

[1] Keller J B 1976 Inverse problems. *Amer. Math. Monthly* 83 no. 2, 107-118.

[3] Kress R 1999 *Linear Integral Equations* 2nd edn (Berlin: Springer)

sound-hard open arc *Comput. Appl. Math.* 71 343-356

[2] Colton D and Kress R 1998 *Inverse Acoustic and Electromagnetic Scattering Theory* 2nd edn

[4] Mönch L 1996 On the numerical solution of the direct scattering problem for a

−1

**Figure 6.** *k* = 3

**9. Conclusion**

the inverse problem.

**Author details**

**10. References**

**Acknowledgements**

(Berlin: Springer)

Kuo-Ming Lee

−0.5

0

0.5

1

1.5

**Figure 4.** *k* = 3

.

**Example 3.** To demonstrate the applicability of our method, we choose a curve which is not a graph of a function. To this end, we take the arc

**Figure 5.** *k* = 1

We see that in the case *k* = 1, the reconstructions are very good both with exact and erroneous data (Fig 5). In the case of more oscillations (*k* = 3, Fig 6), the result is quite good with respect to the bad initial guess for the iteration. We also note that although in this case, the Fréchet derivative of the far field operator is not injective, the actual reconstruction process is still running without any problem.

14 Will-be-set-by-IN-TECH

−0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1

**Example 3.** To demonstrate the applicability of our method, we choose a curve which is not

3*π*

−1

We see that in the case *k* = 1, the reconstructions are very good both with exact and erroneous data (Fig 5). In the case of more oscillations (*k* = 3, Fig 6), the result is quite good with respect to the bad initial guess for the iteration. We also note that although in this case, the Fréchet derivative of the far field operator is not injective, the actual reconstruction process is still

−0.5

0

0.5

1

1.5

−1 −0.8 −0.6 −0.4 −0.2 <sup>0</sup> 0.2 0.4 0.6 0.8 <sup>1</sup> −1

(b) 3% error, *N* = 10

−1.5 −1 −0.5 <sup>0</sup> 0.5 <sup>1</sup> 1.5 <sup>2</sup> 2.5 −1.5

(b) 3% error, *N* = 26

<sup>4</sup> (*<sup>x</sup>* <sup>+</sup> 4/3))), *<sup>x</sup>* <sup>∈</sup> [−1, 1]

−1 −0.8 −0.6 −0.4 −0.2 <sup>0</sup> 0.2 0.4 0.6 0.8 <sup>1</sup> −1

(a) exact data, *N* = 9

a graph of a function. To this end, we take the arc

−1.5 −1 −0.5 <sup>0</sup> 0.5 <sup>1</sup> 1.5 <sup>2</sup> 2.5 −1.5

(a) exact data, *N* = 35

3*π*

<sup>8</sup> (*<sup>x</sup>* <sup>+</sup> 4/3)), <sup>−</sup> sin(

Γ = (2 sin(

−0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1

**Figure 4.** *k* = 3

−1

**Figure 5.** *k* = 1

running without any problem.

−0.5

0

0.5

1

1.5

.

## **9. Conclusion**

In this article, we try to detect a planar crack via iterative Newton's method with the knowledge of the measured far-field pattern. By the one to one correspondence of the scattered field and its far-field, the measurements can be taken anywhere outside the crack. The choice of boundary integral equations method reduces the dimension of the problem by one which makes the computation cheaper. The conceptual simplicity and its numerical accuracy make the Newton's method attractive. Besides, the two variants of the Newton's method (nonlinear integral equations method and two-steps method) on one hand simplifiy the calculation of the Fréchet derivatives and on the other hand help clearifying the idea of the inverse problem.

## **Author details**

Kuo-Ming Lee *Department of Mathematics, National Cheng Kung University, Taiwan*

## **Acknowledgements**

This work is partially supported by the NSC grant NSC-100-2115-M-006-003-MY2.

### **10. References**

	- [5] Mönch L 1997 On the inverse acoustic scattering problem by an open arc: the sound-hard case *Inverse Problems* 13 1379-1392
	- [6] Morozov V A 1966 On the solution of functional equations by the method of regularization *Soviet Math. Doklady* 7 414-417 (English translation)
	- [7] Morozov V A 1967 Choice of parameter for the solution of functional equations by the regularization method *Soviet Math. Doklady* 8 1000-1003 (English translation)
	- [8] Kress R 1995 Inverse scattering from an open arc *Math. Meth. Appl. Sci.* 18 267-293
	- [9] Kress R and Rundell W 2005 Nonlinear integral equations and the iterative solution for an inverse boundary value problem *Inverse Problems* 21 1207-1223
	- [10] Ivanyshyn O and Kress R 2006 Nonlinear integral equations for solving inverse boundary value problems for inclusions and cracks. *J. Integral Equations and Appl.* 18 13-38
	- [11] Ivanyshyn O and Kress R 2005 Nonlinear Integral Equations in Inverse Obstacle Scattering In *Proceedings of the 7th International Workshop on Mathematical Methods in Scattering Theory and Biomedical Engineering, Nymphaio, Greece*
	- [12] Lee K-M 2006 Inverse scattering via nonlinear integral equations for a Neumann crack *Inverse Problems* 22 1989-2000
	- [13] Yan Y and Sloan I H 1988 On integral equations of the first kind with logarithmic kernels *J. Integral Equations Appl.* 1 549-579
	- [14] Potthast R 1994 Fréchet differentiability of boundary integral operators in inverse acoustic scattering *Inverse Problems* 10 431-447

© 2013 Raitman et al., licensee InTech. This is an open access chapter 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.

© 2013 Raitman et al., licensee InTech. This is a paper 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.

© 2013 The Author(s). Licensee InTech. 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,

**Neutron Diffraction on Acoustic Waves in a** 

The effect of crystal vibrations on the scattering of neutrons and X-rays in single crystals has already been studied for several decades. Crystals subjected to ultrasonic excitations have been investigated for some time by X-ray and neutron diffraction methods. The first X-ray diffraction experiments on oscillating crystals were performed in 1931 [1,2] stimulating large discussion to explain the observed increase in intensity of the Laue spots. Neutron experiments go back to the 60's [3-5]. At present time both neutrons and X-rays have become important tools in observing and understanding time-dependent matter-wave optics [6-8]. And vice versa the studying of the neutron and X-rays scattering in the time- and spacemodulated with acoustic waves condensed matter represents a great interest. Applications from focusing effects [9], monochromators with tunable bandwidths [10-12], the characterization of static but tiny strain fields [13,14] have been discussed as well as fundamental questions about the formation of satellites [8] and its applications, inter-branch scattering, gradient crystal effects and the fundamental difference between neutron- and Xray diffraction found their audience [15,16]. Theoreticians have tried to explain the variety of effects and have predicted even more challenging fields for experimentalists, like, for example, the formation of caustics in Laue diffraction [17]. The studies above all involve the spatial characteristics, but also attempts to investigate the temporal parameter have been made earlier in special cases of X-ray [18,19] and neutron diffraction [20]. Some work has been carried out with modulated ultrasonic waves [21-24] impregnating an artificial time

In the case of neutron scattering the effect of energy exchange between a neutron and an acoustic phonon is observed, which stems from the fact that the neutron velocity is comparable with that of ultrasound wave. In such exchange the amount of the energy transmitted is rather small - for an ultrasound wave with a frequency of 100 MHz it is approximately 400 neV. Studying the efficiency of energy exchange at the diffraction of

and reproduction in any medium, provided the original work is properly cited.

**Perfect and Deformed Single Crystals** 

E. Raitman, V. Gavrilov and Ju. Ekmanis

http://dx.doi.org/10.5772/55064

structure to the carrier wave.

**1. Introduction** 

Additional information is available at the end of the chapter
