**3. Acoustic field**

The wave equation for acoustic pressure P*<sup>a</sup> r* !, t � � in a continuous inhomogeneous motionless and stationary medium is well known [16, 17]

$$\frac{1}{\mathbf{c}^2 \left(\overrightarrow{r}\right)} \frac{\partial^2}{\partial \mathbf{t}^2} \mathbf{P}\_a \left(\overrightarrow{r}, \mathbf{t}\right) + \Delta \mathbf{P}\_a \left(\overrightarrow{r}, \mathbf{t}\right) + \left[\nabla \mathbf{P}\_a \left(\overrightarrow{r}, \mathbf{t}\right) + \overrightarrow{\mathbf{f}}\left(\overrightarrow{r}, \mathbf{t}\right)\right] \nabla \ln \rho\_0 \left(\overrightarrow{r}\right) = \nabla \overrightarrow{f}\left(\overrightarrow{r}, \mathbf{t}\right). \tag{15}$$

where *f* ! *r* !, t � � is the density of volumetric external forces that are the source of the acoustic field.

Eq. (1) is obtained by excluding the particle velocity vector from the linearized Euler equations, continuity and state of the medium. If we exclude acoustic pressure from these equations, then we get the equation for the vector of the particle velocity of the acoustic field. For this, we differentiate the equation of state in taking into account the smallness of the acoustic pressure, perturbation of the density of the medium by the *ρ<sup>a</sup> r* !, t � � in comparison with the background values *ρ*<sup>0</sup> *r* !� � and P0 *<sup>r</sup>* !� � the medium. In the inhomogeneous medium, the equation of state P*<sup>c</sup> ρ r* !, t h i � � describes the relationship of the instantaneous local value of pressure and density of the medium. Therefore, it is necessary to use the total time derivative when differentiating the equation of state. Using it, we find in the linear approximation

$$\frac{\mathbf{d}}{\partial \mathbf{t}} \mathbf{P}\_c \left[ \rho \left( \vec{r}, \mathbf{t} \right) \right] = \mathbf{c}^2 \left( \vec{r}, \right) \left[ \frac{\partial}{\partial \mathbf{t}} \rho\_a \left( \vec{r}, \mathbf{t} \right) + \nabla \rho\_0 \left( \vec{r}, \right) \vec{V} \left( \vec{r}, \mathbf{t} \right) \right], \tag{16}$$

$$\begin{aligned} \text{where } & \mathcal{C}\left(\overrightarrow{r}, \textbf{}\right) = \sqrt{\frac{\rho\_{r}[\rho\left(\overrightarrow{r}, \textbf{t}\right)]}{\rho\left(\overrightarrow{r}, \textbf{t}\right)}} \text{ is the local phase speed of sound for the acoustic pressure wave. We used the expansion } \rho\left(\overrightarrow{r}, \textbf{t}\right) = \rho\_{0}\left(\overrightarrow{r}, \textbf{t}\right) + \rho\_{a}\left(\overrightarrow{r}, \textbf{t}\right) \text{ and the concentration } \nabla\rho\_{a}\left(\overrightarrow{r}, \textbf{t}\right) = \rho\_{r}\left(\overrightarrow{r}, \textbf{t}\right) + \rho\_{a}\left(\overrightarrow{r}, \textbf{t}\right) \text{ and the concentration } \nabla\rho\_{r}\left(\overrightarrow{r}, \textbf{t}\right) = \rho\_{r}\left(\overrightarrow{r}, \textbf{t}\right) + \rho\_{a}\left(\overrightarrow{r}, \textbf{t}\right) \text{ and } \nabla\rho\_{r}\left(\overrightarrow{r}, \textbf{t}\right) = \rho\_{r}\left(\overrightarrow{r}, \textbf{t}\right) + \rho\_{a}\left(\overrightarrow{r}, \textbf{t}\right) \text{ the equation of} \\ \text{representation, } \mathbf{P}\_{\theta}\left[\rho\left(\overrightarrow{r}, \textbf{t}\right)\right] = \mathbf{P}\_{0}\left(\overrightarrow{r}\right) + \mathbf{P}\_{a}\left(\overrightarrow{r}, \textbf{t}\right) \text{ the equation of} \\ \text{continuity}$$

$$\frac{\rho}{a\theta}\rho\left(\overrightarrow{r}, \textbf{t}\right) + \nabla\left[\rho\left(\overrightarrow{r}, \textbf{t}\right)\vec{V}\left(\overrightarrow{r}, \textbf{t}\right)\right] = 0 \text{ is reduced to a linearized form} \\ 1. \end{aligned}$$

$$\frac{1}{\rho\_0 \left(\overrightarrow{r}\right)} \frac{\partial}{\partial \textbf{t}} \textbf{P} \left(\overrightarrow{r}, \textbf{t}\right) + \nabla V \left(\overrightarrow{r}, \textbf{t}\right) = \textbf{0} \tag{17}$$

$$\frac{1}{c^2(\vec{r})}\frac{\partial^2}{\partial t^2}\vec{V}(\vec{r},\mathbf{t}) - \Delta \vec{V}(\vec{r},\mathbf{t}) - \nabla \ln[\rho\_0(\vec{r})\mathbf{c}^2(\vec{r})] \nabla \vec{V}(\vec{r},\mathbf{t}) - \nabla \times \nabla \times \vec{V}(\vec{r},\mathbf{t}) = 0$$

$$\mathbf{E} = -\frac{1}{\rho\_0(\vec{r})}\frac{\partial}{\partial \vec{r}}\vec{f}\left(\vec{r},\mathbf{t}\right) \tag{18}$$

$$\begin{aligned} \frac{1}{\varepsilon^2 \left( \overrightarrow{r} \right)} \frac{\partial^2}{\partial t^2} \overrightarrow{V} \left( \overrightarrow{r}, \textbf{t} \right) - \Delta \overrightarrow{V} \left( \overrightarrow{r}, \textbf{t} \right) - \nabla \ln \left[ \rho\_0 \left( \overrightarrow{r} \right) \mathbf{c}^2 \left( \overrightarrow{r} \right) \right] \nabla \overrightarrow{V} \left( \overrightarrow{r}, \textbf{t} \right) + \\ + \nabla \times \left( \left( \nabla \ln \rho\_0 \left( \overrightarrow{r} \right) \right) \times \overrightarrow{V} \left( \overrightarrow{r}, \textbf{t} \right) \right) = \mathbf{0} \end{aligned} \tag{19}$$

$$\left| \frac{\nabla \ln \left( \rho\_0 \left( \overrightarrow{r} \right) \mathbf{c}^2 \left( \overrightarrow{r} \right) \right) \nabla \overrightarrow{\mathbf{V}} \left( \overrightarrow{r}, \mathbf{t} \right)}{\nabla \times \left( \left( \nabla \ln \rho\_0 \left( \overrightarrow{r} \right) \right) \times \overrightarrow{\mathbf{V}} \left( \overrightarrow{r}, \mathbf{t} \right) \right)} \right| \sim \left| \frac{\nabla \ln \left( \rho\_0 \left( \overrightarrow{r} \right) \mathbf{c}^2 \left( \overrightarrow{r} \right) \right)}{\nabla \ln \rho\_0 \left( \overrightarrow{r} \right)} \right| \tag{20}$$

### *A Collection of Papers on Chaos Theory and Its Applications*

media, the relative density gradient of the medium can be large. In these regions, the field of the particle velocity vector and acoustic intensity can have a significant rotational (vortex) component.

At present, the solution of these equations is possible only by numerical methods. If the fourth term in the equations is small, the equations for the vector of particle velocity and acoustic pressure allows one to find analytical expressions connecting the phases and moduli of vector of complex intensity and particle velocity vector, pressure, density of acoustic energy with the density of the medium and the speed of sound in it. Let us do it for equation 19. Using both scalar function Ψ *r* !, t � � and vector *<sup>U</sup>* ! *r* !, t � �

$$P\left(\overrightarrow{r},\textbf{t}\right) = \sqrt{\frac{Z\_{\textbf{p}}\left(\overrightarrow{r}\right)}{Z\_{\textbf{p}}^{0}}} \Psi\left(\overrightarrow{r},\textbf{t}\right) \overrightarrow{V}\left(\overrightarrow{r},\textbf{t}\right) = \sqrt{\frac{Z\_{\textbf{v}}\left(\overrightarrow{r}\right)}{Z\_{\textbf{v}}^{0}}} \overrightarrow{U}\left(\overrightarrow{r'},\textbf{t}\right),\tag{21}$$

we rewrite equations (Eqs. (15) and (19)) in the following form:

$$\frac{1}{c^{2}(r)}\frac{\partial^{2}}{\partial t^{2}}\Psi\left(\vec{r},\mathbf{t}\right)-\Delta\Psi\left(\vec{r},\mathbf{t}\right)+\left[\frac{3\nabla\mathcal{Z}\_{\mathbb{P}}\left(\vec{r}\right)\nabla\mathcal{Z}\_{\mathbb{P}}\left(\vec{r}\right)}{4Z\_{\mathbb{P}}^{2}\left(\vec{r}\right)}-\frac{\Delta\mathcal{Z}\_{\mathbb{P}}\left(\vec{r}\right)}{2\mathcal{Z}\_{\mathbb{P}}\left(\vec{r}\right)}\right]\Psi\left(\vec{r},\mathbf{t}\right)=0\tag{22}$$

$$\frac{1}{c^{2}(r)}\frac{\partial^{2}}{\partial t^{2}}\vec{U}\left(\vec{r},\mathbf{t}\right)-\Delta\vec{U}\left(\vec{r},\mathbf{t}\right)+\left[\frac{3\nabla\mathcal{Z}\_{\mathbb{P}}\left(\vec{r}\right)\nabla\mathcal{Z}\_{\mathbb{P}}\left(\vec{r}\right)}{4Z\_{\mathbb{P}}^{2}\left(\vec{r}\right)}-\frac{\Delta\mathcal{Z}\_{\mathbb{P}}\left(\vec{r}\right)}{2\mathcal{Z}\_{\mathbb{P}}\left(\vec{r}\right)}\right]\vec{U}\left(\vec{r},\mathbf{t}\right)=0,\tag{23}$$

where Zp *r* !� � <sup>¼</sup> *<sup>ρ</sup>*<sup>0</sup> *<sup>r</sup>* !� � and Z*<sup>v</sup> <sup>r</sup>* !� � <sup>¼</sup> <sup>1</sup> *ρ*<sup>0</sup> *r* ! ð Þc2 *<sup>r</sup>* ! ð Þ, and Z<sup>0</sup> <sup>p</sup> ¼ *ρ*<sup>0</sup> *r* ! 0 � �, Z0 *<sup>v</sup>* ¼ 1 *ρ*<sup>0</sup> *r* ! ð Þ<sup>0</sup> c2 *<sup>r</sup>* ! ð Þ<sup>0</sup> are values Z*<sup>p</sup> r* !� � and Z*<sup>v</sup> <sup>r</sup>* !� � at some point in space *<sup>r</sup>* ! 0. For the spectral components Ψ *r* !, *ω* � � and *<sup>U</sup>* ! *r* !,*ω* � � using the Fourier transform of equations (Eqs. (22) and (23)) with respect to the time variable, we obtain the following equations

$$
\Delta\Psi\left(\overrightarrow{r},\,\boldsymbol{\alpha}\right) + \mathbf{k}\_{\varphi}^{2}\left(\overrightarrow{r}\right)\Psi\left(\overrightarrow{r},\,\boldsymbol{\alpha}\right) = \mathbf{0},\tag{24}
$$

$$
\Delta \vec{U} \left( \vec{r}, \alpha \right) + \mathbf{k}\_U^2 \left( \vec{r} \right) \vec{U} \left( \vec{r}, \alpha \right) = \mathbf{0}, \tag{25}
$$

$$\begin{aligned} \text{where } \mathbf{k}\_{\psi}^{2}\left(\overrightarrow{\vec{r}}\right) &= \frac{a^{2}}{\mathbf{c}^{2}\left(\overrightarrow{\vec{r}}\right)} - \frac{3}{4} \left[\frac{\nabla\rho\_{0}\left(\overrightarrow{\vec{r}}\right)}{\rho\_{0}\left(\overrightarrow{\vec{r}}\right)}\right]^{2} + \frac{\Delta\rho\_{0}\left(\overrightarrow{\vec{r}}\right)}{2\rho\_{0}\left(\overrightarrow{\vec{r}}\right)} \\\\ \mathbf{k}\_{U}^{2}\left(\overrightarrow{\vec{r}}\right) &= \frac{a^{2}}{\mathbf{c}^{2}\left(\overrightarrow{\vec{r}}\right)} + \frac{5}{4} \left[\frac{\nabla\rho\_{0}\left(\overrightarrow{\vec{r}}\right)}{\rho\_{0}\left(\overrightarrow{\vec{r}}\right)}\right]^{2} + \frac{\nabla\rho\_{0}\left(\overrightarrow{\vec{r}}\right)\nabla\mathbf{c}\left(\overrightarrow{\vec{r}}\right)}{\rho\_{0}\left(\overrightarrow{\vec{r}}\right)\mathbf{c}\left(\overrightarrow{\vec{r}}\right)} \\\\ &+ 3\left[\frac{\nabla\mathbf{c}\left(\overrightarrow{\vec{r}}\right)}{\mathbf{c}\left(\overrightarrow{\vec{r}}\right)}\right]^{2} - \frac{\Delta\rho\_{0}\left(\overrightarrow{\vec{r}}\right)}{\rho\_{0}\left(\overrightarrow{\vec{r}}\right)} - 2\frac{\Delta\mathbf{c}\left(\overrightarrow{\vec{r}}\right)}{\mathbf{c}\left(\overrightarrow{\vec{r}}\right)} \end{aligned} \tag{26}$$

From expressions (Eq. (26)) it follows that the gradient of the speed of sound affects only the vibrational speed in a medium with a small swirl of the particle velocity field. The acoustic pressure depends only on the gradient of the density of *Green's Function Method for Electromagnetic and Acoustic Fields in Arbitrarily… DOI: http://dx.doi.org/10.5772/intechopen.94852*

the medium and does not depend on the gradient of the speed of sound. This situation is valid for media in which the density gradient of the medium is less than the gradient of the speed of sound. These differences form the phase difference between the acoustic pressure and the vibrational velocity vector during the propagation of an acoustic wave in an inhomogeneous medium. Different field reactions *V* ! *r* !, t � � and *P r*!, t � � to the density gradient of the medium and the gradient of the sound speed form the phase difference between the acoustic pressure Φ*<sup>p</sup>* r !, t � � and the particle velocity Φ*<sup>v</sup>* r !, t � � vector during the propagation of an acoustic wave in an inhomogeneous medium. Solutions to equations (Eqs. (24) and (25)) can be found using the method of successive approximations. For this, we represent these equations in the following form:

$$
\Delta\Psi\left(\stackrel{\rightarrow}{r},\stackrel{\rightarrow}{w}\right) + \mathbf{k}\_0^2\Psi\left(\stackrel{\rightarrow}{r},\stackrel{\rightarrow}{w}\right) = \mathbf{k}\_1\mathbf{1}\_\Psi^2\left(\stackrel{\rightarrow}{r}\right)\Psi\left(\stackrel{\rightarrow}{r},\stackrel{\rightarrow}{w}\right) \tag{27}
$$

$$
\Delta \overrightarrow{U} \left( \overrightarrow{r}, \alpha \right) + \mathbf{k}\_0^2 \overrightarrow{U} \left( \overrightarrow{r}, \alpha \right) = \mathbf{k}\_{1v}^2 \left( \overrightarrow{r} \right) \overleftarrow{U} \left( \overrightarrow{r}, \alpha \right) \tag{28}
$$

where k2 <sup>0</sup> <sup>¼</sup> *<sup>ω</sup>*<sup>2</sup> c2 *r* ! ð Þ<sup>0</sup> , k2 <sup>1</sup><sup>Ψ</sup> *r* !� � <sup>¼</sup> k2 <sup>0</sup> � <sup>k</sup><sup>2</sup> *<sup>ψ</sup> r* !� � and k2 <sup>1</sup>*<sup>U</sup> r* !� � <sup>¼</sup> k2 <sup>0</sup> � k2 *<sup>U</sup> r* !� �

Similarly to the case of an electromagnetic field in the inhomogeneous medium, using the scalar G *r* ! � *r* ! 1 � � and vector *<sup>G</sup>* ! *r* ! � *r* ! 1 � � Green's functions of the Helmholtz equation for a homogeneous unbounded medium. Eqs. (2) and (13) and Eq. (2.14) can be rewritten in the form of the following integral equations:

$$\Psi\left(\vec{r},\,\boldsymbol{\omega}\right) = \Psi\_0\left(\vec{r},\,\boldsymbol{\omega}\right) + \int\_{\vec{\Omega}} \mathbf{G}\left(\vec{r} - \vec{r}\_1\right) \mathbf{k}\_{1\Psi}^2\left(\vec{r}\_1\right) \Psi\left(\vec{r}\_1,\,\boldsymbol{\omega}\right) d\vec{r}\_1 \tag{29}$$

$$U\_i(\overrightarrow{r}, \alpha) = U\_{0i}(\overrightarrow{r}, \alpha) + \int\_{\Omega} G\_i(\overrightarrow{r} - \overrightarrow{r}\_1) \mathbf{k}\_{1U}^2(\overrightarrow{r}\_1) U\_i(\overrightarrow{r}\_1, \alpha) d\overrightarrow{r}\_1 \tag{30}$$

Here *Ui r* ! 1,*ω* � � is the projections of the vector onto the coordinate axes and Ψ<sup>0</sup> *r* !, *ω* � � and *<sup>U</sup>*0*<sup>i</sup> <sup>r</sup>* !, *ω* � � are the solutions of equations (Eq. (27)) and (Eq. (28)) with the right-hand side equal to zero, and *Gi r* ! � *r* ! 1 � � are the components of the vector Green's function. The steps for finding a solution to equations (2.15) and (2.16) by the method of successive refinements can be as follows:


4.To obtain the second more accurate approximation, it is necessary to substitute the first approximations in expressions (Eqs. (29) and (30)) and obtain the second more accurate approximation of the solution.

Similarly, you can get solutions that are more accurate. Integration is performed over the volume of the inhomogeneous medium, the inhomogeneities of which will be secondary, etc. field sources. In a real situation, the volume should be chosen such that its secondary sources make a noticeable contribution to the field.

Let us consider an example that shows how the parameters of an inhomogeneous medium affect the characteristics of the acoustic field. We represent the acoustic pressure, and the vector of the particle velocity of the monochromatic acoustic field by the frequency *ω* in the following form

$$\begin{split}P\left(\overrightarrow{r},t\right)&=P\_0\left(\overrightarrow{r}\right)\exp i\left[\alpha t-\Phi\_p\left(\overrightarrow{r}\right)\right] \text{ and } \overrightarrow{\mathbf{V}}\left(\overrightarrow{r},t\right)=\overrightarrow{\mathbf{V}}\_0\left(\overrightarrow{r}\right)\exp i\left[\alpha t-\Phi\_v\left(\overrightarrow{r}\right)\right].\\ \text{Complex intensity vector will be written as }\overrightarrow{\mathbf{I}}\left(\overrightarrow{r}\right)&=P\left(\overrightarrow{r},t\right)\overrightarrow{\mathbf{V}}^\*\left(\overrightarrow{r},t\right)=\end{split}$$

**I** ! <sup>0</sup> *r* !� � exp *<sup>i</sup>* <sup>Φ</sup>*<sup>v</sup> <sup>r</sup>* !� � � <sup>Φ</sup>*<sup>p</sup> <sup>r</sup>* ! h i � � . In a medium without absorption of the acoustic energy, the phases Φ*<sup>p</sup> r* !� � and <sup>Φ</sup>*<sup>v</sup> <sup>r</sup>* !� �, respectively are equal to the phases <sup>Ψ</sup> *<sup>r</sup>* !, t � � and *U* ! *r* !, t � �. The wave vector of a wave is normal to its phase surface and is determined by the wave phase gradient and the wave number by the modulus of this gradient. For the wavenumbers of the pressure k*<sup>p</sup> r* !� � and the particle velocity vector k*<sup>v</sup> r* !� �, we can take, respectively, the quantities *<sup>k</sup><sup>ψ</sup> <sup>r</sup>* !� � and *kU <sup>r</sup>* !� � if the inequalities *Δψ*<sup>0</sup> *<sup>r</sup>* ! ð Þ *ψ*<sup>0</sup> *r* ! ð Þ � � � � � � � � < < *k*<sup>2</sup> *<sup>ψ</sup> r* !� � � <sup>∇</sup>Φ*<sup>p</sup> <sup>r</sup>* ! � � � � <sup>2</sup> � � � � � � � � and *ΔU* ! <sup>0</sup> *r* ! ð Þ *U*<sup>0</sup> *r* ! ð Þ � � � � � � � � < < k<sup>2</sup> *<sup>U</sup> r* !� � � <sup>∇</sup>Φ*<sup>v</sup> <sup>r</sup>* ! � � � � <sup>2</sup> � � � � � � � � . The refractive indices of the medium for the

acoustic pressure and the particle velocity relative to the point *r* ! <sup>0</sup> are different and accordingly, equal *np r* !� � <sup>¼</sup> *kp <sup>r</sup>* ! ð Þ *<sup>k</sup>*<sup>0</sup> and *nv r* !� � <sup>¼</sup> *kv <sup>r</sup>* ! ð Þ *<sup>k</sup>*<sup>0</sup> , where k0 <sup>¼</sup> *<sup>ω</sup>* c *r* ! ð Þ<sup>0</sup> and *C r*! 0 � � is

the phase velocity of sound for the acoustic pressure wave at the point *r* ! 0. The phase velocities of the acoustic pressure wave and the particle velocity vector become different, which leads to the inequality of the phases of the acoustic pressure and the particle velocity vector when the acoustic wave propagates in an inhomogeneous medium. The absorption of acoustic energy by the medium is taken into account by assuming the speed of sound and the density of the medium to be complex quantities, in which the imaginary part is responsible for the absorption of the energy of the acoustic field. In this case, the phases and acquire an additive

equal to the phases of the values ffiffiffiffiffiffiffiffiffiffi Z*<sup>p</sup> r* ! ð Þ Z0 *p* r and ffiffiffiffiffiffiffiffiffiffi Z*<sup>v</sup> r* ! ð Þ Z0 *v* r . To avoid cumbersome expressions, we restrict ourselves to the first approximation. The proposed example is often implemented in real measurements of the characteristics of the acoustic field. In practice, as a rule, the projections of the vibrational velocity vector and, accordingly, the intensity vector are measured in an orthogonal coordinate system, for example, in a Cartesian one. Consider the projection of a vector *U* ! *r* !,*ω* � � on the OX axis. In this case, the projection will be a function of only the x coordinate, and the Y and Z coordinates will act as parameters and determine the straight line parallel to the OX axis, along which the observation point x changes. The wave numbers k2 <sup>1</sup><sup>Ψ</sup> *r* !� � and k1 2 *<sup>U</sup> r* !� �, accordingly, the solutions of equations (Eqs. (29) and (30))

*Green's Function Method for Electromagnetic and Acoustic Fields in Arbitrarily… DOI: http://dx.doi.org/10.5772/intechopen.94852*

depend on the values of these parameters. In fact, we turn to the case of onedimensional propagation of acoustic radiation along the OX axis passing through the point **r**<sup>0</sup> ð Þ *X*0, *Y*<sup>0</sup> ,*Z*<sup>0</sup> . Let us choose Ψ<sup>0</sup> *r* !, *ω* � � and *<sup>U</sup>* ! <sup>0</sup> *r* !,*ω* � � both in the form of plane waves and propagating along the X axis. We can put the moduli of these plane waves <sup>Ψ</sup>0ð Þ *<sup>ω</sup>* and U! <sup>0</sup>ð Þ *ω* equal to the moduli of the acoustic pressure *P*<sup>0</sup> and the component *v* ! *<sup>x</sup>* of the particle velocity vector on the OX axis. In this case, the component of the vector Green's function will be equal to the one-dimensional Green's function

G xð Þ¼ � x1 1 2ik0 exp ik½ � <sup>0</sup>jx � x1j we find the solution corresponding to the first approximation at the point X

$$\begin{split} \Psi(\mathbf{x}, y\_0, z\_0, \omega) &= P\_0 \exp ik\_0 \mathbf{x} + \int\_{-\infty}^{\infty} \mathbf{k}\_{1\mathbf{W}}^2(\mathbf{x}\_1, y\_0, z\_0) \frac{P\_0}{2ik\_0} \exp(ik\_0 \mathbf{x}) d\mathbf{x}\_1 + \\ &+ \int\_{\mathbf{x}}^{\infty} \mathbf{k}\_{1\mathbf{W}}^2(\mathbf{x}\_1, y\_0, z\_0) \frac{P\_0}{2ik\_0} \exp\left(-ik\_0 \mathbf{x} + 2ik\_0 \mathbf{x}\_1\right) d\mathbf{x}\_1 = P\_0 \exp ik\_0 \mathbf{x} + \Psi\_1(\mathbf{x}) + \Psi\_2(\mathbf{x}) \\ \overrightarrow{U}\_x(\mathbf{x}, y\_0, z\_0, \omega) &= \overrightarrow{V}\_x \exp ik\_0 \mathbf{x} + \int\_{-\infty}^{\infty} k\_{1\mathbf{U}}^2(\mathbf{x}, y\_0, z\_0) \overrightarrow{\overrightarrow{V}\_x} \frac{\overrightarrow{V}\_x \exp(ik\_0 \mathbf{x})}{2ik\_0} d\mathbf{x}\_1 + \\ &+ \int\_{\mathbf{x}}^{\infty} k\_{1\mathbf{U}}^2(\mathbf{x}\_1, y\_0, z\_0) \frac{\overrightarrow{\overrightarrow{V}}\_x}{2ik\_0} \exp\left(-ik\_0 \mathbf{x} + 2ik\_0 \mathbf{x}\_1\right) d\mathbf{x}\_1 = \overrightarrow{V}\_x \exp ik\_0 \mathbf{x} + \overrightarrow{U}\_1(\mathbf{x}) + \overrightarrow{U}\_2(\mathbf{x}) \end{split} \tag{32}$$

Here *P*<sup>0</sup> exp *ik*0*x* и **U** ! <sup>0</sup> exp *ik*0*x* represent the primary radiation, the second terms are the radiation scattered forward in the region �∞ ≤*x*<sup>1</sup> ≤*x*, and the third terms are the radiation scattered back. Solutions (Eqs. (31) and (32)) and the relations (Eq. (21)) allow us, in the first approximation, to find an expression for the projection of the complex intensity vector on the on the OX axis

$$
\vec{\mathbf{T}}\_{x}(\mathbf{x},\boldsymbol{y}\_{0},\boldsymbol{z}\_{0},\boldsymbol{\omega}) = \sqrt{\frac{\mathbf{Z}\_{p}(\mathbf{x},\boldsymbol{y}\_{0},\boldsymbol{x}\_{0})\mathbf{Z}\_{v}(\mathbf{x},\boldsymbol{y}\_{0},\boldsymbol{z}\_{0})}{\mathbf{Z}\_{p}^{0}\mathbf{Z}\_{v}^{0}}} \begin{pmatrix}
\boldsymbol{P}\_{0}\vec{\mathbf{V}}\_{x} + \boldsymbol{+}\boldsymbol{P}\_{0}\vec{\mathbf{U}}\_{1}^{\ast}(\boldsymbol{\omega}) + \boldsymbol{P}\_{0}\vec{\mathbf{U}}\_{2}^{\ast}(\boldsymbol{\omega}) + \boldsymbol{\Psi}\_{1}(\mathbf{x})\vec{\mathbf{V}}\_{x} + \boldsymbol{+}\boldsymbol{\Psi}\_{2}(\mathbf{x})\vec{\mathbf{U}}\_{2}^{\ast}(\boldsymbol{\omega}) \\
+\boldsymbol{\Psi}\_{1}(\mathbf{x})\vec{\mathbf{U}}\_{1}^{\ast}(\boldsymbol{\omega}) + \boldsymbol{\Psi}\_{1}(\mathbf{x})\vec{\mathbf{U}}\_{2}^{\ast}(\boldsymbol{\omega}) + \boldsymbol{\Psi}\_{2}(\mathbf{x})\vec{\mathbf{V}}\_{x} + \boldsymbol{\Psi}\_{2}(\mathbf{x})\vec{\mathbf{V}}\_{2}^{\ast}(\boldsymbol{\omega}) \\
+\boldsymbol{\Psi}\_{2}(\mathbf{x})\vec{\mathbf{U}}\_{1}^{\ast}(\boldsymbol{\omega}) + \boldsymbol{\Psi}\_{2}(\mathbf{x})\vec{\mathbf{U}}\_{2}^{\ast}(\boldsymbol{\omega})
\end{pmatrix} \tag{33}$$

In this expression, the first term describes the complex intensity vector of the primary radiation, the fifth term corresponds to the forward propagating secondary radiation, and the ninth term corresponds to the backscattered radiation. The other terms describe the mutual energy of the primary and scattered radiation. If the field is measured arriving at a point *x* only from the region, *x*<sup>1</sup> ≤*x* then the dependence of the projection of the complex intensity vector on the OX axis takes the following form:

$$
\vec{\mathbf{I}}\_{\mathbf{x}}(\mathbf{x},\mathbf{y}\_{0},\mathbf{z}\_{0}a) \exp i \Phi(\mathbf{x},\mathbf{y}\_{0},\mathbf{z}\_{0}) = \frac{\mathbf{C}\_{0}(\mathbf{x}\_{0},\mathbf{y}\_{0},\mathbf{z}\_{0})}{\mathbf{C}(\mathbf{x},\mathbf{y}\_{0},\mathbf{z}\_{0})} P\_{0} \vec{\mathbf{V}}\_{\mathbf{x}} \begin{bmatrix} 1 + \frac{i}{2k\_{0}} (a\_{\nu}(\mathbf{x}\_{0},\mathbf{y}\_{0},\mathbf{z}\_{0}) - a\_{p}(\mathbf{x},\mathbf{y}\_{0},\mathbf{z}\_{0})) + \\ + \frac{1}{4k\_{0}^{2}} a\_{\nu}(\mathbf{x},\mathbf{y}\_{0},\mathbf{z}\_{0}) a\_{p}(\mathbf{x},\mathbf{y}\_{0},\mathbf{z}\_{0}) \end{bmatrix} . \tag{34}
$$

In this expression *<sup>α</sup>p*ð Þ¼ *<sup>x</sup>* <sup>Ð</sup> *x* �∞ *k*2 <sup>1</sup>Ψð Þ *<sup>x</sup>*<sup>1</sup> *dx*<sup>1</sup> and *<sup>α</sup>v*ð Þ¼ *<sup>x</sup>* <sup>Ð</sup> *x* �∞ *k*2 <sup>1</sup>*U*ð Þ *x*<sup>1</sup> *dx*1. The modulus **I0 <sup>x</sup>**, *<sup>y</sup>***0**, *<sup>z</sup>***0**, **<sup>ω</sup>** � � and phase **<sup>Ф</sup> <sup>x</sup>**, *<sup>y</sup>***0**, *<sup>z</sup>***0**, **<sup>ω</sup>** � � of the complex of acoustic intensity vector are respectively equal:

$$\begin{split} \mathbf{I}\_{0}(\mathbf{x},\boldsymbol{\omega}) &= \frac{\mathbf{C}\_{0}\mathbf{P}\_{0}|\mathbf{V}\_{\mathbf{x}}|}{4k\_{0}^{2}C(\mathbf{x},\boldsymbol{\chi}\_{0},\boldsymbol{z}\_{0})} \sqrt{\left(4k\_{0}^{2} + a\_{p}(\mathbf{x},\boldsymbol{\chi}\_{0},\boldsymbol{z}\_{0})^{2}\right)\left(4k\_{0}^{2} + a\_{v}(\mathbf{x},\boldsymbol{\chi}\_{0},\boldsymbol{z}\_{0})^{2}\right)}, \Phi(\mathbf{x},\boldsymbol{\chi}\_{0},\boldsymbol{z}\_{0},\boldsymbol{\omega}) \\ &= \operatorname{arrgt} \frac{2k\_{0}\left[a\_{p}(\mathbf{x},\boldsymbol{\chi}\_{0},\boldsymbol{z}\_{0}) - a\_{p}(\mathbf{x},\boldsymbol{\chi}\_{0},\boldsymbol{z}\_{0})\right]}{4k\_{0}^{2} + a\_{v}(\mathbf{x},\boldsymbol{\chi}\_{0},\boldsymbol{z}\_{0})a\_{p}(\mathbf{x},\boldsymbol{\chi}\_{0},\boldsymbol{z}\_{0})}. \end{split} \tag{35}$$

The field intensity vector is

$$\operatorname{Re}\frac{1}{2}\stackrel{\dashv}{\mathbf{I}}\_{\mathbf{x}}(\mathbf{x},\boldsymbol{\mathcal{y}}\_{0},\boldsymbol{z}\_{0},\boldsymbol{w}) = \frac{1}{2}\frac{\mathbf{C}\_{0}}{\mathbf{C}(\mathbf{x},\boldsymbol{\mathcal{y}}\_{0},\boldsymbol{z}\_{0})}P\_{0}\stackrel{\dashv}{\mathbf{V}}\_{\mathbf{x}}\left[1 + \frac{1}{4k\_{0}^{2}}a\_{\boldsymbol{\nu}}(\mathbf{x},\boldsymbol{\mathcal{y}}\_{0},\boldsymbol{z}\_{0})a\_{\boldsymbol{\nu}}(\mathbf{x},\boldsymbol{\mathcal{y}}\_{0},\boldsymbol{z}\_{0})\right],\tag{36}$$

and for the average field energy density we have the following expression:

$$\begin{split} \epsilon(\mathbf{x}, \mathbf{y}\_0, \mathbf{z}\_0 w) &= \frac{P(\mathbf{x}, \mathbf{y}\_0, \mathbf{z}\_0 w) P^\*(\mathbf{x}, \mathbf{y}\_0, \mathbf{z}\_0 w)}{\rho(\mathbf{x}, \mathbf{y}\_0, \mathbf{z}\_0) \mathbf{C}^2(\mathbf{x}, \mathbf{y}\_0, \mathbf{z}\_0)} = \\ &= \frac{P\_0^2}{\rho\_0 \mathbf{C}^2(\mathbf{x}, \mathbf{y}\_0, \mathbf{z}\_0)} \left[ 1 + \frac{1}{4k\_0^2} a\_p(\mathbf{x}, \mathbf{y}\_0, \mathbf{z}\_0) a\_p(\mathbf{x}, \mathbf{y}\_0, \mathbf{z}\_0) \right]. \end{split} \tag{37}$$

From expressions (Eqs. (22) and (23)) it is seen that the inhomogeneous nature of the speed of sound will have a more significant effect on the particle velocity vector than on the acoustic pressure. This makes it possible in principle to create methods for separately measuring the contribution to the acoustic field in an inhomogeneous medium of the density of the medium and the speed of sound in it.

In conclusion, we note that the proposed method makes it possible to analytically and numerically solve the problems of mathematical modeling of a shallow sea, remote sensing of natural media, problems of acoustics of a shallow sea, modeling acoustic and optical metamaterials, etc. Note that for applied problems of acoustics, both fields of the particle velocity vector and the intensity vector in any inhomogeneous medium have a vortex character. Therefore, the algorithms for solving applied problems of ocean and especially shallow sea acoustics, problems of modeling the propagation of acoustic energy in composite media and metamaterials should take into account the vortex component of the vector acoustic field intensity and curvature of the streamlines of the acoustic field.
