**1. Introduction**

The chapter discusses the procedure for using the Green's function for the analytical description of electromagnetic and acoustic fields in a stationary isotropic and arbitrarily inhomogeneous medium. In the case of the electromagnetic field, the wave equation for the electric vector of the electromagnetic field in the inhomogeneous medium with conductivity, dielectric and magnetic permeability is used. In the case of the acoustic field, the wave equation proposed by the author for the vector of particle velocity and the well-known equation for acoustic pressure in an inhomogeneous stationary medium are used. Using the Green's function and the method of successive approximations makes it possible to achieve the required accuracy of calculating the electric and magnetic vectors of the electromagnetic field, as well as to calculate the vectors of complex intensity and intensity, density of energy, acoustic pressure and the particle velocity vector of the acoustic field in media with arbitrary spatial variability of the parameters. The approach used allows one to reduce the problem of solving differential wave equations in an arbitrarily inhomogeneous medium to integration. The chapter is divided into two parts. At the beginning of each part, the corresponding wave equations are derived and

next, a method of using the Green's function is described and analytical expressions describing the fields are formulated. At the beginning we will describe the method as applied to the electromagnetic field, and then as applied to the acoustic field.

Research and modeling of the electromagnetic field in spatially inhomogeneous natural and composite media are actively developing in various fields of science and technology, ranging from systems of underground and underwater electromagnetic communication to photonics, metamaterials and metasurfaces [1]. Such a wide field of scientific research requires methods of mathematical modeling of the properties of the electromagnetic field in media with different spatial scales of conductivity, magnetic and dielectric permittivity. At present, analytical methods are applicable in a very limited range of environments. Among the methods of mathematical modeling of the electromagnetic field in the frequency range from fractions of the hertz to optical, various numerical methods and technologies are used [2, 3]. Numerical modeling uses a variety of methods and technologies, for example, parallel computing which are used in electrodynamic modeling programs. Among them, there are also direct and universal methods for solving boundary problems. The drawback of these methods is a large expenditure of computer resources, which leads to a significant simplification of physical models of the environment and mathematical approximations. There is a third class of methods, in which, at the initial stage, analytical methods are used, for example, the Green's function method, which brings the problem to a form that can be solved by fairly simple numerical methods. Below we will use exactly this approach using the Green's function. Green's function is actively used in a wide range of problems [4–6] of describing electromagnetic and other physical fields in various multilayer, chiral and anisotropic media, including inhomogeneous ones. The proposed procedure is also applicable to media with boundaries and arbitrary dependence on the coordinates of conductivity, magnetic and dielectric permittivity. The source of the field in the environment can be the electric current or an external field. The electric current can be located also inside the medium and outside it. The problem of descriptions the electric vector in an inhomogeneous medium by using the Green's function is formulated as the integral equation with its subsequent solution by the method of successive approximations. This procedure uses the equation for the vector of electric field strength in an inhomogeneous medium, with a certain conductivity, magnetic permeability, and dielectric constant.

The acoustic energy flux density vector (intensity vector), basically, until the beginning of the second half of the 20th century was only of theoretical interest. The second half of the 20th century brought about reliable means of synchronous measurement, practically at a single point, of the acoustic pressure and the components of particle velocity vector necessary to determine the intensity vector of acoustic field [7–14]. However, this did not lead to a significant increase in the number and quality of theoretical research methods and modeling of the intensity vector in an inhomogeneous medium. For the complete theoretical description of the acoustic field, knowledge of its acoustic pressure and the particle velocity vector is required. These two quantities make it possible to find the field of the acoustic intensity vector, to describe the energy and phase structure of the acoustic field. Knowledge of these quantities is useful for solving fundamental and applied problems of acoustic tomography and sounding of the geosphere, applied and fundamental hydroacoustics, creation of acoustic metamaterials, technical and architectural acoustics, noise control, etc. [15–17]. The acoustic pressure *P* ! *<sup>a</sup>* **r** !, *t* and the particle velocity vector V! r !, t are interrelated. This connection is obvious for a plane wave and, in the approximation of a continuous medium, has the

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

#### following form: V! r !, t � � ¼ � <sup>1</sup> *ρ*<sup>0</sup> r ! ð Þ Ð *P* ! *<sup>a</sup>* **r** !, *t* � �*dt* , where *<sup>ρ</sup>*<sup>0</sup> <sup>r</sup> !� � is the density of the medium unperturbed by the acoustic field at the point **r** ! and *t* is time, and ∇ is the Nabla operator. This relationship largely determined the development of the theory of sound as a scalar field of acoustic pressure. Currently, there are several directions for the development of methods of calculation and theoretical analysis of the characteristics of the intensity vector. In the first direction, the relationship between the acoustic pressure and the particle velocity vector is used. This approach is applicable when there are mathematical expressions for the acoustic pressure field. As a rule, this is only possible in a homogeneous medium or for simple waveguides [18]. The second direction requires the use of the continuity equation and the equation of state of the inhomogeneous medium, as well as dynamic equations of motion of elementary volumes or particles of the inhomogeneous medium, for example: the Euler or Navier-Stokes equations. These equations are viewed as a system of equations for determining the pressure and the particle velocity vector. This approach is used to model the propagation of waves in various environments, including plasma and stellar atmospheres [19–22]. These equations are widely known, but to find analytical wave solutions of such systems given an arbitrary dependence of the density and speed of sound on the coordinates is a very difficult task. The use of the acoustic energy transfer equation is the third approach [9]. This approach allows one to describe the energy structure of the acoustic field which makes it possible to study the statistical characteristics of the complex intensity vector in a Gaussian delta-correlated inhomogeneous medium with refraction [9]. It is a very difficult task to find solutions to the transport equation in an inhomogeneous media. In turn, numerical methods for modeling metamaterials and propagation of acoustic waves in a medium are usually limited to specific problems [23–25]. None of the listed approaches, including numerical ones, provides the possibility of a complete theoretical description of the characteristics of the acoustic field and their evolution during field propagation in an arbitrary inhomogeneous medium. One of the promising directions is to use two wave equations in an inhomogeneous medium: equations for the acoustic pressure and equations for the particle velocity vector. We use this very approach. It is based on the proposed by authors wave equation for the particle velocity vector and the well-known equation for acoustic pressure in an inhomogeneous stationary medium. The proposed wave equation for the vector of the particle velocity of the acoustic field in a stationary inhomogeneous and isotropic medium is much more complicated than for the acoustic pressure. This makes it difficult to find the analytical solution for inhomogeneous media with an arbitrary spatial dependence of the density of the medium and the speed of sound in it. However, in an inhomogeneous medium, in which the field of the acoustic intensity vector is weakly vortex, the use of the Green's tensor together with the method of successive approximations makes it possible to find analytical solutions for an arbitrary spatial dependence of the speed of sound and density of the inhomogeneous medium.
