Formulation of a T-Matrix Approach for a Piecewise-Homogeneous Anisotropic Medium Excited by a Spherical Sound Wave ^†

Abstract

A piecewise-homogeneous medium, consisting of anisotropic layers, is excited by a primary spherical sound wave due to a point source lying in the exterior of the medium or in one of its layers. The direct scattering problem is formulated by means of a modified scalar Helmholtz equation incorporating the anisotropic characteristics of the problem. The T-matrix of the problem is determined analytically by means of a suitable coordinate transformation. Then, the acoustic fields in all layers are obtained. Finally, specific reductions to special cases are presented.

1. Introduction

The fundamentals of wave propagation in anisotropic media have been established by the groundbreaking work of Biot in fluid-saturated porous solids [1] and in viscoelasticity [2]. Truesdell has also established the foundations upon which the modeling of anisotropic phenomena in elasticity was built [3]. In electromagnetics, the properties of anisotropy have been widely used in many areas, including the bidomain model for the electrical functionality of the heart [4], extraction of wavenumbers in waveguide propagation [5] and in electrostatics [6,7]. Various aspects of vibroacoustic anisotropy find useful applications in multilayered systems excited by plane waves [8], as well as in determining the velocity properties of 3-D wave propagation in poroelastic media [9]. Moreover, the detection of the acoustics axes with strong anisotropy is a key topic for propagation in crystals [10,11,12]. Furthermore, the effects of the dipolar mean field and the collisions in hydrodynamic Fermi gases can be detected by the anisotropy of the sound velocity that they cause [13]. The effects of acoustic anisotropy have been used to localize plastic deformations in different materials [14,15], while acoustic wave excitation is employed in material science for the alignment and ordering of anisotropic particles [16].

However, besides Biot’s and Truesdell’s works, research on the theoretical framework for acoustic anisotropy seems quite limited, especially concerning problems not related to elasticity. For spherically symmetric elastic media, a rigorous theoretical framework based on the continuum laws of elasticity was developed in [17]. A similar framework for linear acoustics from a mathematical standpoint was developed in [18], by employing a formulation based on the fundamental solution of the modified Helmholtz equation and the integral representations of the involved fields. On the other hand, qualitative identification through near field data for acoustic scattering by anisotropic media was investigated from the topological derivative standpoint in [19].

In this work, we follow the framework of [18], and by exploiting the strong equivalence between the anisotropic norm and the standard norm of ${\mathbb{R}}^3$, we devise an analytical algorithm for the solution of the direct scattering problem. We address the generic problem of a spherical scatterer whose layers possess different anisotropic characteristics. Our approach is based on the fundamental solution of the modified Helmholtz equation (Green’s function of the free anisotropic space) and the expansion of the involved fields with respect to the anisotropic norm. The T-matrix of the problem is derived as a composition of submatrices, each stemming from the anisotropic boundary conditions on the medium’s layers, while expressions for calculating the fields in all layers of the medium are obtained.

The rest of the paper is organized as follows. In Section 2, we present the formulation of the problem, while in Section 3, we formulate a T-matrix approach for the solution of the direct scattering problem concerning the excitation of a multilayered anisotropic scatterer by an internal or external point source. Reductions in the obtained results to the corresponding ones for isotropic media are also pointed out. In Section 4, we present explicit solutions for a homogeneous anisotropic sphere surrounded by a (different) anisotropic exterior. The paper closes in Section 5 with conclusions and future work directions.

2. Mathematical Formulation

We identify the piecewise-homogeneous scatterer V as a bounded and open subset of ${\mathbb{R}}^3$ with ${\mathcal{C}}^2$ boundary $S_1$. The interior of V is divided into P nested annuli-like layers $V_p$ by $P-1$, ${\mathcal{C}}^2$ surfaces $S_p$ for $p=2,\dots ,P$. Each layer $V_p$ is filled with an anisotropic material characterized by the positive-definite dyadic ${\tilde{\mathbf{A}}}_p$, mass density ${\rho }_p$, mean compressibility ${\gamma }_p$, and wavenumber $k_p$. The exterior $V_0$ of the scatterer is characterized by the mass density ${\rho }_0$, mean compressibility ${\gamma }_0$, and wavenumber $k_0$ and has an anisotropy, expressed by a positive-definite dyadic ${\tilde{\mathbf{A}}}_0$. We assume that the dyadics in each layer $V_p$ are represented by diagonal matrices with corresponding eigenvalues ${\lambda }_j^p$ for $j=x,y,z$; i.e.,

$${\tilde{\mathbf{A}}}_p={\lambda }_x^p\widehat{\mathbf{x}}\widehat{\mathbf{x}}+{\lambda }_y^p\widehat{\mathbf{y}}\widehat{\mathbf{y}}+{\lambda }_z^p\widehat{\mathbf{z}}\widehat{\mathbf{z}}=\left[\begin{array}{ccc}{\lambda }_x^p& 0& 0\\ 0& {\lambda }_y^p& 0\\ 0& 0& {\lambda }_z^p\end{array}\right].$$

(1)

Assuming the harmonic time dependence $exp(-\mathrm{i}\omega t)$, the fields $u_p\left(\mathbf{r}\right)$ expressing the spatial excess pressure in $V_p$ satisfy the anisotropic scalar Helmholtz equation (see, e.g., (4) of [18]),

$$\nabla ·{\tilde{\mathbf{A}}}_p·\nabla u_p\left(\mathbf{r}\right)+k_p^2{u}_p\left(\mathbf{r}\right)=0,$$

(2)

while the spatial sound velocity ${\mathbf{v}}_p\left(\mathbf{r}\right)$ is related to the spatial excess pressure field $u_p\left(\mathbf{r}\right)$ by

$${\tilde{\mathbf{A}}}_p·\nabla u_p\left(\mathbf{r}\right)=\mathrm{i}\omega {\rho }_p{\mathbf{v}}_p\left(\mathbf{r}\right).$$

(3)

For detailed formulas of the tensor products involving dyadics, see Appendix A.

The scatterer is excited by a point source located at ${\mathbf{r}}_q\in V_q$, with $q\in \{0,1,\dots ,P\}$ generating a primary field $u_q^{\mathrm{pr}}\left(\mathbf{r}\right)$ of the form

$$u_q^{\mathrm{pr}}\left(\mathbf{r}\right)=\frac{e^{\mathrm{i}{k}_q{d}_q(\mathbf{r},{\mathbf{r}}_q)}}{{\mathrm{d}}_q(\mathbf{r},{\mathbf{r}}_q)\sqrt{\det {\tilde{\mathbf{A}}}_q}},$$

(4)

where ${\mathrm{d}}_q(\mathbf{r},{\mathbf{r}}^{\prime })$ denotes the anisotropic distance of $V_q$, given by

$${\mathrm{d}}_q(\mathbf{r},{\mathbf{r}}^{\prime })=\sqrt{{\tilde{\mathbf{A}}}_q^{-1}:(\mathbf{r}-{\mathbf{r}}^{\prime })(\mathbf{r}-{\mathbf{r}}^{\prime })}.$$

(5)

The anisotropic distance is induced by the following anisotropic norm (see (A4) of Appendix A),

$${\mathrm{d}}_q\left(\mathbf{r}\right)=\sqrt{{\tilde{\mathbf{A}}}_q^{-1}:\mathbf{r}\mathbf{r}}.$$

(6)

We note that primary fields (4) satisfy (2) for $\mathbf{r}\in V_q\setminus \left\{{\mathbf{r}}_q\right\}$.

According to Sommerfeld’s method [20], secondary fields $u_p^{\sec }\left(\mathbf{r}\right)$ are generated due to the interaction of the primary field with the scatterer’s layers. The corresponding total fields $u_p\left(\mathbf{r}\right)$ in $V_p$ are given by

$$u_p\left(\mathbf{r}\right)=u_p^{\mathrm{pr}}\left(\mathbf{r}\right){\delta }_{pq}+u_p^{\sec }\left(\mathbf{r}\right),$$

(7)

where ${\delta }_{pq}$ is the Kronecker delta symbol, and $p=0,1,\dots ,P$. The layer (or the exterior) that contains the point source will be referred to as the excitation layer, whereas the other layers will be referred to as propagation layers.

On the boundaries $S_p$ of the scatterer’s layers, the transmission boundary conditions hold

$$\begin{array}{c}\hfill u_{p-1}\left(\mathbf{r}\right)=u_p\left(\mathbf{r}\right),\mathbf{r}\in S_p,\end{array}$$

(8)

$$\begin{array}{c}\hfill \frac{1}{{\rho }_{p-1}}\frac{\partial }{\partial {\mathbf{n}}_{p-1}}{u}_{p-1}\left(\mathbf{r}\right)=\frac{1}{{\rho }_p}\frac{\partial }{\partial {\mathbf{n}}_p}{u}_p\left(\mathbf{r}\right),\mathbf{r}\in S_p,\end{array}$$

(9)

with $\frac{\partial }{\partial {\mathbf{n}}_p}=\widehat{\mathbf{n}}·{\tilde{\mathbf{A}}}_p·\nabla$, and $\widehat{\mathbf{n}}$ is the respective outward unit normal.

We now present the following useful asymptotic expressions for the anisotropic metric and its norm, as $r=\left|\mathbf{r}\right|\to \infty$ (for detailed derivations, we refer to [18]).

$$\begin{array}{c}\hfill d_0(\mathbf{r},{\mathbf{r}}_0)=d_0\left(\mathbf{r}\right)\left(1-{\left(\frac{d_0\left({\mathbf{r}}_0\right)}{d_0\left({\widehat{\mathbf{r}}}_0\right)}\right)}^2\widehat{\mathbf{r}}·{\widehat{\mathbf{r}}}_0\right)+\mathcal{O}\left(\frac{1}{r}\right),\end{array}$$

(10)

$$\begin{array}{c}\hfill \frac{1}{d_0(\mathbf{r},{\mathbf{r}}_0)}=\frac{1}{r{d}_0\left({\widehat{\mathbf{r}}}_0\right)}+\mathcal{O}\left(\frac{1}{r^2}\right),\end{array}$$

(11)

$$\begin{array}{c}\hfill \frac{1}{{\left(d_0(\mathbf{r},{\mathbf{r}}_0)\right)}^2}=\frac{1}{r^2{\left(d_0\left({\widehat{\mathbf{r}}}_0\right)\right)}^2}+\mathcal{O}\left(\frac{1}{r^3}\right),\end{array}$$

(12)

while for the standard metric of ${\mathbb{R}}^3$, it holds that

$$|\mathbf{r}-{\mathbf{r}}_0|=r-r_0\widehat{\mathbf{r}}·{\widehat{\mathbf{r}}}_0+\mathcal{O}\left(\frac{1}{r}\right).$$

(13)

Then, the total field in the exterior $V_0$ has the asymptotic expression, for $r\to \infty$,

$$u_0\left(\mathbf{r}\right)=g\left(\widehat{\mathbf{r}}\right)h_0\left(k_0{\mathrm{d}}_0\left(\mathbf{r}\right)\right)+\mathcal{O}\left(\frac{1}{r^2}\right),$$

(14)

where $h_0\left(x\right)$ is the 0-th order spherical Hankel function of the first kind, while the fields in $V_0$ satisfy the (anisotropic) Sommerfeld radiation condition [18],

$$d_0\left(\widehat{\mathbf{r}}\right)\frac{\partial }{\partial {\mathbf{r}}_0}{u}_0\left(\mathbf{r}\right)-\mathrm{i}{k}_{0}u\left(\mathbf{r}\right)=\mathcal{O}\left(r^{-2}\right).$$

(15)

For the case where the source is lying at ${\mathbf{r}}_0\in V_0$, the primary far-field pattern$g^{\mathrm{pr}}\left(\widehat{\mathbf{r}}\right)$ is of the form

$$g^{\mathrm{pr}}\left(\widehat{\mathbf{r}}\right)=\frac{\mathrm{i}{k}_0}{\sqrt{\det {\tilde{\mathbf{A}}}_0}}\exp \left\{-\mathrm{i}{k}_0{d}_0\left(\widehat{\mathbf{r}}\right)\right\}.$$

(16)

Furthermore, the scattering cross section $\sigma$ is given by

$$\sigma =\frac{1}{k_0^2}{\int }_{S^2}\frac{|g\left(\widehat{\mathbf{r}}\right){|}^2}{{\left(d_0\left(\widehat{\mathbf{r}}\right)\right)}^3}\mathrm{d}s\left(\widehat{\mathbf{r}}\right),$$

(17)

with $S^2$ as the unit sphere of ${\mathbb{R}}^3$.

3. Exact Solution of the Direct Problem

In this section, we devise an analytical T-matrix algorithm for the exact solution of the direct problem in case of a spherical piecewise-homogeneous anisotropic scatterer with each spherical layer $V_p$ defined by $a_{p+1}<r<a_p$. To our knowledge, this is the first such derivation in acoustics.

3.1. Source Located on the z-Axis

We suppose that a point source located at ${\mathbf{r}}_q$ inside $V_q$ and lying on the z-axis excites the scatterer. The Green’s function $G_q$ of the free anisotropic space filled by a material with wavenumber $k_q$ is of the form [18,21]

$$G_q(\mathbf{r},{\mathbf{r}}^{\prime })=\frac{e^{\mathrm{i}{k}_q{\mathrm{d}}_q(\mathbf{r},{\mathbf{r}}^{\prime })}}{{\mathrm{d}}_q(\mathbf{r},{\mathbf{r}}^{\prime })}=\mathrm{i}{k}_q{h}_0\left(k_q{\mathrm{d}}_q(\mathbf{r},{\mathbf{r}}^{\prime })\right).$$

(18)

Utilizing (10.1.45)–(10.1.46) of [22], taking into account the strong equivalence (see Appendix B) between the anisotropic norm and the standard norm of ${\mathbb{R}}^3$, we arrive at the following expansion of the primary field

$$u_q^{\mathrm{pr}}\left(\mathbf{r}\right)=I_q\left\{\begin{array}{c}\sum _{n=0}^{\infty }(2n+1)h_n\left(k_q{d}_q\left({\mathbf{r}}_q\right)\right)j_n\left(k_q{d}_q\left(\mathbf{r}\right)\right)P_n(cos{\theta }_q\left(\widehat{\mathbf{r}}\right)),d_q\left(\mathbf{r}\right)<d_q\left({\mathbf{r}}_q\right)\hfill \\ \sum _{n=0}^{\infty }(2n+1)j_n\left(k_q{d}_q\left({\mathbf{r}}_q\right)\right)h_n\left(k_q{d}_q\left(\mathbf{r}\right)\right)P_n(cos{\theta }_q\left(\widehat{\mathbf{r}}\right)),d_q\left(\mathbf{r}\right)>d_q\left({\mathbf{r}}_q\right)\hfill \end{array}\right.,$$

(19)

with $I_q=\frac{k_q}{{\sqrt{\det \tilde{\mathbf{A}}}}_q}$ and $j_n$ and $h_n$ denote the n-th order spherical Bessel and Hankel functions, respectively, and $P_n$ denote the Legendre polynomials of order n, while ${\theta }_q\left(\widehat{\mathbf{r}}\right)$ is given by (A9), resulting from the coordinate transformation (A7). We have assumed, without loss of generality, that ${\lambda }_z^p$ is the smallest eigenvalue of the tensor ${\tilde{\mathbf{A}}}_p$ so that the arguments in the Legendre polynomials remain in the interval $[-1,1]$.

In a similar way, the expansion for the secondary field in $V_p$ has the form

$$\begin{array}{cc}\hfill u_p^{\sec }\left(\mathbf{r}\right)=I_q\sum _{n=0}^{\infty }& (2n+1)h_n\left(k_q{d}_q\left({\mathbf{r}}_q\right)\right)\times \hfill \\ \hfill & \left(a_n^p{j}_n\left(k_p{d}_p\left(\mathbf{r}\right)\right)+b_n^p{h}_n\left(k_p{d}_p\left(\mathbf{r}\right)\right)\right)P_n(cos{\theta }_p\left(\widehat{\mathbf{r}}\right)),a_{p+1}<r<a_p,\hfill \end{array}$$

(20)

while $a_n^0=0$, and $b_n^P=0$.

Considering the asymptotic relation $h_n\left(x\right)\to {(-\mathrm{i})}^n{h}_0\left(x\right)$, as $\left|x\right|\to \infty$, we arrive at the following expression of the far-field pattern

$$g\left(\widehat{\mathbf{r}}\right)=I_q\sum _{n=0}^{\infty }{(-\mathrm{i})}^n(2n+1)h_n\left(k_q{d}_q\left({\mathbf{r}}_q\right)\right)b_n^0{P}_n(cos{\theta }_0\left(\widehat{\mathbf{r}}\right)).$$

(21)

Furthermore, we note that it holds that $d_p\left(\mathbf{r}\right)=r{d}_p\left(\widehat{\mathbf{r}}\right)$, with $r=\left|\mathbf{r}\right|$. The surface fields, for $r=a_p$, are given by

$$\begin{array}{cc}\hfill u_p^{\sec }\left(\widehat{\mathbf{r}}\right)=I_q\sum _{n=0}^{\infty }& (2n+1)h_n\left(k_q{d}_q\left({\mathbf{r}}_q\right)\right)\times \hfill \\ \hfill & \left(a_n^p{j}_n(k_p{a}_p{d}_p\left(\widehat{\mathbf{r}}\right)+b_n^p{h}_n(k_p{a}_p{d}_p\left(\widehat{\mathbf{r}}\right)\right)P_n(cos{\theta }_p\left(\widehat{\mathbf{r}}\right)).\hfill \end{array}$$

(22)

To extract the transition matrix, we use (22) and impose the boundary conditions (8) and (9) on each surface $S_p$. Then, we multiply each side of the resulting equations by $P_{n^{\prime }}\left(cos{\theta }_p\left(\widehat{\mathbf{r}}\right)\right)sin\theta$ and integrate over $\theta \in [0,\pi ]$, $\varphi \in [0,2\pi ]$ to obtain

$$\begin{array}{c}\hfill \sum _{n=0}^{\infty }\left(a_n^p{G}_{n{n}^{\prime }}^{pp}+b_n^p{F}_{n{n}^{\prime }}^{pp}\right)=\sum _{n=0}^{\infty }\left(a_n^{p-1}{G}_{n{n}^{\prime }}^{(p-1)p}+b_n^{p-1}{F}_{n{n}^{\prime }}^{(p-1)p}\right),\end{array}$$

(23)

$$\begin{array}{c}\hfill \sum _{n=0}^{\infty }\left(a_n^p{K}_{n{n}^{\prime }}^{pp}+b_n^p{L}_{n{n}^{\prime }}^{pp}\right)=\sum _{n=0}^{\infty }\left(a_n^{p-1}{K}_{n{n}^{\prime }}^{(p-1)p}+b_n^{p-1}{L}_{n{n}^{\prime }}^{(p-1)p}\right),\end{array}$$

(24)

with

$$\begin{array}{c}\hfill G_{n{n}^{\prime }}^{sp}={\int }_0^{2\pi }{\int }_0^{\pi }{j}_n\left(k_s{a}_p{d}_s\left(\widehat{\mathbf{r}}\right)\right)P_n\left(cos{\theta }_s\left(\widehat{\mathbf{r}}\right)\right)P_{n^{\prime }}\left(cos{\theta }_p\left(\widehat{\mathbf{r}}\right)\right)sin\theta \mathrm{d}s\left(\widehat{\mathbf{r}}\right),\end{array}$$

(25)

$$\begin{array}{c}\hfill F_{n{n}^{\prime }}^{sp}={\int }_0^{2\pi }{\int }_0^{\pi }{h}_n\left(k_s{a}_p{d}_s\left(\widehat{\mathbf{r}}\right)\right)P_n\left(\cos {\theta }_s\left(\widehat{\mathbf{r}}\right)\right)P_{n^{\prime }}\left(\cos {\theta }_p\left(\widehat{\mathbf{r}}\right)\right)sin\theta \mathrm{d}s\left(\widehat{\mathbf{r}}\right),\end{array}$$

(26)

$$\begin{array}{c}\hfill K_{n{n}^{\prime }}^{sp}=\frac{1}{{\rho }_s}{\int }_0^{2\pi }{\int }_0^{\pi }{\tilde{j}}_n\left(k_s{a}_p{d}_s\left(\widehat{\mathbf{r}}\right)\right)P_n\left(\cos {\theta }_s\left(\widehat{\mathbf{r}}\right)\right)P_{n^{\prime }}\left(\cos {\theta }_p\left(\widehat{\mathbf{r}}\right)\right)sin\theta \mathrm{d}s\left(\widehat{\mathbf{r}}\right),\end{array}$$

(27)

$$\begin{array}{c}\hfill L_{n{n}^{\prime }}^{sp}=\frac{1}{{\rho }_s}{\int }_0^{2\pi }{\int }_0^{\pi }{\tilde{h}}_n\left(k_s{a}_p{d}_s\left(\widehat{\mathbf{r}}\right)\right)P_n\left(\cos {\theta }_s\left(\widehat{\mathbf{r}}\right)\right)P_{n^{\prime }}\left(\cos {\theta }_p\left(\widehat{\mathbf{r}}\right)\right)sin\theta \mathrm{d}s\left(\widehat{\mathbf{r}}\right),\end{array}$$

(28)

where ${\tilde{f}}_n$ denotes the evaluation of $\frac{\partial }{\partial {\mathbf{n}}_s}{f}_n\left(k_s{d}_s\left(\mathbf{r}\right)\right)$, at $r=a_p$, with $f_n$ as the Bessel or Hankel functions of n-th order. Relations (23) and (24) can be written in a matrix form as follows:

$$\left[\begin{array}{cc}\left[\begin{array}{c}{G}^{pp}\end{array}\right]& \left[\begin{array}{c}{F}^{pp}\end{array}\right]\\ \left[\begin{array}{c}{K}^{pp}\end{array}\right]& \left[\begin{array}{c}{L}^{pp}\end{array}\right]\end{array}\right]\left[\begin{array}{c}{\mathbf{a}}^p\\ {\mathbf{b}}^p\end{array}\right]=\left[\begin{array}{cc}\left[\begin{array}{c}{G}^{(p-1)p}\end{array}\right]& \left[\begin{array}{c}{F}^{(p-1)p}\end{array}\right]\\ \left[\begin{array}{c}{K}^{(p-1)p}\end{array}\right]& \left[\begin{array}{c}{L}^{(p-1)p}\end{array}\right]\end{array}\right]\left[\begin{array}{c}{\mathbf{a}}^{p-1}\\ {\mathbf{b}}^{p-1}\end{array}\right],$$

(29)

with $X^{sp}$ for $X\in \{G,F,K,L\}$ and $s\in \{p-1,p\}$ being infinite-size matrices, whose entries $X_{n{n}^{\prime }}$ are given by (25)–(28), and ${\mathbf{a}}^s,{\mathbf{b}}^s$ are infinite-dimensional vectors containing the coefficients-to-be-determined $a_n^s,b_n^s$; e.g., ${\mathbf{a}}^s=(a_1^s,a_2^s,\dots ,a_n^s,\dots )$. More details about these matrices can be found in Appendix D. Matrix form (29) can be further simplified to

$$\left[\begin{array}{c}{\mathbf{a}}^p\\ {\mathbf{b}}^p\end{array}\right]={\mathbb{T}}^p\left[\begin{array}{c}{\mathbf{a}}^{p-1}\\ {\mathbf{b}}^{p-1}\end{array}\right],$$

(30)

with ${\mathbb{T}}^p$ given by

$${\mathbb{T}}^p={\left(\left[\begin{array}{cc}\left[\begin{array}{c}{G}^{pp}\end{array}\right]& \left[\begin{array}{c}{F}^{pp}\end{array}\right]\\ \left[\begin{array}{c}{K}^{pp}\end{array}\right]& \left[\begin{array}{c}{L}^{pp}\end{array}\right]\end{array}\right]\right)}^{-1}\left[\begin{array}{cc}\left[\begin{array}{c}{G}^{(p-1)p}\end{array}\right]& \left[\begin{array}{c}{F}^{(p-1)p}\end{array}\right]\\ \left[\begin{array}{c}{K}^{(p-1)p}\end{array}\right]& \left[\begin{array}{c}{L}^{(p-1)p}\end{array}\right]\end{array}\right].$$

Given the fact that $a_n^0=0$, Equation (30) applies for all propagation layers prior to the excitation layer $V_q$; i.e., for all $p=1,\dots ,q-1$, this leads us to the following relation

$$\left[\begin{array}{c}{\mathbf{a}}^{q-1}\\ {\mathbf{b}}^{q-1}\end{array}\right]={\mathbb{T}}^{(0\to q-1)}\left[\begin{array}{c}\mathbf{0}\\ {\mathbf{b}}^0\end{array}\right],$$

(31)

with $\mathbf{0}$ the infinite-dimensional vector of zeros and

$${\mathbb{T}}^{(0\to q-1)}={\mathbb{T}}^{q-1}{\mathbb{T}}^{q-2}\cdots {\mathbb{T}}^1.$$

(32)

Next, imposing the boundary conditions (8) and (9) on the “upper” boundary $S_q$ of the excitation layer $V_q$, we obtain

$$\left[\begin{array}{c}{\mathbf{a}}^q\\ {\mathbf{b}}^q\end{array}\right]={\mathbb{T}}^q\left[\begin{array}{c}{\mathbf{a}}^{q-1}\\ {\mathbf{b}}^{q-1}\end{array}\right]+\left[\begin{array}{c}\mathbf{0}\\ \mathbf{q}\end{array}\right],$$

(33)

while for the “lower” boundary $S_{q+1}$ of the excitation layer $V_q$, we obtain

$$\left[\begin{array}{c}{\mathbf{a}}^{q+1}\\ {\mathbf{b}}^{q+1}\end{array}\right]={\mathbb{T}}^{q+1}\left[\begin{array}{c}{\mathbf{a}}^q\\ {\mathbf{b}}^q\end{array}\right]+{\mathbb{T}}^{q+1}\left[\begin{array}{c}\mathbf{1}\\ \mathbf{0}\end{array}\right],$$

(34)

with $\mathbf{q}$ an infinite-dimensional vector with elements $q_n$ given by

$$q_n=-\frac{j_n\left(k_q{d}_q\left({\mathbf{r}}_q\right)\right)}{h_n\left(k_q{d}_q\left({\mathbf{r}}_q\right)\right)},$$

(35)

and $\mathbf{1}$ is an infinite-dimensional vector of ones.

Now, relations (33) and (34), combined with (31), yield

$$\left[\begin{array}{c}{\mathbf{a}}^{q+1}\\ {\mathbf{b}}^{q+1}\end{array}\right]={\mathbb{T}}^{(0\to q+1)}\left[\begin{array}{c}\mathbf{0}\\ {\mathbf{b}}^0\end{array}\right]+{\mathbb{T}}^{q+1}\left[\begin{array}{c}\mathbf{1}\\ \mathbf{q}\end{array}\right].$$

(36)

Implementing the same technique for the rest of the propagation layers $V_p$ below the excitation layer, i.e., for $p=q+2,\dots ,P$, and given the fact that $b_n^P=0$, we finally arrive at

$$\left[\begin{array}{c}{\mathbf{a}}^P\\ \mathbf{0}\end{array}\right]={\mathbb{T}}^{(0\to P)}\left[\begin{array}{c}\mathbf{0}\\ {\mathbf{b}}^0\end{array}\right]+{\mathbb{T}}^{(q\to P)}\left[\begin{array}{c}\mathbf{1}\\ \mathbf{q}\end{array}\right].$$

(37)

The last relation, yields the following expression of the vector ${\mathbf{b}}^0$, containing the determinable coefficients $b_n^0$,

$${\mathbf{b}}^0={\left[\begin{array}{c}{\mathbb{T}}_{22}^{(0\to P)}\end{array}\right]}^{-1}·\left[\begin{array}{c}{\mathbb{T}}_{21}^{(q\to P)}\end{array}\right]·\mathbf{1}+{\left[\begin{array}{c}{\mathbb{T}}_{22}^{(0\to P)}\end{array}\right]}^{-1}·\left[\begin{array}{c}{\mathbb{T}}_{22}^{(q\to P)}\end{array}\right]·\mathbf{q},$$

(38)

with ${\mathbb{T}}_{ij}^{(q\to P)}$ denoting the $ij$-submatrix of matrix ${\mathbb{T}}^{(q\to P)}$.

In the case of external excitation, i.e., $q=0$, and core excitation, i.e., $q=P$, relation (38) takes the following form, respectively,

$$\begin{array}{c}\hfill {\mathbf{b}}^0={\left[\begin{array}{c}{\mathbb{T}}_{22}^{(0\to P)}\end{array}\right]}^{-1}·\left[\begin{array}{c}{\mathbb{T}}_{21}^{(0\to P)}\end{array}\right]·\mathbf{1},\end{array}$$

(39)

$$\begin{array}{c}\hfill {\mathbf{b}}^0={\left[\begin{array}{c}{\mathbb{T}}_{22}^{(0\to P)}\end{array}\right]}^{-1}·\mathbf{q}.\end{array}$$

(40)

In the case when all of the scatterer’s layers and the exterior region are composed by isotropic materials, then the results reduce to the corresponding ones of [23]; see Appendix D.

3.2. Source at an Arbitrary Location

Now, we suppose that our source is lying at an arbitrary location ${\mathbf{r}}_q\in V_q$. Then, by the addition theorem [24] for the Legendre polynomials, the primary field is given by

$$u_q^{\mathrm{pr}}\left(\mathbf{r}\right)=I_q\left\{\begin{array}{c}\sum _{n=0}^{\infty }\sum _{m=-n}^n{(-1)}^m{\mathcal{H}}_{n,m}^q\left({\mathbf{r}}_q\right){\mathcal{J}}_{n,-m}^q\left(\mathbf{r}\right),d_q\left(\mathbf{r}\right)<d_q\left({\mathbf{r}}_q\right)\hfill \\ \sum _{n=0}^{\infty }\sum _{m=-n}^n{(-1)}^m{\mathcal{J}}_{n,-m}^q\left({\mathbf{r}}_q\right){\mathcal{H}}_{n,m}^q\left(\mathbf{r}\right),d_q\left(\mathbf{r}\right)>d_q\left({\mathbf{r}}_q\right),\hfill \end{array}\right.$$

(41)

with ${\mathcal{H}}_{n,m}^p,{\mathcal{J}}_{n,m}^p$ being the anisotropic observation functions, originally introduced in [25] for scattering by an isotropic medium,

$$\begin{array}{c}\hfill {\mathcal{H}}_{n,m}^q\left(\mathbf{r}\right)=h_n\left(k_q{d}_q\left(\mathbf{r}\right)\right)Y_n^m({\theta }_q\left(\widehat{\mathbf{r}}\right),{\varphi }_q\left(\widehat{\mathbf{r}}\right)),\end{array}$$

(42)

$$\begin{array}{c}\hfill {\mathcal{J}}_{n,m}^q\left(\mathbf{r}\right)=j_n\left(k_q{d}_q\left(\mathbf{r}\right)\right)Y_n^m({\theta }_q\left(\widehat{\mathbf{r}}\right),{\varphi }_q\left(\widehat{\mathbf{r}}\right)),\end{array}$$

(43)

where $Y_n^m$ denote the spherical harmonics [24],

$$Y_n^m(\theta ,\varphi )=\sqrt{\frac{2n+1}{4\pi }\frac{(n-m)!}{(n+m)!}}{P}_n^m(cos\theta )e^{\mathrm{i}m\varphi },$$

(44)

and $P_n^m$ denote the associated Legendre functions. The exact expressions of ${\theta }_q\left(\widehat{\mathbf{r}}\right)$ and ${\varphi }_q\left(\widehat{\mathbf{r}}\right)$ are given by (A9) and (A10), respectively.

On the other hand, the secondary fields are given by

$$u_p^{\sec }\left(\mathbf{r}\right)=I_q\sum _{n=0}^{\infty }\sum _{m=-n}^n{(-1)}^m{\mathcal{H}}_{n,-m}^q\left({\mathbf{r}}_q\right)\left(a_n^p{\mathcal{J}}_{n,m}^p\left(\mathbf{r}\right)+b_n^p{\mathcal{H}}_{n,m}^p\left(\mathbf{r}\right)\right),a_{p+1}<r<a_p.$$

(45)

In a similar way to (21), we arrive at the expression of the far-field pattern

$$g\left(\widehat{\mathbf{r}}\right)=I_q\sum _{n=0}^{\infty }\sum _{m=-n}^n{(-1)}^m{(-\mathrm{i})}^n{\mathcal{H}}_{n,-m}^q\left({\mathbf{r}}_q\right)b_n^0{Y}_n^m({\theta }_0\left(\widehat{\mathbf{r}}\right),{\varphi }_0\left(\widehat{\mathbf{r}}\right)).$$

(46)

The unknown coefficients $a_n^p,b_n^p$ are determined by employing a similar procedure as in the case of the z-axis excitation. In particular, we impose Equations (8) and (9) on the surface $S_p$, utilize Equation (45), and then multiply each side of the resulting equations by $Y_{n^{\prime }}^{m^{\prime }}({\theta }_p\left(\widehat{\mathbf{r}}\right),{\varphi }_p\left(\widehat{\mathbf{r}}\right))sin\theta$ and integrate over $\theta \in [0,\pi ],\varphi \in [0,2\pi ]$. In this way, we obtain

$$\begin{array}{c}\hfill \sum _{n=0}^{\infty }\sum _{m=-n}^n\left(a_n^p{G}_{n{n}^{\prime }m{m}^{\prime }}^{pp}+b_n^p{F}_{n{n}^{\prime }m{m}^{\prime }}^{pp}\right)=\sum _{n=0}^{\infty }\sum _{m=-n}^n\left(a_n^{p-1}{G}_{n{n}^{\prime }m{m}^{\prime }}^{(p-1)p}+b_n^{p-1}{F}_{n{n}^{\prime }m{m}^{\prime }}^{(p-1)p}\right),\end{array}$$

(47)

$$\begin{array}{c}\hfill \sum _{n=0}^{\infty }\sum _{m=-n}^n\left(a_n^p{K}_{n{n}^{\prime }m{m}^{\prime }}^{pp}+b_n^p{L}_{n{n}^{\prime }m{m}^{\prime }}^{pp}\right)=\sum _{n=0}^{\infty }\sum _{m=-n}^n\left(a_n^{p-1}{K}_{n{n}^{\prime }m{m}^{\prime }}^{(p-1)p}+b_n^{p-1}{L}_{n{n}^{\prime }m{m}^{\prime }}^{(p-1)p}\right),\end{array}$$

(48)

with

$$\begin{array}{c}\hfill G_{n{n}^{\prime }m{m}^{\prime }}^{sp}={\int }_0^{2\pi }{\int }_0^{\pi }{\mathcal{J}}_{n,m}^s\left(a_p\widehat{\mathbf{r}}\right)Y_{n^{\prime }}^{m^{\prime }}({\theta }_p\left(\widehat{\mathbf{r}}\right),{\varphi }_p\left(\widehat{\mathbf{r}}\right))sin\theta \mathrm{d}s\left(\widehat{\mathbf{r}}\right),\end{array}$$

(49)

$$\begin{array}{c}\hfill F_{n{n}^{\prime }m{m}^{\prime }}^{sp}={\int }_0^{2\pi }{\int }_0^{\pi }{\mathcal{H}}_{n,m}^s\left(a_p\widehat{\mathbf{r}}\right)Y_{n^{\prime }}^{m^{\prime }}({\theta }_p\left(\widehat{\mathbf{r}}\right),{\varphi }_p\left(\widehat{\mathbf{r}}\right))sin\theta \mathrm{d}s\left(\widehat{\mathbf{r}}\right),\end{array}$$

(50)

$$\begin{array}{c}\hfill K_{n{n}^{\prime }m{m}^{\prime }}^{sp}=\frac{1}{{\rho }_s}{\int }_0^{2\pi }{\int }_0^{\pi }{\tilde{\mathcal{J}}}_{n,m}^s\left(a_p\widehat{\mathbf{r}}\right)Y_{n^{\prime }}^{m^{\prime }}({\theta }_p\left(\widehat{\mathbf{r}}\right),{\varphi }_p\left(\widehat{\mathbf{r}}\right))sin\theta \mathrm{d}s\left(\widehat{\mathbf{r}}\right),\end{array}$$

(51)

$$\begin{array}{c}\hfill L_{n{n}^{\prime }m{m}^{\prime }}^{sp}=\frac{1}{{\rho }_s}{\int }_0^{2\pi }{\int }_0^{\pi }{\tilde{\mathcal{H}}}_{n,m}^s\left(a_p\widehat{\mathbf{r}}\right)Y_{n^{\prime }}^{m^{\prime }}({\theta }_p\left(\widehat{\mathbf{r}}\right),{\varphi }_p\left(\widehat{\mathbf{r}}\right))sin\theta \mathrm{d}s\left(\widehat{\mathbf{r}}\right),\end{array}$$

(52)

where ${\tilde{\mathcal{F}}}_{n,m}^s$ denote the evaluation of $\frac{\partial }{\partial {\mathbf{n}}_s}{\mathcal{F}}_{n,m}^s\left(\mathbf{r}\right)$ at $r=a_p$ with ${\mathcal{F}}_{n,m}^s$ as the observation functions. This procedure leads to a similar result to (30), only this time, the matrix composition is different, as elaborated in Appendix D. The rest of the process is almost identical to that in Section 3.1, and the results (38)–(40) hold as well.

Semi-analytical solutions for the direct problems of an electrostatic field propagating through an isotropic medium and perturbed by a homogeneous anisotropic and coated anisotropic dielectric sphere were derived in [6,7] by means of a similar coordinate transformation to the one we considered here. In addition, we note that for a piecewise-homogeneous isotropic sphere, the obtained results for an arbitrarily-located source reduce to the corresponding ones of [26].

4. Homogeneous Anisotropic Scatterers

Next, we turn our focus to the case of $P=1$, i.e., a homogeneous, anisotropic scatterer of radius a, in order to provide the reductions in the results of Section 3.2 in a ready-to-use form.

First, we examine the case of the point source being arbitrarily located in the scatterer’s surrounding medium $V_0$. The primary acoustic field is given by (41), for $q=0$, and the secondary fields by (45), for $p=0$ (external field) and $p=1$ (internal field). Equation (29) takes the following form

$$\left[\begin{array}{cc}\left[\begin{array}{c}{G}^1\end{array}\right]& \left[\begin{array}{c}{F}^1\end{array}\right]\\ \left[\begin{array}{c}{K}^1\end{array}\right]& \left[\begin{array}{c}{L}^1\end{array}\right]\end{array}\right]\left[\begin{array}{c}{\mathbf{a}}^1\\ \mathbf{0}\end{array}\right]=\left[\begin{array}{cc}\left[\begin{array}{c}{G}^0\end{array}\right]& \left[\begin{array}{c}{F}^0\end{array}\right]\\ \left[\begin{array}{c}{K}^0\end{array}\right]& \left[\begin{array}{c}{L}^0\end{array}\right]\end{array}\right]\left[\begin{array}{c}\mathbf{1}\\ {\mathbf{b}}^0\end{array}\right],$$

(53)

where we used the simplified notation $X^s$ instead of $X^{s\to p}$ for the submatrices, with $s=0,1$. After some manipulation, for the unknown coefficients, we obtain

$$\begin{array}{c}\hfill {\mathbf{b}}^0=-{\left[{\left(G^1\right)}^{-1}{F}^0-{\left(K^1\right)}^{-1}{L}^0\right]}^{-1}\left[{\left(G^1\right)}^{-1}{G}^0-{\left(K^1\right)}^{-1}{K}^0\right]·\mathbf{1},\end{array}$$

(54)

$$\begin{array}{c}\hfill {\mathbf{a}}^1={\left[{\left(F^0\right)}^{-1}{G}^1-{\left(L^0\right)}^{-1}{K}^1\right]}^{-1}\left[{\left(F^0\right)}^{-1}{G}^0-{\left(L^0\right)}^{-1}{K}^0\right]·\mathbf{1}.\end{array}$$

(55)

Next, for the case of the point source arbitrarily located in the scatterer’s interior $V_1$, the primary acoustic field is given by (41) for $q=1$, and Equation (29) becomes

$$\left[\begin{array}{cc}\left[\begin{array}{c}{G}^1\end{array}\right]& \left[\begin{array}{c}{F}^1\end{array}\right]\\ \left[\begin{array}{c}{K}^1\end{array}\right]& \left[\begin{array}{c}{L}^1\end{array}\right]\end{array}\right]\left[\begin{array}{c}{\mathbf{a}}^1\\ \mathbf{q}\end{array}\right]=\left[\begin{array}{cc}\left[\begin{array}{c}{G}^0\end{array}\right]& \left[\begin{array}{c}{F}^0\end{array}\right]\\ \left[\begin{array}{c}{K}^0\end{array}\right]& \left[\begin{array}{c}{L}^0\end{array}\right]\end{array}\right]\left[\begin{array}{c}\mathbf{0}\\ {\mathbf{b}}^0\end{array}\right],$$

(56)

which finally yields

$$\begin{array}{c}\hfill {\mathbf{b}}^0={\left[{\left(G^1\right)}^{-1}{F}^0-{\left(K^1\right)}^{-1}{L}^0\right]}^{-1}\left[{\left(G^1\right)}^{-1}{F}^1-{\left(K^1\right)}^{-1}{L}^1\right]·\mathbf{q},\end{array}$$

(57)

$$\begin{array}{c}\hfill {\mathbf{a}}^1=-{\left[{\left(F^0\right)}^{-1}{G}^1-{\left(L^0\right)}^{-1}{K}^1\right]}^{-1}\left[{\left(F^0\right)}^{-1}{F}^1-{\left(L^0\right)}^{-1}{L}^1\right]·\mathbf{q}.\end{array}$$

(58)

5. Conclusions and Future Work Directions

We formulated the direct scattering problem of a piecewise-homogeneous anisotropic scatterer excited by a point source located in the scatterer’s exterior or interior. We utilized the Green’s function of the free anisotropic space, i.e., the fundamental solution of the modified (anisotropic) Helmholtz equation, the strong equivalence between the anisotropic norm and the standard norm of ${\mathbb{R}}^3$, and suitable coordinate transformations. By manipulating the equations stemming from the boundary conditions into a T-Matrix formulation, we derived analytical solutions of the formulated problems for the source lying on the z-axis and for the source arbitrarily placed in ${\mathbb{R}}^3$. We explicitly presented the reductions in the results for an anisotropic homogeneous sphere when it is excited by an external or an internal point source.

In this work, we provided the theoretical framework and the foundations of the T-matrix approach for the examined anisotropic problem. Future work directions concern the numerical implementation of the developed approach in specific scattering scenarios.