Plane Dyadic Wave Scattering by a Small Rigid Body and Cavity in 3D Linear Elasticity

Abstract

In this paper, we study the 3D elastic scattering problem of plane dyadic waves for a rigid body and a cavity in linear elasticity. Initially, for each case, we formulate the direct scattering problem in a dyadic form, and we give the corresponding longitudinal and transverse far-field scattering amplitudes. Due to dyadic formulation of the problems, the main outcome of this paper is to establish the necessary energy considerations as well as to present functionals and formulas for the differential and the scattering cross-section in order to measure the disturbance created by the scatterer to the propagation of the plane dyadic incident field. Further, we assume that our incident field is scattered by a “small” rigid body or cavity and relative results for low-frequency scattering are obtained. Finally, we prove similar corresponding expressions for energy functionals in the far-field region, along with expressions for the differential and the total scattering cross-section, which are recovered as special cases.

1. Introduction

Scattering theory deals with the propagation of an incident wave field in an elastic medium in which there exists a given obstacle connected with the disturbance that influences the wave’s propagation. This field of study encompasses two main aspects: the direct scattering problem and the inverse scattering problem. The direct problem evaluates how various geometrical or physical irregularities affect wave propagation in a known field, while the inverse problem aims to identify the physical or geometrical characteristics of a scattering obstacle. The significance of this approach lies in the fact that the information that can be retrieved originates from the mechanism of wave–obstacle interaction that, in general, does not affect the geometric or the physical traits of the obstacle. Consequently, scattering theory provides the essential instrument for non-invasive assessment in its general sense. Noninvasive medical assessments, RADAR and SONAR technologies, geophysical investigation, and non-invasive industrial testing are just a handful of the fields with broad applications of scattering theory.

A detailed presentation and analysis of the mathematical theory of elasticity in three dimensions were presented by Kupradze in [1]. This excellent bibliographic source includes fundamental solutions, integral representations, uniqueness theorems, and radiation conditions. An additional contribution to the field includes Barrat and Collins’ [2] expressions for far-field patterns and scattering cross-sections for harmonic elastic waves. Further, low-frequency scattering for elastic waves was discussed by Dassios and Kiriaki in [3], whereas a more analytical discussion of this theory for “small” obstacles was presented in [4] (see also the references therein). Concerning now electromagnetic wave scattering, three-dimensional electromagnetic scattering scenarios are addressed in Twersky’s work on dyadic scattering in electromagnetics [5]. The cases concerning scattering of a spherical dyadic field by a small rigid sphere are given in [6], while research on the scattering of dyadic fields by a small spherical cavity is discussed in [7].

At this point, we mention that results for low-frequency approximation in acoustic wave scattering by point sources has been performed in the work by Dassios and his co-workers [8,9,10], whereas for low-frequency electromagnetic scattering, it has been performed in the work by Athanasiadis, Martin and Stratis, in [11,12]. An excellent source on low-frequency methods for three different cases, i.e., acoustics, electromagnetics and elastic wave scattering, can be found in [13].

The paper at hand focuses on the elastic wave scattering problem based on plane dyadic wave propagation in three-dimensional linear elasticity, with disturbances caused by either a rigid scatterer or a cavity within the medium. Our obstacles are considered “small” in the sense that their characteristic dimension is much less than the wave length of the incident field. We study boundary value problems for two cases, one case where the displacement field is nullified at the scatterer’s surface (the rigid case), and the other case where surface interaction occurs at the cavity boundary. Our mathematical representations will be structured within a dyadic formalism that enhances symmetry and uniformity. This dyadic approach simplifies the resolution of scattering issues by removing the dependence on the polarization direction since it can be recovered by the direction of propagation.

In this work, we extend the ideas presented in [7,14], providing essential energy considerations to derive relevant functional expressions for the differential and scattering cross-section related to a plane incident dyadic field. Furthermore, we will develop analogous results [4,7], relevant to low-frequency scattering of dyadic incident fields influenced by obstacles, which is particularly applicable in real-world situations where scatterers are significantly smaller than the incident wavelength. Indeed, in modern real-life applications, the low-frequency assumption is necessary since the size of the scatterers is much smaller than the incident wavelength. For example, the wavelength of a seismic wave ranges from 300 m to 1500 m, while the effect that this seismic wave has on a building of typical dimension, 10 m, justifies the operation at low frequencies. On the other hand, a small cavity within a metallic matrix has a low-frequency reaction to any elastic wave, even with a wavelength of a few cms. Hence, the low-frequency theory still nowadays stands for a strong analytical tool that provides assistance to the mathematical manipulations of related physical problems without reducing the generality of the methodology. At this point, we mention to the reader that general properties of dyadics will be used, and an excellent source for dyadic formulas is the book by Tai [15].

A potential extension of this research is to enhance the practical applicability of the proposed solution for fully three-dimensional problems, with some numerical techniques. One of several established numerical or computational methods for solving linear boundary-value problems that arise in mathematics, the physical sciences, and engineering, is the boundary element method (BEM) [16]. Furthermore, another interesting numerical technique could be the isogeometric analysis (IGA). This method is basically an extension of the finite element method (FEM) using basis functions not only representing the solution space but also the geometry [17].

The structure of our paper is the following: In Section 2, dyadic scattering problems due to a plane dyadic incident field are formulated. Our scatterer is irradiated by a complete dyadic field, which is decomposed into a plane longitudinal wave and a plane transverse one. In Section 3, the integral representation for the scattered field is given, and dyadic forms of the longitudinal and transverse far-field patterns are presented. In Section 4, we provide expressions for energy functionals in the far-field region along with expressions for the differential and the total scattering cross-section. Furthermore, in Section 5, we exploit the low-frequency approximation theory for scattering of dyadic incident field by a small rigid body or a cavity. Last but not least, a corresponding expression for the differential scattering cross-section is also established. The paper ends up with Section 6, in which we briefly summarize our research effort.

2. Setting up the Scattering Problems

The scattering of an incoming plane dyadic wave field by an obstacle in 3D linear elasticity is considered. We denote the bounded connected domain as $D_i$ with boundary $\partial D$, which is characterized as a ${\mathrm{C}}^2$ bounded surface. The set $D_i$ will be referred to as the scatterer, which is irradiated by a known incident wave field. The external region is represented by $D={\mathrm{R}}^3\setminus {\overline{D}}_i$, where ${\overline{D}}_i=D_i\cup \partial D$, and is composed of isotropic and homogeneous elastic material specified by Lamé constants $\lambda >0$ and $\mu >0$, along with a mass density $\rho$. The interior domain $D_i$ is filled with a material specified by constant Lamé constants ${\lambda }_i$, ${\mu }_i$, and mass density ${\rho }_i$.

Assuming time harmonic dependence of the form $\exp \{-i\omega t\}$, where $\omega >0$ is the angular frequency, we excite our obstacle using a dyadic field that propagates in a specified direction $\widehat{\mathbf{d}}$, which is decomposed into a plane longitudinal wave and a plane transverse represented as follows:

$${\tilde{\mathbf{u}}}^{inc}\left(\mathbf{r}\right)=\widehat{\mathbf{d}}\otimes \widehat{\mathbf{d}}{e}^{i{k}_p\mathbf{r}·\widehat{\mathbf{d}}}+(\tilde{\mathbf{I}}-\widehat{\mathbf{d}}\otimes \widehat{\mathbf{d}})e^{i{k}_s\mathbf{r}·\widehat{\mathbf{d}}},\mathbf{r}\in D,$$

(1)

where $\widehat{\mathbf{d}}=($sin${\theta }_1$cos${\theta }_2,$sin${\theta }_1$sin${\theta }_2,-cos{\theta }_1)$ indicates the propagation direction, specified by angular coordinates ${\theta }_1\in \left[0,{\displaystyle \frac{\pi }{2}}\right)$ and ${\theta }_2\in \left[0,2\pi \right)$. In relation (1), $\tilde{\mathbf{I}}$ denotes the identity dyadic, and “⊗” is the juxtaposition between two vectors, i.e., this gives a dyadic. In relation (1), we also denote $k_p=\omega \sqrt{\rho /\lambda +2\mu }$ and $k_s=\omega \sqrt{\rho /\mu }$ for the wave numbers for the longitudinal P and transverse S-waves, respectively. The overtilde “∼” notation is utilized to indicate dyadic fields.

The dyadic field (1) can be expressed as two dyadic superpositions: the first representing the longitudinal wave’s effects, while the second corresponds to the transverse waves, using the incident orthogonal basis $\{\widehat{\mathbf{d}},{\widehat{\mathbf{d}}}^{\perp }\}$, i.e.,

$${\tilde{\mathbf{u}}}^{inc}(\mathbf{r};\widehat{\mathbf{d}})={\mathbf{u}}_p^{inc}(\mathbf{r};\widehat{\mathbf{d}},\widehat{\mathbf{d}})\otimes \widehat{\mathbf{d}}+{\mathbf{u}}_s^{inc}(\mathbf{r};\widehat{\mathbf{d}},{\widehat{\mathbf{d}}}^{\perp })\otimes {\widehat{\mathbf{d}}}^{\perp },\mathbf{r}\in D,$$

(2)

where

$${\mathbf{u}}_p^{inc}(\mathbf{r};\widehat{\mathbf{d}},\widehat{\mathbf{d}})=\widehat{\mathbf{d}}{e}^{i{k}_p\mathbf{r}·\widehat{\mathbf{d}}},{\mathbf{u}}_s^{inc}(\mathbf{r};\widehat{\mathbf{d}},{\widehat{\mathbf{d}}}^{\perp })={\widehat{\mathbf{d}}}^{\perp }{e}^{i{k}_s\mathbf{r}·\widehat{\mathbf{d}}},\mathbf{r}\in D.$$

(3)

In relation (2), the first argument corresponds to the position vector of observation, while the second signifies the propagation vector, and the third represents the polarization vector.

Illuminating the obstacle with the dyadic incident field (2), the scattered field ${\tilde{\mathbf{u}}}^{sct}$ is generated and its representation due to the Helmholtz decomposition [18] is given by

$${\tilde{\mathbf{u}}}^{sct}\left(\mathbf{r}\right)={\tilde{\mathbf{u}}}_p^{sct}\left(\mathbf{r}\right)+{\tilde{\mathbf{u}}}_s^{sct}\left(\mathbf{r}\right),\mathbf{r}\in D,$$

(4)

where ${\tilde{\mathbf{u}}}_p^{sct}$ is the irrotational (P-wave) and ${\tilde{\mathbf{u}}}_s^{sct}$ the solenoidal (S-wave). From relation (2), we can imply the form of the scattered dyadic field, given by

$${\tilde{\mathbf{u}}}^{sct}(\mathbf{r};\widehat{\mathbf{d}})={\mathbf{u}}_p^{sct}(\mathbf{r};\widehat{\mathbf{d}},\widehat{\mathbf{d}})\otimes \widehat{\mathbf{d}}+{\mathbf{u}}_s^{sct}(\mathbf{r};\widehat{\mathbf{d}},{\widehat{\mathbf{d}}}^{\perp })\otimes {\widehat{\mathbf{d}}}^{\perp },\mathbf{r}\in D,$$

(5)

where the first vector of the dyads corresponds to the vector scattered fields generated by the first components of ${\tilde{\mathbf{u}}}^{inc}$, respectively.

By using the linearity of the incident field (2), we can express the total displacement field (5) as follows:

$${\tilde{\mathbf{u}}}^{tot}\left(\mathbf{r};\widehat{\mathbf{d}}\right)={\tilde{\mathbf{u}}}^{inc}\left(\mathbf{r};\widehat{\mathbf{d}}\right)+{\tilde{\mathbf{u}}}^{sct}\left(\mathbf{r};\widehat{\mathbf{d}}\right),\mathbf{r}\in D,$$

(6)

which satisfies the dyadic Navier equation:

$$c_s^2\Delta {\tilde{\mathbf{u}}}^{tot}\left(\mathbf{r}\right)+(c_p^2-c_s^2)\mathrm{grad}\mathrm{div}{\tilde{\mathbf{u}}}^{tot}\left(\mathbf{r}\right)+{\omega }^2\tilde{\mathbf{I}}{\tilde{\mathbf{u}}}^{tot}\left(\mathbf{r}\right)=\tilde{\mathbf{0}},\mathbf{r}\in D.$$

(7)

In the above Equation (7), the phase velocities of the longitudinal and transverse waves denoted by $c_p,c_s$ are given by

$$c_p=\sqrt{{\displaystyle \frac{\lambda +2\mu }{\rho }}},c_s=\sqrt{{\displaystyle \frac{\mu }{\rho }}},$$

(8)

respectively. Next, we will outline the mathematical formulation of the problem, beginning with the description of the two types of boundary conditions applied to the surface of the scatterer.

The first scattering problem due to a rigid scatterer is outlined as follows: For an incident plane dyadic field ${\tilde{\mathbf{u}}}^{inc}$, and under the influence of zero body forces, we seek a solution ${\tilde{\mathbf{u}}}^{tot}\in {[C^2({\mathrm{R}}^3\setminus {\overline{D}}_i)\cap C^1({\mathrm{R}}^3\setminus D_i)]}^2$ satisfying (7), such that for a rigid scatterer, the total displacement field vanishes on the boundary:

$${\tilde{\mathbf{u}}}^{tot}\left(\mathbf{r};\widehat{\mathbf{d}}\right)=\tilde{\mathbf{0}},\mathbf{r}\in \partial D,$$

(9)

while for the second scaterring problem due to a cavity, we require that the action of the surface stress operator on the displacement field be zero:

$$T{\tilde{\mathbf{u}}}^{tot}\left(\mathbf{r};\widehat{\mathbf{d}}\right)=\tilde{\mathbf{0}},\mathbf{r}\in \partial D,$$

(10)

where T indicates the surface stress operator defined as follows:

$$T=2\mu \widehat{\mathbf{n}}·\mathrm{grad}+\lambda \widehat{\mathbf{n}}\mathrm{div}+\mu \widehat{\mathbf{n}}\times \mathrm{curl}.$$

(11)

In Equation (11), $\widehat{\mathbf{n}}$ represents the outward unit normal vector on the $C^2$–boundary $\partial D$ at the point $\mathbf{r}$.

To ensure the well posedness of the above problems, each displacement field that appears as a first vector of each dyad in (5) must satisfy the radiation conditions at infinity for three-dimensional scenarios, as stipulated by Dassios [6],

$$\underset{r\to \infty }{lim}r({\displaystyle \frac{\partial {\mathbf{u}}_a^{sct}}{\partial r}}-i{k}_a{\mathbf{u}}_a^{sct})=\mathbf{0},a=p,s$$

(12)

uniformly for all directions $r=\left|\mathbf{r}\right|$.

The free-space fundamental dyadic in ${\mathrm{R}}^3$ is given by [1]

$$\begin{array}{ccc}\hfill \tilde{\mathsf{\Gamma }}{(\mathbf{r},\mathbf{r}}^{\prime })=& & -{\displaystyle \frac{i{k}_p}{4\pi \rho {\omega }^2}}{\mathrm{grad}}_{\mathbf{r}}{\mathrm{grad}}_{\mathbf{r}}^{\top }{\displaystyle \frac{e^{i{k}_p\left|\mathbf{r}-{\mathbf{r}}^{\prime }\right|}}{i{k}_p\left|\mathbf{r}-{\mathbf{r}}^{\prime }\right|}}\hfill \\ & & +{\displaystyle \frac{i{k}_s}{4\pi \rho {\omega }^2}}\left({\mathrm{grad}}_{\mathbf{r}}{\mathrm{grad}}_{\mathbf{r}}^{\top }+k_s^2\tilde{\mathbf{I}}\right){\displaystyle \frac{e^{i{k}_p\left|\mathbf{r}-{\mathbf{r}}^{\prime }\right|}}{i{k}_p\left|\mathbf{r}-{\mathbf{r}}^{\prime }\right|}},\hfill \end{array}$$

(13)

which is a solution to the following equation:

$$c_s^2{\Delta }_{\mathbf{r}}\tilde{\mathsf{\Gamma }}{(\mathbf{r},\mathbf{r}}^{\prime })+(c_p^2-c_s^2){\mathrm{grad}}_{\mathbf{r}}{\mathrm{div}}_{\mathbf{r}}\tilde{\mathsf{\Gamma }}{(\mathbf{r},\mathbf{r}}^{\prime })+{\omega }^2\tilde{\mathbf{I}}\tilde{\mathsf{\Gamma }}{(\mathbf{r},\mathbf{r}}^{\prime })=-\tilde{\mathbf{I}}\delta \left({\mathbf{r}-\mathbf{r}}^{\prime }\right),\mathbf{r}\in D,$$

(14)

where “⊤” denotes transportation and ${\displaystyle \frac{e^{i{k}_a\epsilon }}{i{k}_a\epsilon }}\equiv h_0^{\left(1\right)}\left(k_a\epsilon \right),\epsilon =\left|\mathbf{r}-{\mathbf{r}}^{\prime }\right|,a=p,s$ are the spherical Hankel functions of the first kind and zero order.

Using asymptotic analysis to the dyadic $\tilde{\mathsf{\Gamma }}(\mathbf{r},{\mathbf{r}}^{\prime })$ as $r=\left|\mathbf{r}\right|\to \infty$, we can obtain the following expressions for the far-field patterns associated with the longitudinal and transverse components of the fundamental dyadic $\tilde{\mathsf{\Gamma }}(\mathbf{r},{\mathbf{r}}^{\prime })$:

$${\tilde{\mathsf{\Gamma }}}_{\infty }^p(\mathbf{r},{\mathbf{r}}^{\prime })={\displaystyle \frac{i{k}_p}{\lambda +2\mu }}\widehat{\mathbf{r}}\otimes \widehat{\mathbf{r}}{e}^{-i{k}_p\widehat{\mathbf{r}}·{\mathbf{r}}^{\prime }}$$

(15)

and

$${\tilde{\mathsf{\Gamma }}}_{\infty }^s(\mathbf{r},{\mathbf{r}}^{\prime })={\displaystyle \frac{i{k}_s}{\mu }}(\tilde{\mathbf{I}}-\widehat{\mathbf{r}}\otimes \widehat{\mathbf{r}})e^{-i{k}_s\widehat{\mathbf{r}}·{\mathbf{r}}^{\prime }},$$

(16)

where $(\widehat{\mathbf{r}}\otimes \widehat{\mathbf{r}})$ and $(\tilde{\mathbf{I}}-\widehat{\mathbf{r}}\otimes \widehat{\mathbf{r}})$ in (15) and (16) are dyadics, which corresponds to the radial and tangential behavior of the longitudinal and transverse parts, respectively, of the fundamental dyadic $\tilde{\mathsf{\Gamma }}(\mathbf{r},{\mathbf{r}}^{\prime })$ far away from the scatterer at the radiation zone. The three-dimensional elastic scattering problem of a dyadic wave can be found in [4,6].

3. Integral Representations

It is recognized that, following either the direct method, based on Betti’s formulae [19], or the indirect method using the layer potentials, we can reformulate the direct scattering problem in integral form. The well posedness of the solution of the boundary integral equations, derived from the layer theoretic approach for the rigid, cavity, or transmission problem, can be found in [20,21].

In what follows, we applying Betti’s formulae in order to derive an integral representation for the displacement field in three-dimensional elasticity. Therefore, we obtain an integral representation for the radiating solutions ${\tilde{\mathbf{u}}}^{sct}\in {[C^2({\mathrm{R}}^3\setminus {\overline{D}}_i)\cap C^1({\mathrm{R}}^3\setminus D_i)]}^2$ of the Navier equation, expressed as follows:

$${\tilde{\mathbf{u}}}^{sct}\left(\mathbf{r};\widehat{\mathbf{d}}\right)={\int }_{\partial D}\left[{\tilde{\mathbf{u}}}^{sct}({\mathbf{r}}^{\prime };\widehat{\mathbf{d}})·{\left(T_{{\mathbf{r}}^{\prime }}\tilde{\mathsf{\Gamma }}\left({\mathbf{r},\mathbf{r}}^{\prime }\right)\right)}^{\top }-\tilde{\mathsf{\Gamma }}\left({\mathbf{r},\mathbf{r}}^{\prime }\right)·T_{{\mathbf{r}}^{\prime }}{\tilde{\mathbf{u}}}^{sct}({\mathbf{r}}^{\prime };\widehat{\mathbf{d}})\right]ds\left({\mathbf{r}}^{\prime }\right),$$

(17)

$\mathbf{r}\in {\mathrm{R}}^3\setminus {\overline{D}}_i$. The relations that exist for the far-field patterns of the scattered field at the radiation zone can be expressed using asymptotic analysis, i.e.,

$${\tilde{\mathbf{u}}}^{sct}\left(\mathbf{r};\widehat{\mathbf{d}}\right)={\tilde{\mathbf{u}}}_{\infty }^p\left(\widehat{\mathbf{r}};\widehat{\mathbf{d}}\right){\displaystyle \frac{e^{i{k}_{p}r}}{i{k}_{p}r}}+{\tilde{\mathbf{u}}}_{\infty }^s\left(\widehat{\mathbf{r}};\widehat{\mathbf{d}}\right){\displaystyle \frac{e^{i{k}_{s}r}}{i{k}_{s}r}}+O\left(r^{-2}\right),r=\left|\mathbf{r}\right|\to \infty ,$$

(18)

uniformly with respect to $\widehat{\mathbf{r}}={\displaystyle \frac{\mathbf{r}}{\left|\mathbf{r}\right|}}\in \Omega$, with $\Omega :=\{\mathbf{r}\in {\mathrm{R}}^3:\left|\mathbf{r}\right|=1\}$. The second argument $\widehat{\mathbf{d}}$ (propagation vector) in (18) will no longer be used for notation simplicity; therefore, we use ${\tilde{\mathbf{u}}}^{sct}\left(\mathbf{r}\right)$ rather than ${\tilde{\mathbf{u}}}^{sct}\left(\mathbf{r};\widehat{\mathbf{d}}\right)$. The corresponding far-field patterns are the coefficients of the terms ${\displaystyle \frac{e^{i{k}_{a}r}}{i{k}_{a}r}}$, $a=p,s$, the dyadics ${\tilde{\mathbf{u}}}_{\infty }^p$ and ${\tilde{\mathbf{u}}}_{\infty }^s$ are defined on the unit sphere $\Omega$ in ${\mathrm{R}}^3$ known as longitudinal and transverse far-field pattern, and they are expressed by

$${\tilde{\mathbf{u}}}_{\infty }^p\left(\widehat{\mathbf{r}}\right)={\displaystyle \frac{k_p^2}{4\pi {\omega }^2}}{\int }_{\partial D}\left({\left[T_{{\mathbf{r}}^{\prime }}{(\widehat{\mathbf{r}}\otimes \widehat{\mathbf{r}})}^{\top }{e}^{-i{k}_p\widehat{\mathbf{r}}·{\mathbf{r}}^{\prime }}\right]}^{\top }{\tilde{\mathbf{u}}}^{tot}\left({\mathbf{r}}^{\prime }\right)-{(\widehat{\mathbf{r}}\otimes \widehat{\mathbf{r}})}^{\top }{e}^{-i{k}_p\widehat{\mathbf{r}}·{\mathbf{r}}^{\prime }}{T}_{{\mathbf{r}}^{\prime }}{\tilde{\mathbf{u}}}^{tot}\left({\mathbf{r}}^{\prime }\right)\right)ds\left({\mathbf{r}}^{\prime }\right),$$

(19)

and

$${\tilde{\mathbf{u}}}_{\infty }^s\left(\widehat{\mathbf{r}}\right)={\displaystyle \frac{k_s^2}{4\pi {\omega }^2}}{\int }_{\partial D}\left({\left[T_{{\mathbf{r}}^{\prime }}(\tilde{\mathbf{I}}-\widehat{\mathbf{r}}\otimes {\widehat{\mathbf{r}}}^{\top })e^{-i{k}_s\widehat{\mathbf{r}}·{\mathbf{r}}^{\prime }}\right]}^{\top }{\tilde{\mathbf{u}}}^{tot}\left({\mathbf{r}}^{\prime }\right)-(\tilde{\mathbf{I}}-\widehat{\mathbf{r}}\otimes {\widehat{\mathbf{r}}}^{\top })e^{-i{k}_s\widehat{\mathbf{r}}·{\mathbf{r}}^{\prime }}{T}_{{\mathbf{r}}^{\prime }}{\tilde{\mathbf{u}}}^{tot}\left({\mathbf{r}}^{\prime }\right)\right)ds\left({\mathbf{r}}^{\prime }\right),$$

(20)

where $k_p,k_s$ are the wavenumbers as defined above (see page 3). The dyadics $\widehat{\mathbf{r}}\otimes \widehat{\mathbf{r}}$ and $\tilde{\mathbf{I}}-\widehat{\mathbf{r}}\otimes \widehat{\mathbf{r}}$ present the radial and tangential behaviour of the longitudinal and transverse parts, respectively, of the scattered field far away from the scatterer.

So far, we have examined scattering problems for the cases of a rigid body and a cavity. Extending the ideas in [4,7] to the three-dimensional elastic situation is our goal in the section that follows. The necessary energy considerations will be specifically covered, along with related functionals and formulas for the differential and the scattering cross-section caused by the plane incident dyadic field.

4. The Scattering Cross-Section

We will examine the scenarios of a rigid body and a cavity illuminated by an incident plane dyadic field in this section. The disturbance caused by the scatterer to the propagation of the incident wave is measured by the scattering cross-section. Furthermore, we will establish an expression for the scattering cross-section, caused by an incident plane dyadic field.

The definition of the energy flux vector for the scattered field in the radiation zone is the following:

$${\mathbf{P}}^{sct}\left(\mathbf{r}\right)={\widetilde{\tilde{\mathbf{E}}}}^{sct}\left(\mathbf{r}\right):\mathbf{c}\otimes \mathbf{c},\mathbf{r}\in D,$$

(21)

where ${\widetilde{\tilde{\mathbf{E}}}}^{sct}$ represents the energy triadic and is given by

$${\widetilde{\tilde{\mathbf{E}}}}^{sct}\left(\mathbf{r}\right)=\omega \Im \left[{\left({\overline{{\tilde{\mathbf{u}}}^{sct}\left(\mathbf{r}\right)}}^{\top }·{\widetilde{\tilde{\mathbf{S}}}}^{sct}\left(\mathbf{r}\right)\right)}^{213}\right],\mathbf{r}\in D,$$

(22)

where an arbitrary constant vector is denoted by $\mathbf{c}$. The stress triadic ${\widetilde{\tilde{\mathbf{S}}}}^{sct}$, generated by the scattered field ${\tilde{\mathbf{u}}}^{sct}$ due to the incident plane dyadic field, is defined by the relation

$${\widetilde{\tilde{\mathbf{S}}}}^{sct}\left(\mathbf{r}\right)=\lambda \tilde{\mathbf{I}}\otimes \left({\nabla }_{\mathbf{r}}·{\tilde{\mathbf{u}}}^{sct}\left(\mathbf{r}\right)\right)+\mu {\nabla }_{\mathbf{r}}\otimes {\tilde{\mathbf{u}}}^{sct}\left(\mathbf{r}\right)+\mu {\left({\nabla }_{\mathbf{r}}\otimes {\tilde{\mathbf{u}}}^{sct}\left(\mathbf{r}\right)\right)}^{213},$$

(23)

where “213” signifies the order of the tensorial product associated with the respective triadic. In addition, we obtain

$$\nabla \otimes \tilde{\mathbf{V}}={\widehat{\mathbf{x}}}_1\otimes {\displaystyle \frac{\partial \tilde{\mathbf{V}}}{\partial x_1}}+{\widehat{\mathbf{x}}}_2\otimes {\displaystyle \frac{\partial \tilde{\mathbf{V}}}{\partial x_2}}+{\widehat{\mathbf{x}}}_3\otimes {\displaystyle \frac{\partial \tilde{\mathbf{V}}}{\partial x_3}},$$

(24)

where $\tilde{\mathbf{V}}=({\mathbf{V}}_1,{\mathbf{V}}_2,{\mathbf{V}}_3)$ is a third-rank tensor.

Next, we perform calculations to evaluate ${\nabla }_{\mathbf{r}}·{\tilde{\mathbf{u}}}^{sct}$ and ${\nabla }_{\mathbf{r}}\otimes {\tilde{\mathbf{u}}}^{sct}$ from relation (23), particularly as $r\to \infty ,$ and we obtain the following expressions:

$${\nabla }_{\mathbf{r}}·{\tilde{\mathbf{u}}}^{sct}\left(\mathbf{r}\right)=i{k}_p\widehat{\mathbf{r}}·{\tilde{\mathbf{u}}}_{\infty }^p\left(\widehat{\mathbf{r}}\right){\displaystyle \frac{e^{i{k}_{p}r}}{i{k}_{p}r}}+O\left(r^{-2}\right),$$

(25)

and

$${\nabla }_{\mathbf{r}}\otimes {\tilde{\mathbf{u}}}^{sct}\left(\mathbf{r}\right)=i{k}_p\widehat{\mathbf{r}}\otimes {\tilde{\mathbf{u}}}_{\infty }^p\left(\widehat{\mathbf{r}}\right){\displaystyle \frac{e^{i{k}_{p}r}}{i{k}_{p}r}}+i{k}_s\widehat{\mathbf{r}}\otimes {\tilde{\mathbf{u}}}_{\infty }^s\left(\widehat{\mathbf{r}}\right){\displaystyle \frac{e^{i{k}_{s}r}}{i{k}_{s}r}}+O\left(r^{-2}\right).$$

(26)

By substitution of (25) and (26) in Equation (23), we obtain the following relation, for $r\to \infty$,

$$\begin{array}{ccc}\hfill {\widetilde{\tilde{\mathbf{S}}}}^{sct}\left(\mathbf{r}\right)& =& i{k}_p(\lambda \tilde{\mathbf{I}}\otimes \widehat{\mathbf{r}}+2\mu \widehat{\mathbf{r}}\otimes \tilde{\mathbf{I}})·{\tilde{\mathbf{u}}}_{\infty }^p\left(\widehat{\mathbf{r}}\right){\displaystyle \frac{e^{i{k}_{p}r}}{i{k}_{p}r}}\hfill \\ & & +i{k}_s\left[\mu \widehat{\mathbf{r}}\otimes {\tilde{\mathbf{u}}}_{\infty }^s\left(\widehat{\mathbf{r}}\right)+\mu {\left(\widehat{\mathbf{r}}\otimes {\tilde{\mathbf{u}}}_{\infty }^s\left(\widehat{\mathbf{r}}\right)\right)}^{213}\right]{\displaystyle \frac{e^{i{k}_{s}r}}{i{k}_{s}r}}+O\left(r^{-2}\right).\hfill \end{array}$$

(27)

The energy flux of the scattered wave at point $\widehat{\mathbf{r}}$ in the direction of observation, with the aid of (21) and (22), is given by

$$\begin{array}{ccc}\hfill \widehat{\mathbf{r}}·{\mathbf{P}}^{sct}\left(\mathbf{r}\right)& =& \widehat{\mathbf{r}}·{\widetilde{\tilde{\mathbf{E}}}}^{sct}\left(\mathbf{r}\right):\mathbf{c}\otimes \mathbf{c}\hfill \\ & =& \widehat{\mathbf{r}}·\omega \Im \left[{\left({\overline{{\tilde{\mathbf{u}}}^{sct}\left(\mathbf{r}\right)}}^{\top }·{\widetilde{\tilde{\mathbf{S}}}}^{sct}\left(\mathbf{r}\right)\right)}^{213}\right]:\mathbf{c}\otimes \mathbf{c}.\hfill \end{array}$$

(28)

Hence, in view of (18), and after some calculations, we obtain

$$\begin{array}{ccc}\hfill {\overline{{\tilde{\mathbf{u}}}^{sct}\left(\mathbf{r}\right)}}^{\top }·{\widetilde{\tilde{\mathbf{S}}}}^{sct}\left(\mathbf{r}\right)& =& {\displaystyle \frac{i{k}_p}{r^2}}{\overline{{\tilde{\mathbf{u}}}_{\infty }^p\left(\widehat{\mathbf{r}}\right)}}^{\top }·\left(\lambda \tilde{\mathbf{I}}\otimes \widehat{\mathbf{r}}+2\mu \widehat{\mathbf{r}}\otimes \tilde{\mathbf{I}}\right)·{\tilde{\mathbf{u}}}_{\infty }^p\left(\widehat{\mathbf{r}}\right)\hfill \\ \hfill \\ & & +{\displaystyle \frac{\mu i{k}_s}{r^2}}{\overline{{\tilde{\mathbf{u}}}_{\infty }^s\left(\widehat{\mathbf{r}}\right)}}^{\top }·\left[\widehat{\mathbf{r}}\otimes {\tilde{\mathbf{u}}}_{\infty }^s\left(\widehat{\mathbf{r}}\right)+\mu {\left(\widehat{\mathbf{r}}\otimes {\tilde{\mathbf{u}}}_{\infty }^s\left(\widehat{\mathbf{r}}\right)\right)}^{213}\right]\hfill \\ \hfill \\ & & +{\displaystyle \frac{\mu i{k}_s{e}^{-i(k_p-k_s)r}}{r^2}}{\overline{{\tilde{\mathbf{u}}}_{\infty }^s\left(\widehat{\mathbf{r}}\right)}}^{\top }·\left[\widehat{\mathbf{r}}\otimes {\tilde{\mathbf{u}}}_{\infty }^s\left(\widehat{\mathbf{r}}\right)+\mu {\left(\widehat{\mathbf{r}}\otimes {\tilde{\mathbf{u}}}_{\infty }^s\left(\widehat{\mathbf{r}}\right)\right)}^{213}\right]\hfill \\ \hfill \\ & & +{\displaystyle \frac{i{k}_p{e}^{-i(k_s-k_p)r}}{r^2}}{\overline{{\tilde{\mathbf{u}}}_{\infty }^s\left(\widehat{\mathbf{r}}\right)}}^{\top }·\left(\lambda \tilde{\mathbf{I}}\otimes \widehat{\mathbf{r}}+2\mu \widehat{\mathbf{r}}\otimes \tilde{\mathbf{I}}\right)·{\tilde{\mathbf{u}}}_{\infty }^p\left(\widehat{\mathbf{r}}\right)\hfill \\ \hfill \\ & & +O\left(r^{-2}\right),r\to \infty .\hfill \end{array}$$

(29)

In Equation (28), we need now to contract the observation vector $\widehat{\mathbf{r}}$ with the energy triadic ${\tilde{\mathbf{E}}}^{sct}$. From the above relation, (29), and using the property of the left transpose of a triadic [15], i.e.,

$${(\mathbf{a}\otimes \mathbf{b}\otimes \mathbf{c})}^{213}=\mathbf{b}\otimes \mathbf{a}\otimes \mathbf{c},$$

(30)

where $\mathbf{a},\mathbf{b},\mathbf{c}$ are any vectors, we can obtain the following, as $r\to \infty$

$$\widehat{\mathbf{r}}·{\widetilde{\tilde{\mathbf{E}}}}^{sct}\left(\mathbf{r}\right)={\displaystyle \frac{\rho }{r^2}}\Re \left(c_p^3{k}_p^2{\overline{{\tilde{\mathbf{u}}}_{\infty }^p\left(\widehat{\mathbf{r}}\right)}}^{\top }·{\tilde{\mathbf{u}}}_{\infty }^p\left(\widehat{\mathbf{r}}\right)+c_s^3{k}_s^2{\overline{{\tilde{\mathbf{u}}}_{\infty }^s\left(\widehat{\mathbf{r}}\right)}}^{\top }·{\tilde{\mathbf{u}}}_{\infty }^s\left(\widehat{\mathbf{r}}\right)\right)+O\left(r^{-2}\right).$$

(31)

By substitution, (31) into (28), the outward energy flux at the point $\mathbf{r}$ in the direction $\widehat{\mathbf{r}}$ is given by

$$\widehat{\mathbf{r}}·{\mathbf{P}}^{sct}\left(\mathbf{r}\right)={\displaystyle \frac{\rho {\omega }^3}{r^2}}\left({\displaystyle \frac{1}{k_p}}\parallel {\tilde{\mathbf{u}}}_{\infty }^p\left(\widehat{\mathbf{r}}\right){·\mathbf{c}\parallel }^2+{\displaystyle \frac{1}{k_s}}{\parallel {\tilde{\mathbf{u}}}_{\infty }^s\left(\widehat{\mathbf{r}}\right)·\mathbf{c}\parallel }^2\right)+O\left(r^{-2}\right),r\to \infty .$$

(32)

In the sequel, we will define the corresponding differential scattering cross-section $\sigma \left(\widehat{\mathbf{r}}\right)$, and, for that reason, it is necessary to deal first with the energy triadic and the energy flux vector for the plane dyadic incidence. ${\tilde{\mathbf{u}}}^{inc}$ generates the stress triadic ${\widetilde{\tilde{\mathbf{S}}}}^{inc}$ given by

$${\widetilde{\tilde{\mathbf{S}}}}^{inc}\left(\mathbf{r}\right)=\lambda \tilde{\mathbf{I}}\otimes \left({\nabla }_{\mathbf{r}}·{\tilde{\mathbf{u}}}^{inc}\left(\mathbf{r}\right)\right)+\mu {\nabla }_{\mathbf{r}}\otimes {\tilde{\mathbf{u}}}^{inc}\left(\mathbf{r}\right)+\mu {\left({\nabla }_{\mathbf{r}}\otimes {\tilde{\mathbf{u}}}^{inc}\left(\mathbf{r}\right)\right)}^{213},$$

(33)

$\left(\mathrm{we} \mathrm{remind} \mathrm{that}{\tilde{\mathbf{u}}}^{inc}\left(\mathbf{r}\right)=\widehat{\mathbf{d}}\otimes \widehat{\mathbf{d}}{e}^{i{k}_p\mathbf{r}·\widehat{\mathbf{d}}}+(\tilde{\mathbf{I}}-\widehat{\mathbf{d}}\otimes \widehat{\mathbf{d}})e^{i{k}_s\mathbf{r}·\widehat{\mathbf{d}}}\right)$. After that, we follow the previous steps in the same way for ${\widetilde{\tilde{\mathbf{S}}}}^{sct}$ in view of (33), and, after some calculations, we obtain the following expressions:

$$\tilde{\mathbf{I}}\otimes {\nabla }_{\mathbf{r}}·{\tilde{\mathbf{u}}}^{inc}\left(\mathbf{r}\right)=i{k}_p{e}^{i{k}_p\mathbf{r}·\widehat{\mathbf{d}}}\tilde{\mathbf{I}}\otimes \widehat{\mathbf{d}},$$

(34)

$$\begin{array}{ccc}\hfill {\nabla }_{\mathbf{r}}\otimes {\tilde{\mathbf{u}}}^{inc}\left(\mathbf{r}\right)& =& \sum _{i=1}^3{\widehat{\mathbf{x}}}_i\otimes {\displaystyle \frac{\partial {\tilde{\mathbf{u}}}^{inc}\left(\mathbf{r}\right)}{\partial x_i}}\hfill \\ \hfill \\ & =& {\widehat{\mathbf{x}}}_1\otimes {\displaystyle \frac{\partial {\tilde{\mathbf{u}}}^{inc}\left(\mathbf{r}\right)}{\partial x_1}}+{\widehat{\mathbf{x}}}_2\otimes {\displaystyle \frac{\partial {\tilde{\mathbf{u}}}^{inc}\left(\mathbf{r}\right)}{\partial x_2}}+{\widehat{\mathbf{x}}}_3\otimes {\displaystyle \frac{\partial {\tilde{\mathbf{u}}}^{inc}\left(\mathbf{r}\right)}{\partial x_3}}\hfill \\ \hfill \\ & =& i{k}_p{e}^{i{k}_p\mathbf{r}·\widehat{\mathbf{d}}}\widehat{\mathbf{d}}\otimes \widehat{\mathbf{d}}\otimes \widehat{\mathbf{d}}+i{k}_s{e}^{i{k}_s\mathbf{r}·\widehat{\mathbf{d}}}\widehat{\mathbf{d}}\otimes \left(\tilde{\mathbf{I}}-\widehat{\mathbf{d}}\otimes \widehat{\mathbf{d}}\right),\hfill \end{array}$$

(35)

with $\mathbf{r}=(x_1,x_2,x_3)\in {\mathrm{R}}^3$. Hence, ${\tilde{\mathbf{u}}}^{inc}$ generates the stress triadic, which is given by

$$\begin{array}{ccc}\hfill {\widetilde{\tilde{\mathbf{S}}}}^{inc}\left(\mathbf{r}\right)& =& i{k}_p\left(\lambda \tilde{\mathbf{I}}+2\mu \widehat{\mathbf{d}}\otimes \widehat{\mathbf{d}}\right)\otimes \widehat{\mathbf{d}}{e}^{i{k}_p\mathbf{r}·\widehat{\mathbf{d}}}\hfill \\ \hfill \\ & & +i{k}_s\mu \left(\widehat{\mathbf{d}}\otimes \tilde{\mathbf{I}}+{(\widehat{\mathbf{d}}\otimes \tilde{\mathbf{I}})}^{213}-2\widehat{\mathbf{d}}\otimes \widehat{\mathbf{d}}\otimes \widehat{\mathbf{d}}\right)e^{i{k}_s\mathbf{r}·\widehat{\mathbf{d}}},\mathbf{r}\in D,\hfill \end{array}$$

(36)

and the corresponding energy triadic ${\widetilde{\tilde{\mathbf{E}}}}^{inc}$ is defined by

$${\widetilde{\tilde{\mathbf{E}}}}^{inc}\left(\mathbf{r}\right)=\omega \Im \left[{\left({\overline{{\tilde{\mathbf{u}}}^{inc}\left(\mathbf{r}\right)}}^{\top }·{\widetilde{\tilde{\mathbf{S}}}}^{inc}\left(\mathbf{r}\right)\right)}^{213}\right],\mathbf{r}\in D,$$

(37)

where the above quantity in the bracket is calculated to be

$$\begin{array}{ccc}\hfill {\left({\overline{{\tilde{\mathbf{u}}}^{inc}\left(\mathbf{r}\right)}}^{\top }·{\widetilde{\tilde{\mathbf{S}}}}^{inc}\left(\mathbf{r}\right)\right)}^{213}& =& i{k}_p(\lambda +2\mu )\widehat{\mathbf{d}}\otimes \widehat{\mathbf{d}}\otimes \widehat{\mathbf{d}}\hfill \\ \hfill \\ & & +i{k}_s\mu \widehat{\mathbf{d}}\otimes (\tilde{\mathbf{I}}-\widehat{\mathbf{d}}\otimes \widehat{\mathbf{d}})\hfill \\ \hfill \\ & & +i{k}_p\lambda (\tilde{\mathbf{I}}-\widehat{\mathbf{d}}\otimes \widehat{\mathbf{d}})\otimes \widehat{\mathbf{d}}{e}^{i(k_p-k_s)\mathbf{r}·\widehat{\mathbf{d}}}\hfill \\ \hfill \\ & & +i{k}_s\mu {\left(\widehat{\mathbf{d}}\otimes (\tilde{\mathbf{I}}-\widehat{\mathbf{d}}\otimes \widehat{\mathbf{d}})\right)}^{213}{e}^{-i(k_p-k_s)\mathbf{r}·\widehat{\mathbf{d}}}.\hfill \end{array}$$

(38)

The following relation denotes the energy flux vector

$${\mathbf{P}}^{inc}\left(\mathbf{r}\right)={\widetilde{\tilde{\mathbf{E}}}}^{inc}:\mathbf{c}\otimes \mathbf{c},\mathbf{r}\in D,$$

(39)

which is relevant to our research goal. By utilizing Equations (38) and (39), we can compute the quantity $\widehat{\mathbf{d}}·{\widetilde{\tilde{\mathbf{E}}}}^{inc}$. Notably, we can conclude that

$$\widehat{\mathbf{d}}·\left[\left(\tilde{\mathbf{I}}-\widehat{\mathbf{d}}\otimes \widehat{\mathbf{d}}\right)\otimes \widehat{\mathbf{d}}\right]=\tilde{\mathbf{0}}\mathrm{and}\widehat{\mathbf{d}}·{\left[\widehat{\mathbf{d}}\otimes (\tilde{\mathbf{I}}-\widehat{\mathbf{d}}\otimes \widehat{\mathbf{d}})\right]}^{213}=\tilde{\mathbf{0}},$$

(40)

leading us to the calculation of the plane energy flux, i.e.,

$$\begin{array}{ccc}\hfill \widehat{\mathbf{d}}·{\mathbf{P}}^{inc}\left(\mathbf{r}\right)& =& \widehat{\mathbf{d}}·{\widetilde{\tilde{\mathbf{E}}}}^{inc}\left(\mathbf{r}\right):\mathbf{c}\otimes \mathbf{c}\hfill \\ \hfill \\ & =& \widehat{\mathbf{d}}·\left[\omega \Im {\left({\overline{{\tilde{\mathbf{u}}}^{inc}\left(\mathbf{r}\right)}}^{\top }·{\widetilde{\tilde{\mathbf{S}}}}^{inc}\left(\mathbf{r}\right)\right)}^{213}:\mathbf{c}\otimes \mathbf{c}\right]\hfill \\ \hfill \\ \hfill \\ & =& \rho {\omega }^2\left[c_p\widehat{\mathbf{d}}\otimes \widehat{\mathbf{d}}:\mathbf{c}\otimes \mathbf{c}+c_s(\tilde{\mathbf{I}}-\widehat{\mathbf{d}}\otimes \widehat{\mathbf{d}}):\mathbf{c}\otimes \mathbf{c}\right].\hfill \end{array}$$

(41)

After that, we define the differential scattering cross-section caused by plane dyadic incidence, following an analogous procedure in [13], as follows

$$\sigma \left(\widehat{\mathbf{r}}\right)=\underset{r\to \infty }{lim}{\displaystyle \frac{2\pi r{\widehat{\mathbf{r}}·\mathbf{P}}^{sct}\left(\mathbf{r}\right)}{\widehat{\mathbf{d}}·{\mathbf{P}}^{inc}\left(\mathbf{r}\right)}},$$

(42)

where the direction of observation is given by $\widehat{\mathbf{r}}$, ${\mathbf{P}}^{inc}$ is the energy flux vector for plane incidence, and the product ${\widehat{\mathbf{r}}·\mathbf{P}}^{sct}$ denotes the outward normal energy flux in the far-field. The denominator $\widehat{\mathbf{d}}·{\mathbf{P}}^{inc}$ represents the energy flux in the direction of propagation $\widehat{\mathbf{d}}$.

Additionally, with regard to Equations (32) and (41) and the definition (42), we derive the following equations for rigid or cavity scatterers:

$$\sigma \left(\widehat{\mathbf{r}}\right)={\displaystyle \frac{2\pi \left(\frac{1}{k_p}\parallel {\tilde{\mathbf{u}}}_{\infty }^p\left(\widehat{\mathbf{r}}\right){·\mathbf{c}\parallel }^2+\frac{1}{k_s}{\parallel {\tilde{\mathbf{u}}}_{\infty }^s\left(\widehat{\mathbf{r}}\right)·\mathbf{c}\parallel }^2\right)}{\frac{1}{k_p}\left(\widehat{\mathbf{d}}\otimes \widehat{\mathbf{d}}:\mathbf{c}\otimes \mathbf{c}\right)+\frac{1}{k_s}\left(\tilde{\mathbf{I}}-\widehat{\mathbf{d}}\otimes \widehat{\mathbf{d}}\right):\mathbf{c}\otimes \mathbf{c}}}+O\left(r^{-2}\right),r\to \infty ,$$

(43)

where ${\tilde{\mathbf{u}}}_{\infty }^p\mathrm{and}{\tilde{\mathbf{u}}}_{\infty }^s$ are the far-field presented by relations (19) and (20). If now we take the integral for $\sigma \left(\widehat{\mathbf{r}}\right)$ over the unit circle $\Omega$, we can define a measure of the disturbance caused by the scatterer to the propagation of the plane dyadic incident field, which denotes the scattering cross-section or total cross-section, i.e.,

$${\sigma }^{sct}={\displaystyle \frac{1}{2\pi }}{\int }_{\Omega }\sigma \left(\widehat{\mathbf{r}}\right)ds\left(\widehat{\mathbf{r}}\right),$$

(44)

and, finally, we can easily get the total scattering cross-section from above relation (44), as

$${\sigma }^{sct}={\displaystyle \frac{{\int }_{\Omega }\left(\frac{1}{k_p}\parallel {\tilde{\mathbf{u}}}_{\infty }^p\left(\widehat{\mathbf{r}}\right){·\mathbf{c}\parallel }^2+\frac{1}{k_s}{\parallel {\tilde{\mathbf{u}}}_{\infty }^s\left(\widehat{\mathbf{r}}\right)·\mathbf{c}\parallel }^2\right)ds\left(\widehat{\mathbf{r}}\right)}{\frac{1}{k_p}\left(\widehat{\mathbf{d}}\otimes \widehat{\mathbf{d}}:\mathbf{c}\otimes \mathbf{c}\right)+\frac{1}{k_s}\left(\tilde{\mathbf{I}}-\widehat{\mathbf{d}}\otimes \widehat{\mathbf{d}}\right):\mathbf{c}\otimes \mathbf{c}}}.$$

(45)

5. Far-Field Low-Frequency Approximation

Herein, we provide the low-frequency scattering approximations of dyadic fields in three-dimensional elastic media. Bearing in mind that the total displacement field is a function of the wave number $k_p$ or $k_s$, which is analytic in a neighborhood of the origin, we can expand it via converging integral power series by virtue of the wave number $k_p$, i.e.,

$${\tilde{\mathbf{u}}}^{tot}\left(\mathbf{r}\right)=\sum _{n=0}^{+\infty }{\left(i{k}_p\right)}^n{\tilde{\mathbf{u}}}_n^{tot}\left(\mathbf{r}\right),\mathbf{r}\in D,$$

(46)

where

$${\tilde{\mathbf{u}}}_n^{tot}\left(\mathbf{r}\right)={\tilde{\mathbf{u}}}_n^{inc}\left(\mathbf{r}\right)+{\tilde{\mathbf{u}}}_n^{sct}\left(\mathbf{r}\right),n\ge 0,\mathbf{r}\in D.$$

(47)

Towards this direction, we can obtain the corresponding low-frequency expansions for the incident field

$${\tilde{\mathbf{u}}}^{inc}\left(\mathbf{r}\right)=\sum _{n=0}^{\infty }{\left(i{k}_p\right)}^n{\tilde{\mathbf{u}}}_n^{inc}\left(\mathbf{r}\right),\mathbf{r}\in D,$$

(48)

and for the scattered field,

$${\tilde{\mathbf{u}}}^{sct}\left(\mathbf{r}\right)=\sum _{n=0}^{+\infty }{\left(i{k}_p\right)}^n{\tilde{\mathbf{u}}}_n^{sct}\left(\mathbf{r}\right),\mathbf{r}\in D.$$

(49)

Obviously, similar expansions are available if we choose powers of $k_s$, instead of $k_p$; however, we proceed with the series expansions of (46) with (47) and (48), and (49) without loss of generality.

The plane dyadic incident field (1) is written as

$${\tilde{\mathbf{u}}}^{inc}\left(\mathbf{r}\right)=\sum _{n=0}^{+\infty }{\left(i{k}_p\right)}^n\left[\widehat{\mathbf{d}}\otimes \widehat{\mathbf{d}}+{\tau }^{-n}\left(\tilde{\mathbf{I}}-\widehat{\mathbf{d}}\otimes \widehat{\mathbf{d}}\right)\right]{\displaystyle \frac{{\left(\widehat{\mathbf{d}}·\mathbf{r}\right)}^n}{n!}},$$

(50)

in which $\tau =k_p/k_s$, wherein the low-frequency components ${\tilde{\mathbf{u}}}^{inc}\left(\mathbf{r}\right)$, $n\ge 0$ are identified by matching relationships (48) and (50). Next, substituting expansion (46) into the Navier Equation (7) and rearranging properly indexes, we equate function coefficients of the same low-frequency powers, yielding

$${\tau }^2{\Delta }_{\mathbf{r}}{\tilde{\mathbf{u}}}_n^{tot}\left(\mathbf{r}\right)+(1-{\tau }^2){\nabla }_{\mathbf{r}}{\nabla }_{\mathbf{r}}·{\tilde{\mathbf{u}}}_n^{tot}\left(\mathbf{r}\right)-{\tilde{\mathbf{u}}}_{n-2}^{tot}\left(\mathbf{r}\right)=\tilde{\mathbf{0}},n=2,3,\dots ,$$

(51)

for $n=0$

$${\tau }^2{\Delta }_{\mathbf{r}}{\tilde{\mathbf{u}}}_0^{tot}\left(\mathbf{r}\right)+(1-{\tau }^2){\nabla }_{\mathbf{r}}{\nabla }_{\mathbf{r}}·{\tilde{\mathbf{u}}}_0^{tot}\left(\mathbf{r}\right)=\tilde{\mathbf{0}},$$

(52)

while for $n=1$

$${\tau }^2{\Delta }_{\mathbf{r}}{\tilde{\mathbf{u}}}_1^{tot}\left(\mathbf{r}\right)+(1-{\tau }^2){\nabla }_{\mathbf{r}}{\nabla }_{\mathbf{r}}·{\tilde{\mathbf{u}}}_1^{tot}\left(\mathbf{r}\right)=\tilde{\mathbf{0}},$$

(53)

being valid for every $\mathbf{r}\in D$. It is easily verified that Formulae (51)–(53) are also satisfied by the sequence of primary components ${\tilde{\mathbf{u}}}^{inc}$ and of scattering components ${\tilde{\mathbf{u}}}^{sct}$ for any $n\ge 0$. On the other hand, applying the same procedure at low frequencies to the classical boundary conditions that refer to the rigid body (see (9)) and to the cavity (see (10)), we arrive at

$${\tilde{\mathbf{u}}}_n^{tot}\left(\mathbf{r}\right)=\tilde{\mathbf{0}},n\ge 0,\mathbf{r}\in \partial D$$

(54)

and

$$T{\tilde{\mathbf{u}}}_n^{inc}\left(\mathbf{r}\right)=\tilde{\mathbf{0}},n\ge 0,\mathbf{r}\in \partial D,$$

(55)

respectively, for each low-frequency term. Apparently, the boundary conditions (54) and (55) include the total fields, given by (47). We remark that the limiting condition (12) is secured by definition for every low-frequency order.

Likewise, in the sequel and in view of the above, we aim to find a handy formula for the low-frequency expansion of the Navier fundamental solution (13) in 3D-linear elasticity. On that account and as a first step, it is necessary to further process on (13) by means of applying the double-gradient operator. Hence, by definition of

$$\mathbf{R}=\mathbf{r}-{\mathbf{r}}^{\prime }\mathrm{with}R=\left|\mathbf{R}\right|\mathrm{and}\widehat{\mathbf{R}}={\displaystyle \frac{\mathbf{R}}{R}},$$

(56)

we differentiate (13); so, after some long and tedious, but straightforward, calculations, we recover the fundamental solution as

$$\begin{array}{ccc}\hfill \tilde{\mathsf{\Gamma }}\left(\mathbf{r},{\mathbf{r}}^{\prime }\right)& =& {\displaystyle \frac{1}{\left(\lambda +2\mu \right)k_p^2{R}^3}}\left[k_p^2\mathbf{R}\otimes \mathbf{R}+\left(1-i{k}_{p}R\right)\left(\tilde{\mathbf{I}}-3\widehat{\mathbf{R}}\otimes \widehat{\mathbf{R}}\right)\right]e^{i{k}_{p}R}\hfill \\ \hfill \\ & & -{\displaystyle \frac{1}{\mu k_s^2{R}^3}}\left\{\left[k_s^2\mathbf{R}\otimes \mathbf{R}+\left(1-i{k}_{s}R\right)\left(\tilde{\mathbf{I}}-3\widehat{\mathbf{R}}\otimes \widehat{\mathbf{R}}\right)\right]-k_s^2{R}^2\tilde{\mathbf{I}}\right\}{e}^{i{k}_{s}R},\hfill \end{array}$$

(57)

while the dyadic (57) generates the stress dyadic on any material point ${\mathbf{r}}^{\prime }$, given by the action of $T_{{\mathbf{r}}^{\prime }}$ on $\tilde{\mathsf{\Gamma }}$, rendering

$$\begin{array}{ccc}\hfill T_{{\mathbf{r}}^{\prime }}\tilde{\mathsf{\Gamma }}\left(\mathbf{r},{\mathbf{r}}^{\prime }\right)& =& {\displaystyle \frac{1-i{k}_{p}R}{\left(\lambda +2\mu \right)R^2}}\left[\widehat{{\mathbf{n}}^{\prime }}·\left(\lambda \tilde{\mathbf{I}}+2\mu \widehat{\mathbf{R}}\otimes \widehat{\mathbf{R}}\right)\otimes \widehat{\mathbf{R}}\right]e^{i{k}_{p}R}\hfill \\ \hfill \\ & & +{\displaystyle \frac{1-i{k}_{s}R}{R^2}}\left[\widehat{\mathbf{R}}\otimes \left(\tilde{\mathbf{I}}-\widehat{\mathbf{R}}\otimes \widehat{\mathbf{R}}\right)+\left(\tilde{\mathbf{I}}-\widehat{\mathbf{R}}\otimes \widehat{\mathbf{R}}\right)\otimes \widehat{\mathbf{R}}\right]·\widehat{{\mathbf{n}}^{\prime }}{e}^{i{k}_{s}R}\hfill \\ \hfill \\ & & -{\displaystyle \frac{2\mu }{\rho {\omega }^2{R}^4}}\left[\left(k_p^2{R}^2+3i{k}_{p}R-3\right)e^{i{k}_{p}R}-\left(k_s^2{R}^2+3i{k}_{s}R-3\right)e^{i{k}_{s}R}\right]\hfill \\ \hfill \\ & & \times \left\{\widehat{{\mathbf{n}}^{\prime }}·\left(\tilde{\mathbf{I}}-3\widehat{\mathbf{R}}\otimes \widehat{\mathbf{R}}\right)\otimes \widehat{\mathbf{R}}+\left[\widehat{\mathbf{R}}\otimes \left(\tilde{\mathbf{I}}-\widehat{\mathbf{R}}\otimes \widehat{\mathbf{R}}\right)+\left(\tilde{\mathbf{I}}-\widehat{\mathbf{R}}\otimes \widehat{\mathbf{R}}\right)\otimes \widehat{\mathbf{R}}\right]·\widehat{{\mathbf{n}}^{\prime }}\right\},\hfill \end{array}$$

(58)

wherein we have used the fact that $\rho {\omega }^2=\mu k_s^2=\left(\lambda +2\mu \right)k_p^2$. The low-frequency expansions of the dyadic fields (57) and (59) follow as much as we expand the exponentials $\exp \left(i{k}_{p}R\right)$ and $\exp \left(i{k}_{s}R\right)=\exp \left(i{k}_{p}R/\tau \right)$ via Taylor series around zero, both in terms of the expansion parameter that is the wave number $k_p$. Once we do that the fundamental solution (57) becomes

$$\tilde{\mathsf{\Gamma }}\left(\mathbf{r},{\mathbf{r}}^{\prime }\right)=\sum _{n=0}^{+\infty }{\left(i{k}_p\right)}^n{\tilde{\gamma }}_n\left(\mathbf{r},{\mathbf{r}}^{\prime }\right),$$

(59)

where

$${\tilde{\gamma }}_n\left(\mathbf{r},{\mathbf{r}}^{\prime }\right)={\displaystyle \frac{R^{n-1}}{n!\mu {\tau }^n}}\left[\left(1+{\displaystyle \frac{{\tau }^{n+2}-1}{n+2}}\right)\tilde{\mathbf{I}}+\left(n-1\right){\displaystyle \frac{{\tau }^{n+2}-1}{n+2}}\widehat{\mathbf{R}}\otimes \widehat{\mathbf{R}}\right]\mathrm{for}n\ge 0$$

(60)

at low frequencies, while the surface traction that operates on the fundamental dyadic (see (59) for instance) has the low-frequency form

$$T_{{\mathbf{r}}^{\prime }}\tilde{\mathsf{\Gamma }}\left(\mathbf{r},{\mathbf{r}}^{\prime }\right)=\sum _{n=0}^{+\infty }{\left(i{k}_p\right)}^n{T}_{{\mathbf{r}}^{\prime }}{\tilde{\gamma }}_n\left(\mathbf{r},{\mathbf{r}}^{\prime }\right),$$

(61)

where

$$\begin{array}{ccc}\hfill T_{{\mathbf{r}}^{\prime }}{\tilde{\gamma }}_n\left(\mathbf{r},{\mathbf{r}}^{\prime }\right)& =& -{\displaystyle \frac{\left(n-1\right)R^{n-2}}{\left(n+2\right)n!\mu {\tau }^n}}\{\mu \left(2{\tau }^{n+2}+n\right)\left[\left(\widehat{{\mathbf{n}}^{\prime }}·\widehat{\mathbf{R}}\right)\tilde{\mathbf{I}}+\widehat{\mathbf{R}}\otimes \widehat{{\mathbf{n}}^{\prime }}\right]\hfill \\ \hfill \\ & & +2\mu \left(n-3\right)\left({\tau }^{n+2}-1\right)\left[\left(\widehat{{\mathbf{n}}^{\prime }}·\widehat{\mathbf{R}}\right)\widehat{\mathbf{R}}\otimes \widehat{\mathbf{R}}\right]\hfill \\ \hfill \\ & & +\left[\lambda \left(n+2\right){\tau }^{n+2}+2\mu \left({\tau }^{n+2}-1\right)\right]\widehat{{\mathbf{n}}^{\prime }}\otimes \widehat{\mathbf{R}}\}\mathrm{for}n\ge 0,\hfill \end{array}$$

(62)

which concludes this task. Moreover, the low-frequency expansion of the integral representation of the scattered displacement dyadic field ${\tilde{\mathbf{u}}}^{sct}$ is recovered trivially if we substitute (49) and (59) with (60) and (61) with (62) into (17); consequently, we end up with the double-series expansion

$${\tilde{\mathbf{u}}}^{sct}\left(r\right)=\sum _{n,m=0}^{+\infty }\left\{{\int }_{\partial D}\left[{\tilde{\mathbf{u}}}_n^{sct}(\mathbf{r},{\mathbf{r}}^{\prime })·{\left(T_{{\mathbf{r}}^{\prime }}{\tilde{\gamma }}_m\left({\mathbf{r},\mathbf{r}}^{\prime }\right)\right)}^{\top }-{\tilde{\gamma }}_m\left({\mathbf{r},\mathbf{r}}^{\prime }\right)·T_{{\mathbf{r}}^{\prime }}{\tilde{\mathbf{u}}}_n^{sct}(\mathbf{r},{\mathbf{r}}^{\prime })\right]ds\left({\mathbf{r}}^{\prime }\right)\right\}{\left(i{k}_p\right)}^{n+m},$$

(63)

in which the interchange between integral and series is possible due to the uniform convergence of the implicated power series.

Otherwise, under the aim of finding similar low-frequency expansions for the asymptotic forms (15) and (16), we expand via classical Maclaurin series the exponentials $\exp \left[i{k}_p\left(\widehat{\mathbf{r}}·{\mathbf{r}}^{\prime }\right)\right]$ and $\exp \left[i{k}_s\left(\widehat{\mathbf{r}}·{\mathbf{r}}^{\prime }\right)\right]=\exp \left[i{k}_p\left(\widehat{\mathbf{r}}·{\mathbf{r}}^{\prime }\right)/\tau \right]$, concluding to the following low-frequency expansions

$${\tilde{\mathsf{\Gamma }}}_{\infty }^p(\mathbf{r},{\mathbf{r}}^{\prime })=\sum _{n=1}^{+\infty }{\left(i{k}_p\right)}^n{\tilde{\gamma }}_{\infty ,n}^p\left(\mathbf{r},{\mathbf{r}}^{\prime }\right)$$

(64)

and

$${\tilde{\mathsf{\Gamma }}}_{\infty }^s(\mathbf{r},{\mathbf{r}}^{\prime })=\sum _{n=1}^{+\infty }{\left(i{k}_p\right)}^n{\tilde{\gamma }}_{\infty ,n}^s\left(\mathbf{r},{\mathbf{r}}^{\prime }\right),$$

(65)

for the longitudinal and transverse far-field patterns of the fundamental dyadic $\tilde{\mathsf{\Gamma }}$, respectively. The leading function coefficients within (64) and (65) are proved to have the expressions

$${\tilde{\gamma }}_{\infty ,n}^p(\mathbf{r},{\mathbf{r}}^{\prime })={\displaystyle \frac{\widehat{\mathbf{r}}\otimes \widehat{\mathbf{r}}}{\lambda +2\mu }}\left[{\displaystyle \frac{{(-1)}^{n-1}{\left(\widehat{\mathbf{r}}·{\mathbf{r}}^{\prime }\right)}^{n-1}}{(n-1)!}}\right]\mathrm{for}n\ge 1$$

(66)

and

$${\tilde{\gamma }}_{\infty ,n}^s(\mathbf{r},{\mathbf{r}}^{\prime })={\displaystyle \frac{\tilde{\mathbf{I}}-\widehat{\mathbf{r}}\otimes \widehat{\mathbf{r}}}{\mu }}\left[{\displaystyle \frac{{(-1)}^{n-1}{\left(\widehat{\mathbf{r}}·{\mathbf{r}}^{\prime }\right)}^{n-1}}{\tau n(n-1)!}}\right]\mathrm{for}n\ge 1,$$

(67)

correspondingly. In particular, the relationships that provide us with the longitudinal and transverse far-field patterns in the low-frequency regime can be retrieved once formulae (19) and (20) are rewritten in a more convenient fashion with respect to (15) and (16), that is

$${\tilde{\mathbf{u}}}_{\infty }^p\left(\widehat{\mathbf{r}}\right)={\displaystyle \frac{\left(\lambda +2\mu \right)\left(i{k}_p\right)}{4\pi {\omega }^2}}{\int }_{\partial D}\left[{\left({\tilde{\mathsf{\Gamma }}}_{\infty }^p(\mathbf{r},{\mathbf{r}}^{\prime })\right)}^{\top }·\left(T_{{\mathbf{r}}^{\prime }}{\tilde{\mathbf{u}}}^{tot}\left({\mathbf{r}}^{\prime }\right)\right)-{\left(T_{{\mathbf{r}}^{\prime }}{\tilde{\mathsf{\Gamma }}}_{\infty }^p(\mathbf{r},{\mathbf{r}}^{\prime })\right)}^{\top }·{\tilde{\mathbf{u}}}^{tot}\left({\mathbf{r}}^{\prime }\right)\right]ds\left({\mathbf{r}}^{\prime }\right)$$

(68)

and

$${\tilde{\mathbf{u}}}_{\infty }^s\left(\widehat{\mathbf{r}}\right)={\displaystyle \frac{\mu \left(i{k}_p\right)}{4\pi {\omega }^2\tau }}{\int }_{\partial D}\left[{\left({\tilde{\mathsf{\Gamma }}}_{\infty }^s(\mathbf{r},{\mathbf{r}}^{\prime })\right)}^{\top }·\left(T_{{\mathbf{r}}^{\prime }}{\tilde{\mathbf{u}}}^{tot}\left({\mathbf{r}}^{\prime }\right)\right)-{\left(T_{{\mathbf{r}}^{\prime }}{\tilde{\mathsf{\Gamma }}}_{\infty }^s(\mathbf{r},{\mathbf{r}}^{\prime })\right)}^{\top }·{\tilde{\mathbf{u}}}^{tot}\left({\mathbf{r}}^{\prime }\right)\right]ds\left({\mathbf{r}}^{\prime }\right),$$

(69)

respectively. Then, we incorporate the low-frequency expansions (46), (64), and (65) into (68) and (69) in order to obtain

$$\begin{array}{ccc}\hfill {\tilde{\mathbf{u}}}_{\infty }^p\left(\widehat{\mathbf{r}}\right)& =& {\displaystyle \frac{\left(\lambda +2\mu \right)}{4\pi {\omega }^2}}\{\sum _{n,m=1}^{+\infty }{\left(i{k}_p\right)}^{n+m}{\int }_{\partial D}\left[{\left({\tilde{\gamma }}_{\infty ,m}^p(\mathbf{r},{\mathbf{r}}^{\prime })\right)}^{\top }·\left(T_{{\mathbf{r}}^{\prime }}{\tilde{\mathbf{u}}}_{n-1}^{tot}\left({\mathbf{r}}^{\prime }\right)\right)\right]ds\left({\mathbf{r}}^{\prime }\right)\hfill \\ \hfill \\ & & -\sum _{n,m=1}^{+\infty }{\left(i{k}_p\right)}^{n+m}{\int }_{\partial D}\left[{\left(T_{{\mathbf{r}}^{\prime }}{\tilde{\gamma }}_{\infty ,m}^p(\mathbf{r},{\mathbf{r}}^{\prime })\right)}^{\top }·{\tilde{\mathbf{u}}}_{n-1}^{tot}\left({\mathbf{r}}^{\prime }\right)\right]ds\left({\mathbf{r}}^{\prime }\right)\},\hfill \end{array}$$

(70)

and

$$\begin{array}{ccc}\hfill {\tilde{\mathbf{u}}}_{\infty }^s\left(\widehat{\mathbf{r}}\right)& =& {\displaystyle \frac{\mu }{4\pi {\omega }^2}}\{\sum _{n,m=1}^{+\infty }{\left(i{k}_p\right)}^{n+m}{\int }_{\partial D}\left[{\left({\tilde{\gamma }}_{\infty ,m}^s(\mathbf{r},{\mathbf{r}}^{\prime })\right)}^{\top }·\left(T_{{\mathbf{r}}^{\prime }}{\tilde{\mathbf{u}}}_{n-1}^{tot}\left({\mathbf{r}}^{\prime }\right)\right)\right]ds\left({\mathbf{r}}^{\prime }\right)\hfill \\ \hfill \\ & & -\sum _{n,m=1}^{+\infty }{\left(i{k}_p\right)}^{n+m}{\int }_{\partial D}\left[{\left(T_{{\mathbf{r}}^{\prime }}{\tilde{\gamma }}_{\infty ,m}^s(\mathbf{r},{\mathbf{r}}^{\prime })\right)}^{\top }·{\tilde{\mathbf{u}}}_{n-1}^{tot}\left({\mathbf{r}}^{\prime }\right)\right]ds\left({\mathbf{r}}^{\prime }\right)\},\hfill \end{array}$$

(71)

wherein ${\tilde{\gamma }}_{\infty ,m}^p$ and ${\tilde{\gamma }}_{\infty ,m}^s$ for $m\ge 1$ are given by (66) and (67), respectively. The separate cases of the rigid body (${\tilde{\mathbf{u}}}_{n-1}^{tot}\left({\mathbf{r}}^{\prime }\right)=\tilde{0}$ for $n\ge 1$) and the cavity ($T_{{\mathbf{r}}^{\prime }}{\tilde{\mathbf{u}}}_{n-1}^{tot}\left({\mathbf{r}}^{\prime }\right)=\tilde{0}$ for $n\ge 1$) are recovered from (70) and (71), leading to more simplified integral representations for the longitudinal and transverse far fields.

Finally, the total scattering cross-section ${\sigma }^{sct}$ due to the disturbance that is caused by the scatterer to the propagation of the incident plane wave yields the three-dimensional integral

$${\sigma }^{sct}={\displaystyle \frac{1}{2\pi }}{\int }_{\Omega }\sigma \left(\widehat{\mathbf{r}}\right)ds\left(\widehat{\mathbf{r}}\right),$$

(72)

being given in terms of the differential scattering cross-section $\sigma$, which assumes the relation

$$\sigma \left(\widehat{\mathbf{r}}\right)\simeq {\displaystyle \frac{2\pi \left(\parallel {\tilde{\mathbf{u}}}_{\infty }^p\left(\widehat{\mathbf{r}}\right){\parallel }^2+\tau {\parallel {\tilde{\mathbf{u}}}_{\infty }^s\left(\widehat{\mathbf{r}}\right)\parallel }^2\right)}{\left(\widehat{\mathbf{d}}\otimes \widehat{\mathbf{d}}:\mathbf{c}\otimes \mathbf{c}\right)+\tau \left(\tilde{\mathbf{I}}-\widehat{\mathbf{d}}\otimes \widehat{\mathbf{d}}\right):\mathbf{c}\otimes \mathbf{c}}},$$

(73)

reminding that $\tau =k_p/k_s$. The low-frequency behaviour of the far-field patterns ${\tilde{\mathbf{u}}}_{\infty }^p$ and ${\tilde{\mathbf{u}}}_{\infty }^s$ is given in (70) and (71), respectively, for both the cases of a rigid body and a cavity, as discussed earlier. Thereafter, implying (72) and (73), the low-frequency expansion for the scattering cross-section of a rigid body or a cavity follows in an easily amenable manner.

6. Conclusions, Remarks, and Suggestions

In this paper, the elastic scattering problem of plane dyadic waves for a rigid body or a cavity in three-dimensional linear elasticity was considered. The formulation of our problem was in a dyadic form due to the dyadic nature of the free-space fundamental solution. We also make the following remarks:

(i)

The dyadic formulation was also based (except for the aforementioned reason) on the symmetry that we gain for our scattering problem, i.e., the direction of propagation alone $\widehat{\mathbf{d}}$ suffices to determine the polarization vector ${\widehat{\mathbf{d}}}^{\perp }$ as well (see Equation (1)).

(ii)

In this work, two types of boundary value problems were addressed: one with the displacement field vanished on the surface of the scatterer, i.e., the rigid problem, and a second one with the case where no surface traction appears on the boundary, i.e., the cavity problem. For these cases, functionals for the differential and the scattering cross-section were established and analogous results were proved for the low-frequency scattering case. Some modern real-life applications enable low-frequency assumption where the size of the scatterers is much smaller than the incident wavelength.

(iii)

Last, but not least, we mention that essential expressions for energy functionals in the far-field region along with expressions for the differential and the total scattering cross-section were provided; also, an expression of the differential scattering cross-section for the rigid and cavity obstacle case was established.

(iv)

The authors suggest, as a future work, the extension of this work to the transmission case (a penetrable obstacle where transmission boundary conditions occurred) and exploit similar results.