# Factorization method for inverse time-harmonic elastic scattering with a
single plane wave

Guanqiu Ma111Department of Applied Mathematics, Beijing Computational Science
Research Center, Beijing 100193, P. R. China (guanqium@csrc.ac.cn). Guanghui
Hu222School of Mathematical Sciences and LPMC, Nankai Universtiy, Tianjin
300071, China<EMAIL_ADDRESS>corresponding author).

###### Abstract

This paper is concerned with the factorization method with a single far-field
pattern to recover an arbitrary convex polygonal scatterer/source in linear
elasticity. The approach also applies to the compressional (resp. shear) part
of the far-field pattern excited by a single compressional (resp. shear) plane
wave. The one-wave factorization is based on the scattering data for a priori
given testing scatterers. It can be regarded as a domain-defined sampling
method and does not require forward solvers. We derive the spectral system of
the far-field operator for rigid disks and show that, using testing disks, the
one-wave factorization method can be justified independently of the classical
factorization method.

Keywords: factorization method, inverse elastic scattering, linear elasticity,
single far-field pattern, polygonal scatterers, corner scattering.

##  1 Introduction

The purpose of inverse time-harmonic elastic scattering is to recover the
position, shape and physical properties of an elastic body by using
information of the scattered wave generated by time-harmonic plane and point
source waves. We refer to [21, 6, 2] for a comprehensive introduction of
mathematical theory and inverse problems in linear elasticity.

Over the last twenty years, sampling-type methods have attracted much
attention, because forward solvers and good initial approximations of the
target are no longer required, in contrast with the iterative approaches. The
multi-wave sampling methods do not require a priori information on physical
and geometrical properties of the scatterer, but usually need far-field data
for a large number of incident waves. Here we give an incomplete list of the
applications to the Navier equation, including linear sampling and
factorization methods [3, 1, 7, 8, 15], singular source method [11],
orthogonal/direct sampling method [19], enclosure method[13] and the Reverse
time-migration method in the frequency domain [9]. On the other hand, there
also exists the so-called one-wave sampling methods, which are usually
designed to test the analytic extensibility of the scattered field; see the
monograph [29, Chapter 15] for detailed discussions on scalar equations, for
instance, range test and no-response test [22, 24, 27] and enclosure method
[16, 17]. The one-wave method requires only a single far-field pattern or one-
pair Cauchy data, but one must pre-assume the absence of an analytical
continuation across the scattering interface.

If a single far-field pattern is available only, the inverse scattering
problems become severely ill-posed and thus more challenging. This paper is
concerned with the one-wave factorization method for recovering a convex rigid
elastic body of polygonal type from a single elastic far-field pattern. Such a
method was earlier discussed in [12] for inverse elastic scattering from rigid
polygonal bodies but without too much details. It is closet to the extended
linear sampling method [25, 26] and the one-wave range test method [22]. In
the authors’ previous work [28], the one-wave factorization method for the
Helmholtz equation was rigorously established with the help of corner
scattering theory. The connections to the range test and extended linear
sampling were also discussed there. The one-wave factorization method is a
both data-driven and model driven method, and could lead to an explicit
characterization of an arbitrary convex scatterer of polygonal type if the
testing scatterers are chosen as disks. In this sense, it inherits merits of
the classical factorization method for precisely characterizing targets [20]
but restricted to convex polygonal scatterers/sources. The purpose of this
paper is to generalize the mathematical theory of [28] to the Navier equation.
The following items can be considered as complementary contributions to the
previous work [12]: i) One-wave factorization method using only compressional
or shear waves in linear elasticity; ii) Explicit expression of the spectral
data for elastic far-field operators corresponding to rigid disks and a
straightforward verification of the one-wave factorization method by using
testing disks.

This paper is organized as follows. In Section 2, we introduce basic concepts
of the direct and inverse elastic scattering problems. In Section 3, the
multi-wave factorization method for recovering a rigid scatterers will be
briefly reviewed. In Section 4, we present a rigorous justification of the
one-wave method by combining the classical factorization method and elastic
corner scattering theory. Explicit examples by using testing disks will be
presented in Section 5, including derivation of an eigensystem of the far-
field operator for a rigid disk. Finally, we describe our imaging schemes in
Section 6.

##  2 Preliminaries

In this paper, we will consider the scattering of elastic waves in two-
dimensional space $\mathbb{R}^{2}$. Let $D\subset\mathbb{R}^{2}$ be a bounded
rigid elastic body with connected exterior
$D^{c}\coloneqq\mathbb{R}^{2}\backslash\bar{D}$. Let $D^{c}$ be filled with a
homogeneous and isotropic elastic medium. Suppose that a time-harmonic elastic
plane wave of the form

$u^{i}\left(x;d,c_{p},c_{s}\right)=c_{p}d\text{e}^{ik_{p}x\cdot
d}+c_{s}d^{\bot}\text{e}^{ik_{s}x\cdot d},\quad
c_{p},c_{s}\in\mathbb{C},\quad|c_{p}|+|c_{s}|\neq 0,$ (2.1)

is incident onto the scatterer $D$. Here
$d=(\cos\theta_{d},\sin\theta_{d})^{T},\theta_{d}\in\left[0,2\pi\right)$ is
the incident direction;
$d^{\bot}\coloneqq(-\cos\theta_{d},\sin\theta_{d})^{T}$ is a vector orthogonal
to $d$; $\omega>0$ is the frequency;
$k_{p}\coloneqq\omega/\sqrt{\lambda+2\mu}$ and
$k_{s}\coloneqq\omega/\sqrt{\mu}$ are the compressional and shear wave
numbers, respectively. Note that for simplicity the density of the background
medium has been normalized to be one and the Lame constants $\lambda$ and
$\mu$ satisfy $\mu>0$ and $\lambda+2\mu>0$ in two dimensions. The propagation
of time-harmonic elastic waves in $D^{c}$ is governed by the Navier equation
(or system)

$\Delta^{\ast}u+\omega^{2}u\coloneqq\mu\vartriangle
u+\left(\lambda+\mu\right)\triangledown\left(\triangledown\cdot
u\right)+\omega^{2}u=0\quad\text{in}\quad D^{c},\quad
u=\left(u_{1},u_{2}\right)^{T},$ (2.2)

where $u=u^{i}+u^{s}$ donotes the total displacement field. By Hodge
decomposition, any solution $u$ to (2.2) can be decomposed into the from

$u=u_{p}+u_{s},\quad u_{p}\coloneqq-\frac{1}{k^{2}_{p}}\ \text{grad}\
\text{div}\ u,\quad u_{s}\coloneqq\frac{1}{k^{2}_{s}}\ \text{curl}\
\overrightarrow{\text{curl}}\ u,$ (2.3)

where $u_{p}$ and $u_{s}$ are called compressional and shear waves
respectively. Note that in (2.3) the two curl operators are defined as

$\overrightarrow{\text{curl}}\
u\coloneqq\partial_{2}u_{1}-\partial_{1}u_{2},\quad\text{curl}\
f=\left(-\partial_{2}f,\partial_{1}f\right)^{T},$ (2.4)

satisfying the relation

$\text{curl}\ \overrightarrow{\text{curl}}\ u=-\Delta u+\text{grad}\
\text{div}\ u.$ (2.5)

Moreover, $u_{\alpha}\left(\alpha=p,s\right)$ satisfies the vector Helmholtz
equations $\left(\Delta+k^{2}_{\alpha}\right)u_{\alpha}=0$ and
$\overrightarrow{\text{curl}}\ u_{p}=\text{div}\ u_{s}=0$ in $D^{c}$. In this
paper, we require $u^{s}$ to fulfill the Kupradze radiation condition

$\partial_{r}u^{s}_{\alpha}-ik_{\alpha}u^{s}_{\alpha}=o\left(r^{-\frac{1}{2}}\right)\quad\text{as}\quad
r=|x|\to\infty,\quad\alpha=p,s,$ (2.6)

uniformly in all directions $\hat{x}=x/|x|$ on the unit circle
$\mathbb{S}\coloneqq\left\\{x\in\mathbb{R}^{2}:|x|=1\right\\}.$ It is well
known that the direct scattering problem admits one solution $u\in
C^{2}({\mathbb{R}}^{2}\backslash\overline{D})\cap
C^{1}({\mathbb{R}}^{2}\backslash D)$ if $\partial D$ is of $C^{2}$-smooth (see
[21]) and $u\in(H^{1}_{loc}({\mathbb{R}}^{2}\backslash\overline{D}))^{2}$ if
$\partial D$ is Lipschitz (see e.g., [4, 23]).

This paper is concerned with the inverse scattering problem of recovering
$\partial D$ from the information of the far-field pattern of a single
incoming plane wave. The compressional and shear parts
$u^{s}_{\alpha}\left(\alpha=p,s\right)$ of the radiating solution $u^{s}$
admit an asymptotic behavior of the form [12, 14]

$\begin{split}u^{s}_{p}\left(x\right)&=\frac{\text{e}^{ik_{p}r}}{\sqrt{r}}\left\\{u^{\infty}_{p}\left(\hat{x}\right)\hat{x}+\mathcal{O}\left(\frac{1}{r}\right)\right\\},\\\
u^{s}_{s}\left(x\right)&=\frac{\text{e}^{ik_{s}r}}{\sqrt{r}}\left\\{u^{\infty}_{s}\left(\hat{x}\right)\hat{x}^{\perp}+\mathcal{O}\left(\frac{1}{r}\right)\right\\}\end{split}$
(2.7)

as $r=|x|\to\infty$, where $u^{\infty}_{p}$ and $u^{\infty}_{s}$ are both
scalar functions defined on $\mathbb{S}$. Hence, a Kupradze radiating solution
has the asymptotic behavior

$u^{s}\left(x\right)=\frac{\text{e}^{ik_{p}r}}{\sqrt{r}}u^{\infty}_{p}\left(\hat{x}\right)\hat{x}+\frac{\text{e}^{ik_{s}r}}{\sqrt{r}}u^{\infty}_{s}\left(\hat{x}\right)\hat{x}^{\perp}+\mathcal{O}\left(\frac{1}{r^{\frac{3}{2}}}\right)\quad\text{as}\quad
r\to\infty.$ (2.8)

The far-field pattern $u^{\infty}$ of $u^{s}$ is defined as

$u^{\infty}\left(\hat{x}\right)\coloneqq
u^{\infty}_{p}\left(\hat{x}\right)\hat{x}+u^{\infty}_{s}\left(\hat{x}\right)\hat{x}^{\perp}.$
(2.9)

Then the compressional and shear parts of the far field are uniquely
determined by $u^{\infty}$ as

$u^{\infty}_{p}\left(\hat{x}\right)=u^{\infty}\left(\hat{x}\right)\cdot\hat{x},\quad
u^{\infty}_{s}\left(\hat{x}\right)=u^{\infty}\left(\hat{x}\right)\cdot\hat{x}^{\perp}.$
(2.10)

Introduce the compressional and shear parts of $u^{i}$ by

$u^{i}_{p}\left(x;d\right)\coloneqq u^{i}\left(x;d,1,0\right),\
u^{i}_{s}\left(x;d\right)\coloneqq u^{i}\left(x;d,0,1\right).$ (2.11)

Obviously,

$\displaystyle
u^{i}(x;d,c_{p},c_{s})=c_{p}u^{i}_{p}(x;d)+c_{s}u^{i}_{s}(x;d).$ (2.12)

Correspondingly, for the far-filed pattern, we have

$\begin{split}u^{\infty}_{pp}\left(\hat{x};d\right)&\coloneqq
u^{\infty}_{p}\left(\hat{x};d,1,0\right),\\\
u^{\infty}_{sp}\left(\hat{x};d\right)&\coloneqq
u^{\infty}_{s}\left(\hat{x};d,1,0\right),\\\
u^{\infty}_{ps}\left(\hat{x};d\right)&\coloneqq
u^{\infty}_{p}\left(\hat{x};d,0,1\right),\\\
u^{\infty}_{ss}\left(\hat{x};d\right)&\coloneqq
u^{\infty}_{s}\left(\hat{x};d,0,1\right).\end{split}$ (2.13)

Thus, we obtain

$u^{\infty}(\hat{x};d,c_{p},c_{s})=\left(c_{p}u^{\infty}_{pp}(\hat{x})+c_{s}u^{\infty}_{ps}(\hat{x})\right)\hat{x}+\left(c_{p}u^{\infty}_{sp}(\hat{x})+c_{s}u^{\infty}_{ss}(\hat{x})\right)\hat{x}^{\perp},$
(2.14)

where the dependence of $u^{\infty}_{\alpha\beta}$ $(\alpha,\beta=p,s)$ on $d$
has been omitted for simplicity. In this paper, the following inverse elastic
scattering problems will be considered:

IP-P: Reconstruct the shape and position of the scattere $D$ from knowledge of
the compressional part $u^{\infty}_{pp}(\hat{x})$ of the far-field pattern due
to one incident compressional wave $u^{i}_{p}$.

IP-S: Reconstruct the shape and position of $D$ from knowledge of the shear
part $u^{\infty}_{ss}(\hat{x})$ of the far-field pattern due to one incident
shear wave $u^{i}_{s}$.

IP-F: Reconstruct the shape and position of $D$ from using the entire far-
field pattern $u^{\infty}(\hat{x})$ due to one incident wave $u^{i}$.

##  3 Factorization method with infinitely many plane waves

###  3.1 Review of the classical Factorization method for inverse elastic
scattering

Given a vector field
$g(d)=g_{p}(d)d+g_{s}(d)d^{\bot}\in\left(L^{2}(\mathbb{S})\right)^{2}$, the
superposition of plane waves

$v_{g}(x):=\text{
e}^{-i\frac{\pi}{4}}\int_{{\mathbb{S}}}\left\\{\sqrt{\frac{k_{p}}{\omega}}d\text{e}^{ik_{p}x\cdot
d}g_{p}(d)+\sqrt{\frac{k_{s}}{\omega}}d^{\bot}\text{e}^{ik_{s}x\cdot
d}g_{s}(d)\right\\}ds(d)$ (3.1)

is denoted as the elastic Herglotz wave function with density $g$. The Green’s
tensor of the Navier equation in free space, also called Kupradze’s tensor
(see e.g., [3]), is denoted by

$\Gamma(x,y)\coloneqq\frac{i}{4\mu}H^{(1)}_{0}(k_{s}|x-y|){\bf
I}+\frac{i}{4\omega^{2}}\triangledown^{\bot}_{x}\triangledown_{x}\left(H^{(1)}_{0}(k_{s}|x-y|)-H^{(1)}_{0}(k_{p}|x-y|)\right),\
x,y\in\mathbb{R}^{2},\ x\neq y,$ (3.2)

where $H^{(1)}_{0}$ is the Hankel function of the first kind and of order $n$.
For any $y\in{\mathbb{R}}^{2}$ and any direction $a\in{\mathbb{S}}$, an
elastic point source in $y$ with the polarization $a$ is given by

$u(x)=\Gamma(x,y)a,\quad x\in{\mathbb{R}}^{2}\backslash\\{y\\}.$ (3.3)

The far-field pattern $\Gamma^{\infty}(\cdot,y;a)$ of this point source is
given by

$\Gamma^{\infty}(\hat{x},y;a)=\frac{k^{2}_{p}}{\omega^{2}}\frac{\text{e}^{i\frac{\pi}{4}}}{\sqrt{8\pi
k_{p}}}\text{e}^{-ik_{p}\hat{x}\cdot y}(\hat{x}\cdot
a)\hat{x}+\frac{k^{2}_{s}}{\omega^{2}}\frac{\text{e}^{i\frac{\pi}{4}}}{\sqrt{8\pi
k_{s}}}\text{e}^{-ik_{s}\hat{x}\cdot y}(\hat{x}^{\bot}\cdot a)\hat{x}^{\bot},$
(3.4)

with the compressional and shear parts:

$\Gamma^{\infty}_{p}(\hat{x},y;a)=\Gamma^{\infty}(\hat{x},y;a)\cdot\hat{x},\quad\Gamma^{\infty}_{s}(\hat{x},y;a)=\Gamma^{\infty}(\hat{x},y;a)\cdot\hat{x}^{\bot}.$
(3.5)

In this paper we define the elastic far-field operator as follows.

######  Definition 3.1.

The far-field operator
$F_{D}:\left(L^{2}(\mathbb{S})\right)^{2}\to\left(L^{2}(\mathbb{S})\right)^{2}$
is defined by

$\begin{split}(F_{D}g)(\hat{x})&\coloneqq\text{
e}^{-i\frac{\pi}{4}}\int_{\mathtt{S}}u^{\infty}_{D}\left(\hat{x};d,\sqrt{\frac{k_{p}}{\omega}}g_{p}(d),\sqrt{\frac{k_{s}}{\omega}}g_{s}(d)\right)ds(d)\\\
&=\text{
e}^{-i\frac{\pi}{4}}\int_{\mathtt{S}}\left\\{\sqrt{\frac{k_{p}}{\omega}}u^{\infty}_{D}(\hat{x};d,1,0)g_{p}(d)+\sqrt{\frac{k_{s}}{\omega}}u^{\infty}_{D}(\hat{x};d,0,1)g_{s}(d)\right\\}ds(d).\end{split}$
(3.6)

For rigid elastic bodies, it is well known that $F_{D}$ is a normal operator.
It was proved in [3, Theorem 4.3] that the operator $F_{D}$ can be decomposed
into the form

$F_{D}=-\sqrt{8\pi\omega}G_{D}S^{\ast}_{D}G^{\ast}_{D}.$ (3.7)

Here the data-to-patten operator $G_{D}:\left(H^{1/2}(\partial
D)\right)^{2}\to\left(L^{2}(\mathbb{S})\right)^{2}$ is defined by
$G_{D}h\coloneqq u^{\infty}$, where $u^{\infty}$ is the far-field pattern of
the solution to the Dirichlet boundary value problem of the Navier equation
with the boundary value $h$. The operator
$S^{\ast}_{D}:\left(H^{-1/2}(\partial D)\right)^{2}\to\left(H^{1/2}(\partial
D)\right)^{2}$ is the adjoint of the elastic single layer potential operator
$S_{D}$, given by

$S_{D}\phi(x)\coloneqq\int_{\partial D}\Gamma(x,y)\phi(y)ds(y),\quad
x\in\partial D.$ (3.8)

By the Factorization method, the far-field pattern
$\Gamma^{\infty}(\cdot,z;a)$ belongs to the range of $G_{D}$ if and only if
$z\in D$ (see [3, Theorem 4.7]). Moreover, the $(F^{\ast}F)^{1/4}$-method (see
[3, Theorem 4.8]) verifies the relation
$\text{Range}(G_{D})=\text{Range}((F^{\ast}_{D}F_{D})^{1/4})$ if $\omega^{2}$
is not an eigenvalue of $-\Delta^{\ast}$ over $D$. Hence, by the Picard
theorem, the scatterer $D$ can be characterized by the spectra of $F_{D}$ as
follows.

######  Theorem 3.2.

([3, Theorem 4.8]) Assume that $\omega^{2}$ is not a Dirichlet eigenvalue of
$-\Delta^{\ast}$ over $D$. Denote by $(\lambda^{(n)}_{D},\varphi^{(n)}_{D})$ a
spectrum system of the far-field operator
$F_{D}:\left(L^{2}(\mathbb{S})\right)^{2}\rightarrow\left(L^{2}(\mathbb{S})\right)^{2}$.
Then,

$z\in D\Longleftrightarrow
I(z)\coloneqq\sum\limits_{n\in\mathbb{Z}}\frac{\left|\left\langle\Gamma^{\infty}(\cdot,z;a),\varphi_{D}^{(n)}\right\rangle_{\mathbb{S}}\right|^{2}}{\left|\lambda_{D}^{(n)}\right|}<+\infty.$
(3.9)

By Theorem 3.2, the sign of the indicator function $I(z)$ can be regarded as
the characteristic function of $D$. We note that in (3.9),
$z\in{\mathbb{R}}^{2}$ are the sampling variables/points and the spectral data
$(\lambda^{(n)}_{D},\varphi^{(n)}_{D})$ are determined by the far-field
patterns $u_{D}^{\infty}(\hat{x},d)$ over all observation and incident
directions $\hat{x},d\in{\mathbb{S}}$.

Below we state the Factorization method which involves only the compressional
or shear plane waves. Introduce the projection space
$L^{2}_{p}({\mathbb{S}})\coloneqq\left\\{g_{p}:{\mathbb{S}}\to{\mathbb{C}},g_{p}(d)=g(d)\cdot
d,|g_{p}|\in L^{2}({\mathbb{S}})\right\\}$ and
$L^{2}_{s}({\mathbb{S}})\coloneqq\left\\{g_{s}:{\mathbb{S}}\to{\mathbb{C}},g_{s}(d)=g(d)\cdot
d^{\bot},|g_{s}|\in L^{2}({\mathbb{S}})\right\\}$. Define the projection
operators $P_{p}:\left(L^{2}({\mathbb{S}})\right)^{2}\to
L^{2}_{p}({\mathbb{S}})$ and $P_{s}:\left(L^{2}({\mathbb{S}})\right)^{2}\to
L^{2}_{s}({\mathbb{S}})$ (see e.g., [15]) by

$P_{p}g(d):=g_{p}(d),\quad P_{s}g(d):=g_{s}(d).$ (3.10)

Thus, we can define the P-part and S-part of the far-field operator $F_{D}$.

######  Definition 3.3.

The far-field operator $F^{(p)}_{D}:L^{2}_{p}({\mathbb{S}})\to
L^{2}_{p}({\mathbb{S}})$ is defined by

$F^{(p)}_{D}g_{p}(d)\coloneqq P_{p}F_{D}P^{\ast}_{p}g_{p}(d).$ (3.11)

Analogously, the far-field operator $F^{(s)}_{D}:L^{2}_{s}({\mathbb{S}})\to
L^{2}_{s}({\mathbb{S}})$ is defined by

$F^{(s)}_{D}g_{s}(d)\coloneqq P_{s}F_{D}P^{\ast}_{s}g_{s}(d).$ (3.12)

Here $P^{\ast}_{p}$ and $P^{\ast}_{s}$ are the adjoint operators of $P_{p}$
and $P_{s}$, respectively.

By (3.7), we have the factorization

$F^{(\alpha)}_{D}=-\sqrt{8\pi\omega}(P_{\alpha}G_{D})S^{\ast}_{D}(P_{\alpha}G_{D})^{\ast},$
(3.13)

where $\alpha=p,s$. Based on (3.13), the $F_{\\#}$-method (see [15, Lemma
3.5]) verifies the relation
$\text{Range}(P_{\alpha}G_{D})=\text{Range}((F^{(\alpha)}_{D,\\#})^{1/2})$,
provided $\omega^{2}$ is not a Dirichlet eigenvalue of $-\Delta^{\ast}$ over
$D$. Here the operator $F_{\\#}$ is defined by

$F_{\\#}\coloneqq|\text{Re}F|+|\text{Im}F|,\quad\text{Re}F\coloneqq\frac{1}{2}[F+F^{\ast}],\quad\text{Im}F\coloneqq\frac{1}{2i}[F-F^{\ast}].$
(3.14)

Hence, the scatterer $D$ can be characterized by the spectra of
$F^{(\alpha)}_{D}$ as follows.

######  Theorem 3.4.

([15, Theorem 3.7, 3.8]) Assume that $\omega^{2}$ is not a Dirichlet
eigenvalue of $-\Delta^{\ast}$ over $D$. Denote by
$(\lambda^{(n)}_{D,\alpha},\varphi^{(n)}_{D,\alpha})$ a spectrum system of the
positive operator $F^{(\alpha)}_{D,\\#}$. Then,

$z\in D\Longleftrightarrow
I^{(\alpha)}(z)\coloneqq\sum\limits_{n\in\mathbb{Z}}\frac{\left|\left\langle\Gamma^{\infty}_{\alpha}(\cdot,z;a),\varphi_{D,\alpha}^{(n)}\right\rangle_{\mathbb{S}}\right|^{2}}{\left|\lambda_{D,\alpha}^{(n)}\right|}<+\infty,\quad\alpha=p,s.$
(3.15)

###  3.2 Further discussions on Factorization method

Before stating the one-wave version of the factorization method for inverse
elastic scattering, we first present a corollary of Theorems 3.2. Denote by
$\Omega\subset{\mathbb{R}}^{2}$ a convex and bounded Lipschitz domain which
represents a rigid elastic scatterer. Here we use a new notation $\Omega$ in
order to distinguish from our target scatterer $D$. The far-field operator
$F_{\Omega}:\left(L^{2}(\mathbb{S})\right)^{2}\rightarrow\left(L^{2}(\mathbb{S})\right)^{2}$
corresponding to $\Omega$ is therefore defined by

$(F_{\Omega}g)(\hat{x})\coloneqq\text{
e}^{-i\frac{\pi}{4}}\int_{\mathbb{S}}\left\\{\sqrt{\frac{k_{p}}{\omega}}u^{\infty}_{\Omega}(\hat{x};d,1,0)g_{p}(d)+\sqrt{\frac{k_{s}}{\omega}}u^{\infty}_{\Omega}(\hat{x};d,0,1)g_{s}(d)\right\\}ds(d),$
(3.16)

where $u^{\infty}_{\Omega}(\hat{x};d,1,0)$ and
$u^{\infty}_{\Omega}(\hat{x};d,0,1)$ are the far-field patterns corresponding
to the elastic plane waves $u^{i}_{p}(x;d)$ and $u^{i}_{s}(x;d)$ incident onto
$\Omega$, respectively. The eigenvalues and eigenfunctions of $F_{\Omega}$
will be denoted by $(\lambda^{(n)}_{\Omega},\varphi^{(n)}_{\Omega})$. The
following Corollaries can be derived straightforwardly from the classical
Factorization method in the previous subsection.

######  Corollary 3.5.

Let $v^{\infty}\in\left(L^{2}({\mathbb{S}})\right)^{2}$ and assume that
$\omega^{2}$ is not a Dirichlet eigenvalue of $-\Delta^{\ast}$ over $\Omega$.
Then

$I(\Omega)=\sum_{n\in\mathbb{Z}}\frac{\left|\left\langle
v^{\infty},\varphi^{(n)}_{\Omega}\right\rangle_{\mathbb{S}}\right|^{2}}{\left|\lambda^{(n)}_{\Omega}\right|}<+\infty$
(3.17)

if and only if $v^{\infty}$ is the far-filed pattern of some Kupradze
radiating solution $v^{s}$, where $v^{s}$ satisfies the Navier equation

$\Delta^{\ast}v^{s}+\omega^{2}v^{s}=0\qquad\text{in}\quad\mathbb{R}^{2}\backslash\overline{\Omega},$
(3.18)

with the boundary data
$v^{s}|_{\partial\Omega}\in\left(H^{1/2}(\partial\Omega)\right)^{2}.$

###### Proof.

By (3.7), we have
$F_{\Omega}=-\sqrt{8\pi\omega}G_{\Omega}S^{*}_{\Omega}G^{*}_{\Omega}$, where
$G_{\Omega}:\left(H^{1/2}(\partial\Omega)\right)^{2}\to\left(L^{2}(\mathbb{S})\right)^{2}$
is the data-to-pattern operator corresponding to $\Omega$. Obviously,
$I(\Omega)<+\infty$ if and only if
$v^{\infty}\in\text{Range}((F^{*}_{\Omega}F_{\Omega})^{1/4})$. Since
$\text{Range}((F^{*}_{\Omega}F_{\Omega})^{1/4})=\text{Range}(G_{\Omega})$, we
get $v^{\infty}\in\text{Range}(G_{\Omega})$ if and only if
$I(\Omega)<+\infty$. Recalling the definition of $G_{\Omega}$, it follows that
$v^{s}$ satisfies the Navier equation (3.18) and the Kupradze radiation
condition (2.6) with the boundary data
$v^{s}|_{\partial\Omega}\in\left(H^{1/2}(\partial\Omega)\right)^{2}.$ ∎

######  Corollary 3.6.

Let $w^{\infty}_{\alpha\alpha}\in L^{2}_{\alpha}({\mathbb{S}})$($\alpha=p,s$)
and assume that $\omega^{2}$ is not a Dirichlet eigenvalue of $-\Delta^{\ast}$
over $\Omega$. Denote by
$(\lambda^{(n)}_{\Omega,\alpha},\varphi^{(n)}_{\Omega,\alpha})$ a spectrum
system of the positive operator $F^{(\alpha)}_{\Omega,\\#}$. Then

$I^{(\alpha)}(\Omega)=\sum_{n\in\mathbb{Z}}\frac{\left|\left\langle
w^{\infty}_{\alpha\alpha},\varphi^{(n)}_{\Omega,\alpha}\right\rangle_{\mathbb{S}}\right|^{2}}{\left|\lambda^{(n)}_{\Omega,\alpha}\right|}<+\infty$
(3.19)

if and only if $w^{\infty}_{\alpha\alpha}=P_{\alpha}v^{\infty}$, where
$v^{\infty}$ is the far-field pattern of some Kupradze radiating solution
$v^{s}$, which is defined in $\mathbb{R}^{2}\backslash\overline{\Omega}$ and
$v^{s}|_{\partial\Omega}\in\left(H^{1/2}(\partial\Omega)\right)^{2}$. That is,
$w^{\infty}_{\alpha\alpha}$ is the far-field pattern of some Sommerfeld
radiating solution $w^{s}_{\alpha\alpha}$, fulfilling the relations

$w_{pp}^{s}=-\frac{1}{k_{p}^{2}}{\rm
div\,}v^{s}\quad\mbox{if}\quad\alpha=p;\quad
w_{ss}^{s}=\frac{1}{k_{s}^{2}}{\rm curl\,}v^{s}\quad\mbox{if}\quad\alpha=s,$
(3.20)

where $v^{s}$ satisfies the boundary value problem of the Navier equation

$\Delta^{\ast}v^{s}+\omega^{2}v^{s}=0\ \text{in}\
\mathbb{R}^{2}\backslash\overline{\Omega},\quad
v^{s}|_{\partial\Omega}\in\left(H^{1/2}(\partial\Omega)\right)^{2}.$ (3.21)

Similar to Corollary 3.5, Corollary 3.6 can be proved by using the projection
operator $P_{\alpha}$ and the $F_{\\#}$-method.

###### Proof.

Without loss of generality, we assume that the relation (3.19) holds with
$\alpha=p$. By the Picard theorem,
$w^{\infty}_{pp}\in\text{Range}(F^{(p)}_{\Omega,\\#})=\text{Range}(P_{p}G_{\Omega})$
if and only if the indicator function $I^{(p)}(\Omega)<+\infty$. This means
that $w^{\infty}_{pp}=P_{p}v^{\infty}$, where $v^{\infty}$ is the far-field
pattern of a Kupradze’s radiating solution $v^{s}$ to the boundary value
problem (3.21). Using the Hodge decomposition, we see $w^{\infty}_{pp}$ is the
far-field pattern of $w^{s}_{pp}$, where
$w^{s}_{pp}=-\frac{1}{k_{p}^{2}}\text{div}v^{s}$.

On the other hand, let $v^{s}$ be a solution to the boundary value problem
(3.21) and define $w_{pp}^{s}:=-\frac{1}{k_{p}^{2}}{\rm div\,}v^{s}$. Suppose
that $w^{\infty}_{pp}$ is the far-field pattern of $w^{s}_{pp}$. Then, by the
Hodge decomposition it follows that
$w^{\infty}_{pp}=P_{p}v^{\infty}=P_{p}G_{\Omega}(v^{s}|_{\partial\Omega})$,
implying that
$w^{\infty}_{pp}\in\text{Range}(P_{p}G_{\Omega})=\text{Range}(F^{(p)}_{\Omega,\\#})$.
Applying the Picard theorem yields $I^{(p)}(\Omega)<+\infty$. ∎

######  Remark 3.7.

It follows from (3.20) that the restrictions of $w^{s}_{pp}$, $w^{s}_{ss}$ to
$\partial\Omega$ lie in the space $H^{-1/2}(\partial\Omega)$.

##  4 Factorization method with one plane waves

In our applications of Corollary 3.5 (resp. Corollary 3.6), we will take
$v^{\infty}$ to be the measurement data
$u_{D}^{\infty}(\hat{x};d_{0},c_{p},c_{s})$ (resp.
$u_{D,\alpha\alpha}^{\infty}(\hat{x};d_{0})$, $\alpha=p,s$) corresponding to
our target elastic scatterer $D$ and the incident elastic plane wave
$u^{i}(x;d_{0},c_{p},c_{s})$ (resp. $u^{i}_{\alpha}(\hat{x};d_{0})$) for some
fixed $d_{0}\in{\mathbb{S}}$. We shall omit the dependance on $d_{0}$ if it is
always clear from the context. Our purpose is to extract the geometrical
information on $D$ from the domain-defined indicator functions $I(\Omega)$ and
$I^{(\alpha)}(\Omega)$ ($\alpha=p,s$). By Corollary 3.5, $I(\Omega)<\infty$ if
the scattered field $u^{s}_{D}(x)=u^{s}(x;d_{0},D)$ can be extended to the
domain ${\mathbb{R}}^{2}\backslash\overline{\Omega}$ as a solution to the
Navier equation. Below we shall discuss the absence of the analytic extension
of $u^{s}_{D}$, $u^{s}_{D,p}$ and $u^{s}_{D,s}$ around a planer corner point
of $D$.

######  Lemma 4.1.

Assume that $D$ is a rigid elastic scatterer occupying a convex polygon. Then
the scattered field $u^{s}_{D}(x;d_{0})$, $u^{s}_{D,p}(x;d_{0})$ and
$u^{s}_{D,s}(x;d_{0})$ cannot be analytically extended from
$\mathbb{R}^{2}\backslash\overline{D}$ into $D$ across any corner of $D$.

###### Proof.

We shall carry out the proof by contradiction.

(i) Assume on the contrary that $u^{s}_{D}(x;d_{0})$ can be analytically
continued across a corner of $\partial D$. By coordinate translation and
rotation, we may suppose that this corner coincides with the origin, so that
$u^{s}_{D}(x;d_{0})$ and also the total field
$u_{D}=u^{s}_{D}(x;d_{0})+u^{i}(x;d_{0},c_{p},c_{s})$ satisfy the Navier
equation (2.2) in $B_{\epsilon}(O)$ for some $\epsilon>0$. Since $u_{D}$ is
real analytic in $({\mathbb{R}}^{2}\backslash\overline{D})\cup
B_{\epsilon}(O)$ and $D$ is a convex polygon, $u_{D}$ satisfies the Navier
equation on the closure of an infinite sector
$\Sigma\subset{\mathbb{R}}^{2}\backslash\overline{D}$ which extends the finite
sector $B_{\epsilon}(O)\cap D$ to ${\mathbb{R}}^{2}\backslash\overline{D}$. In
particular, the total field $u_{D}$ fulfills the Dirichlet boundary condition
on the two half lines $\partial\Sigma$ starting from the corner point $O$.
Since $u^{s}_{D}(x;d_{0})$ fulfills the Kupradze’s radiation condition, it
holds that

$\lim_{|x|\rightarrow\infty}u^{i}(x;d_{0},c_{p},c_{s})=-\lim_{|x|\rightarrow\infty}u^{s}_{D}(x;d_{0},c_{p},c_{s})=0,\quad
x\in\partial\Sigma.$

However, this is impossible for an elastic plane wave incidence of the form
(2.1).

(ii) Suppose that $u^{s}_{D,p}$ can be analytically extended from
${\mathbb{R}}\backslash\bar{D}$ into $D$ across a corner $O$ of $D$. That is,
$u^{s}_{D,p}$ extends to a function which satisfies the vector Helmholtz
equation $\Delta u^{s}_{D,p}+k_{p}^{2}u^{s}_{D,p}=0$ in $B_{\epsilon}(O)$ for
some $\epsilon>0$. By the Hodge decomposition of the total field
$u_{D}=u_{D,p}+u_{D,s}=\triangledown\varphi+{\rm curl\,}\psi$, the function
$\varphi$ can be also extended to $B_{\epsilon}(O)$ as a solution to the
scalar Helmholtz equation. Here we have used the fact that the incident wave
$u^{i}$ is an entire solution to the Navier equation. In particular, this
implies that the normal and tangential derivatives
$(\partial_{n}\varphi,\partial_{\tau}\varphi)$ of $\varphi$ are both piecewise
analytic on $\partial D\cap B_{\epsilon}(O)$. Recalling the Dirichlet boundary
condition of $u_{D}$, we have

$\left\\{\begin{split}u_{D}\cdot n&=\frac{\partial\varphi}{\partial
n}+\frac{\partial\psi}{\partial\tau}=0\\\
u_{D}\cdot\tau&=\frac{\partial\varphi}{\partial\tau}-\frac{\partial\psi}{\partial
n}=0\end{split}\right.\quad\text{on}\ \partial D\cap B_{\epsilon}(O).$ (4.1)

Hence,
$(\partial_{\tau}\psi,\partial_{n}\psi)=(-\partial_{n}\varphi,\partial_{\tau}\varphi)$
on $\partial D\cap B_{\epsilon}(O)$ and thus the Cauchy data of $\psi$ are
also piecewise analytic. By the Cauchy-Kovalevskaya theorem, the function
$\psi$ admits an extension from
$B_{\epsilon}(O)\cap({\mathbb{R}}^{2}\backslash\overline{D})$ to
$B_{\epsilon}(O)\cap D$, as a solution to the scalar Helmholtz equation with
the wave number $k_{s}$. Hence, the total field can be continued to
$B_{\epsilon}(O)\cap D$, which however is impossible by the first part of the
proof.

(iii) The case of $u^{s}_{D,s}$ can be proved similarly to the second step for
$u^{s}_{D,p}$. ∎

As in the acoustic case [28], our approach applies to inverse source problems
as well. For this purpose, we need to justify the absence of analytical
extension for elastic source scattering problems in a corner domain, which is
closet to studies of non-radiating elastic sources given in [5].

######  Lemma 4.2.

Let $\chi_{D}$ be the characteristic function for the convex polygon $D$. If
$u\in\left(H^{2}_{loc}(\mathbb{R}^{2})\right)^{2}$ is a radiating solution to

$\Delta^{\ast}u(x)+\omega^{2}u(x)=\chi_{D}(x)f(x)\quad\mbox{in}\quad\mathbb{R}^{2},$
(4.2)

where $f\in L^{\infty}({\mathbb{R}}^{2})$ is Hölder continuous near the corner
point $O$ of $D$ satisfying $f(O)\neq 0$. Then $u$ cannot be analytically
extended from $\mathbb{R}^{2}\backslash\overline{D}$ to $D$ across the corner
$O$.

Figure 4.1: Illustration of a convex polygonal source term where $O$ is corner
point of $D$.

###### Proof.

Without loss of generality, the corner point $O$ is supposed to coincide with
the origin. Set $v^{\pm}=u|_{D^{\pm}}$ where
$D^{+}:=B_{\delta}(O)\cap({\mathbb{R}}^{2}\backslash\overline{D})$ and
$D^{-}:=B_{\delta}(O)\cap D$. Assume that $v^{+}$ can be analytically extended
from ${\mathbb{R}}^{2}\backslash\overline{D}$ to $B_{\delta}(O)$ for some
$\delta>0$ (see Figure 4.1), as a solution of the Navier equation. This
implies that $\Delta^{\ast}v^{+}+\omega^{2}v^{+}=0$ in $B_{\delta}(O)$, with
the Cauchy data

$v^{-}=v^{+},\quad\mathcal{T}_{\nu}v^{-}=\mathcal{T}_{\nu}v^{+}\quad\mbox{on}\quad\Gamma:=\partial
D\cap B_{\delta}(O).$

Here, the boundary traction operator $\mathcal{T}_{\nu}$ is defined as

$\mathcal{T}_{\nu}u=2\mu\frac{\partial
u}{\partial\nu}+\lambda\nu\triangledown\cdot
u+\mu\nu^{\bot}(\partial_{2}u_{1}-\partial_{1}u_{2})$

in the two-dimensional case. Since $\Delta^{\ast}v^{-}+\omega^{2}v^{-}=f$ in
$B_{\delta}(O)\cap D$, the difference $w:=u^{-}-u^{+}$ is a solution to

$\Delta^{\ast}w+\omega^{2}w=f\quad\mbox{in}\quad B_{\delta}(O)\cap D,\quad
w=\mathcal{T}_{\nu}w=0\quad\mbox{on}\quad\Gamma.$ (4.3)

By [5, Proposition 3.2], it follows that $f(O)=0$, which is in contradiction
with our assumption that $f(O)\neq 0$. ∎

To state the one-wave factorization method, we shall restrict our discussions
to a convex polygonal rigid elastic scatterer $D$. Let $\Omega$ be another
convex rigid scatterer for detecting $D$ such that $\omega^{2}$ is not the
Dirichlet eigenvalue of $-\Delta^{\ast}$ in $\Omega$. Denote by
$(\lambda^{(n)}_{\Omega},\varphi^{(n)}_{\Omega})$ the eigenvalues and
eigenfunctions of the far-field operator $F_{\Omega}$. Below we characterize
the inclusion relationship between our target scatterer $D$ and the test
domain $\Omega$ by the measurement data $u_{D}^{\infty}$ and the spectra of
$F_{\Omega}$.

######  Theorem 4.3.

Define

$W(\Omega):=\sum_{n\in\mathbb{Z}}\frac{\left|\left\langle
u_{D}^{\infty},\varphi^{(n)}_{\Omega}\right\rangle_{\mathbb{S}}\right|^{2}}{\left|\lambda^{(n)}_{\Omega}\right|}.$
(4.4)

Then $W(\Omega)<\infty$ if and only if $D\subseteq\Omega$.

###### Proof.

By Corollary 3.5, $W(\Omega)<+\infty$ implies that $u^{s}_{D}(x;d_{0})$ is
analytic in $\mathbb{R}^{2}\backslash\overline{\Omega}$. If
$D\nsubseteq\Omega$, three cases might happen: (i) $\Omega\subset D$; (ii)
$\Omega\cap D=\emptyset$; (iii) $\Omega\cap D\neq\emptyset$ and
$\Omega\cap({\mathbb{R}}^{2}\backslash\overline{D})\neq\emptyset$. In either
of these cases, we observe that there is always a corner $O\in D$ and
$O\notin\Omega$ by the convexity of both $\Omega$ and $D$. Then,
$u^{s}_{D}(x;d_{0})$ can be analytically continued from
${\mathbb{R}}^{2}\backslash\overline{D}$ to $D$ across the corner $O$ of
$\partial D$, which however is impossible by Lemma 4.1. This proves the
relationship $D\subseteq\Omega$.

Now assume that $D\subseteq\Omega$. Then the scattered field
$u^{s}_{D}(x;d_{0})$ satisfies the Navier equation
$\Delta^{\ast}u^{s}_{D}(x;d_{0})+\omega^{2}u^{s}_{D}(x;d_{0})=0$ in
$\mathbb{R}^{2}\backslash\overline{\Omega}$ with the boundary data
$f:=u^{s}_{D}(x;d_{0})|_{\partial\Omega}\in\left(H^{1/2}(\partial\Omega)\right)^{2}$.
This implies that $u_{D}^{\infty}=G_{\Omega}(f)$. Hence, we get
$W(\Omega)<+\infty$ by applying Corollary 3.5. ∎

For the operator $F^{(\alpha)}_{\Omega}$($\alpha=p,s$), denote by
$(\lambda^{(n)}_{\Omega,\alpha},\varphi^{(n)}_{\Omega,\alpha})$ the
eigenvalues and eigenfunctions of the positive operator
$F^{(\alpha)}_{\Omega,\\#}$. Using Corollary 3.6 and arguing analogously to
the proof of Theorem 4.3, we obtain

######  Theorem 4.4.

Define

$W^{(\alpha)}(\Omega):=\sum_{n\in\mathbb{Z}}\frac{\left|\left\langle
u_{D,\alpha\alpha}^{\infty},\varphi^{(n)}_{\Omega,\alpha}\right\rangle_{\mathbb{S}}\right|^{2}}{\left|\lambda^{(n)}_{\Omega,\alpha}\right|},$
(4.5)

where $u_{D,\alpha\alpha}^{\infty}$ is defined the same as (2.13) for the
scatterer $D$, $\alpha=p,s$. Then $W^{(\alpha)}(\Omega)<\infty$ if and only if
$D\subseteq\Omega$.

##  5 Explicit examples when $\Omega$ is a disk

Theorems 4.3 and 4.4 rely essentially on the factorization form (see e.g.,
(3.7)) of the far-field operator for the elastic scatterer $\Omega$. Below we
show that the results of Theorems 4.3 and 4.4 can be justified independently
of the factorization form, as long as the test domain $\Omega$ is chosen to be
a rigid elastic disk. This is mainly due to the explicit form of the far-field
pattern for a rigid disk in terms of special functions; see Subsection 5.1
below. Then we can get an explicit spectral system of the far-field operator
$F_{B_{R}}$ in Subsection 5.2. The proofs Corollaries 3.5 and 3.6 will be
shown in Subsection 5.3. Note that this section is of independent interests,
since as shown in the subsequent sections, the derivation of eigenvalues and
eigenfunctions of $F_{B_{R}}$ turns out to be non-trivial, which is in
contrast to the acoustic case of the Helmholtz equation.

###  5.1 Far-field pattern of a rigid disk $B_{R}$

Assume that $B_{R}\coloneqq\left\\{x:|x|<R\right\\}$ is a rigid disk centered
at the origin with radius $R>0$. Let
$\hat{x}=\left(\cos\theta_{x},\sin\theta_{x}\right)^{T}\in{\mathbb{S}}$ be the
observation direction (or variable) of the elastic far-field pattern.
According to the Hodge decomposition (2.3), we can introduce scalar functions
$\varphi\left(r,\theta_{x}\right)$ and $\psi\left(r,\theta_{x}\right)$ such
that in polar coordinates $x=(r,\theta_{x})$,

$u^{s}(r,\theta_{x})=\text{grad}\ \varphi(r,\theta_{x})+\text{curl}\
\psi(r,\theta_{x}).$ (5.1)

Recall the relationship between the Cartesian and polar coordinates for
gradient:

$\begin{pmatrix}\partial_{1}\\\
\partial_{2}\end{pmatrix}=\begin{pmatrix}\cos\theta&-\frac{1}{r}\sin\theta\\\
\sin\theta&\frac{1}{r}\cos\theta\end{pmatrix}\begin{pmatrix}\partial_{r}\\\
\partial_{\theta}\end{pmatrix}.$ (5.2)

Noting that $\varphi$ and $\psi$ are both Sommerfeld radiating solutions, we
make the following ansatz:

$\varphi=\sum_{n\in\mathbb{Z}}\frac{1}{\sqrt{k_{p}}}A_{n}H^{(1)}_{n}(k_{p}r)\text{e}^{in\theta_{x}},\quad\psi=\sum_{n\in\mathbb{Z}}\frac{1}{\sqrt{k_{s}}}B_{n}H^{(1)}_{n}(k_{s}r)\text{e}^{in\theta_{x}}.$

We can get from (5.1) that

$\begin{split}u^{s}(r,\theta_{x})&=\begin{pmatrix}\cos\theta_{x}\partial_{r}\varphi-\frac{1}{r}\sin\theta_{x}\partial_{\theta_{x}}\varphi\\\
\sin\theta_{x}\partial_{r}\varphi+\frac{1}{r}\cos\theta_{x}\partial_{\theta_{x}}\varphi\end{pmatrix}+\begin{pmatrix}-\sin\theta_{x}\partial_{r}\psi-\frac{1}{r}\cos\theta_{x}\partial_{\theta_{x}}\psi\\\
\cos\theta_{x}\partial_{r}\psi-\frac{1}{r}\sin\theta_{x}\partial_{\theta_{x}}\psi\end{pmatrix}\\\
&=\sum_{n\in\mathbb{Z}}\frac{\text{e}^{in\theta_{x}}}{r}\left[\left(\sqrt{k_{p}}rH^{(1)\prime}_{n}(k_{p}r)A_{n}-\frac{in}{\sqrt{k_{s}}}H^{(1)}_{n}(k_{s}r)B_{n}\right)\begin{pmatrix}\cos\theta_{x}\\\
\sin\theta_{x}\end{pmatrix}\right.\\\
&\quad\left.+\left(\frac{in}{\sqrt{k_{p}}}H^{(1)}_{n}(k_{p}r)A_{n}+\sqrt{k_{s}}rH^{(1)\prime}_{n}(k_{s}r)B_{n}\right)\begin{pmatrix}-\sin\theta_{x}\\\
\cos\theta_{x}\end{pmatrix}\right].\end{split}$ (5.3)

Using the asymptotic property of Hankel functions

$\begin{split}H_{n}^{(1)}(z)&=\sqrt{\frac{2}{\pi
z}}\text{e}^{i(z-\frac{n\pi}{2}-\frac{\pi}{4})}\left(1+\mathcal{O}(\frac{1}{z})\right),\
z\to\infty,\\\ H_{n}^{(1)\prime}(z)&=\sqrt{\frac{2}{\pi
z}}\text{e}^{i(z-\frac{n\pi}{2}+\frac{\pi}{4})}\left(1+\mathcal{O}(\frac{1}{z})\right),\
z\to\infty,\end{split}$ (5.4)

we have

$u^{s}(x)=\frac{\text{e}^{ik_{p}r}}{\sqrt{r}}\sqrt{\frac{2}{\pi}}\sum_{n\in\mathbb{Z}}\text{e}^{-i(\frac{n\pi}{2}-\frac{\pi}{4})}A_{n}\text{e}^{in\theta_{x}}\hat{x}+\frac{\text{e}^{ik_{s}r}}{\sqrt{r}}\sqrt{\frac{2}{\pi}}\sum_{n\in\mathbb{Z}}\text{e}^{-i(\frac{n\pi}{2}-\frac{\pi}{4})}B_{n}\text{e}^{in\theta_{x}}\hat{x}^{\bot}+\mathcal{O}(\frac{1}{r^{3/2}}).$
(5.5)

Thus,

$\begin{split}u^{\infty}_{p}(\hat{x})=\sqrt{\frac{2}{\pi}}\text{e}^{i\frac{\pi}{4}}\sum_{n\in\mathbb{Z}}i^{-n}A_{n}\text{e}^{in\theta_{x}},\quad
u^{\infty}_{s}(\hat{x})=\sqrt{\frac{2}{\pi}}\text{e}^{i\frac{\pi}{4}}\sum_{n\in\mathbb{Z}}i^{-n}B_{n}\text{e}^{in\theta_{x}}.\end{split}$
(5.6)

Now, we set

$t_{\alpha}=k_{\alpha}R,\ \alpha=p,s,$ (5.7)

to get

$\begin{split}u^{s}(r,\theta_{x})|_{r=R}&=\sum_{n\in\mathbb{Z}}\left[\left(\frac{t_{p}}{\sqrt{k_{p}}}H^{(1)\prime}_{n}(t_{p})A_{n}-\frac{in}{\sqrt{k_{s}}}H^{(1)}_{n}(t_{s})B_{n}\right)\begin{pmatrix}\cos\theta_{x}\\\
\sin\theta_{x}\end{pmatrix}\right.\\\
&\quad+\left.\left(\frac{in}{\sqrt{k_{p}}}H^{(1)}_{n}(t_{p})A_{n}+\frac{t_{s}}{\sqrt{k_{s}}}H^{(1)\prime}_{n}(t_{s})B_{n}\right)\begin{pmatrix}-\sin\theta_{x}\\\
\cos\theta_{x}\end{pmatrix}\right]\frac{\text{e}^{in\theta_{x}}}{R}\\\
&=\sum_{n\in\mathbb{Z}}\begin{pmatrix}\nu&\tau\end{pmatrix}\begin{pmatrix}t_{p}H^{(1)\prime}_{n}(t_{p})&-inH^{(1)}_{n}(t_{s})\\\
inH^{(1)}_{n}(t_{p})&t_{s}H^{(1)\prime}_{n}(t_{s})\end{pmatrix}\begin{pmatrix}\frac{1}{\sqrt{k_{p}}}&0\\\
0&\frac{1}{\sqrt{k_{s}}}\end{pmatrix}\begin{pmatrix}A_{n}\\\
B_{n}\end{pmatrix}\frac{\text{e}^{in\theta_{x}}}{R},\end{split}$ (5.8)

where $\nu=(\cos\theta_{x},\sin\theta_{x})^{T}$ and
$\tau=(-\sin\theta_{x},\cos\theta_{x})^{T}$ are tangential and normal
directions of $\partial B_{R}$ respectively, and that $(\nu\quad\tau)$ is thus
a 2-by-2 matrix. Set

$\mathcal{H}_{n}\coloneqq\begin{pmatrix}t_{p}H^{(1)\prime}_{n}(t_{p})&-inH^{(1)}_{n}(t_{s})\\\
inH^{(1)}_{n}(t_{p})&t_{s}H^{(1)\prime}_{n}(t_{s})\end{pmatrix}\text{ and
}Q\coloneqq\begin{pmatrix}\frac{1}{\sqrt{k_{p}}}&0\\\
0&\frac{1}{\sqrt{k_{s}}}\end{pmatrix}.$ (5.9)

From (5.8) and (5.9), we have

$u^{s}(r,\theta_{x})|_{r=R}=\sum_{n\in\mathbb{Z}}\begin{pmatrix}\nu&\tau\end{pmatrix}\mathcal{H}_{n}Q\begin{pmatrix}A_{n}\\\
B_{n}\end{pmatrix}\frac{\text{e}^{in\theta_{x}}}{R}.$ (5.10)

######  Remark 5.1.

By [4, Lemma 2.11] and [23, Remark 3.2], it follows that
$|\mathcal{H}_{n}|=t_{p}t_{s}H_{n}^{(1)\prime}(t_{p})H_{n}^{(1)\prime}(t_{s})-n^{2}H_{n}^{(1)}(t_{p})H_{n}^{(1)}(t_{s})\neq
0$ for any $n\in\mathbb{Z}$. Hence the matrix $\mathcal{H}_{n}$ is always
invertible.

Having expanded the scattered field into the series (5.10), now we need to
represent an elastic plane wave in terms of the special functions on $|x|=R$.
Let $d=\left(\cos\theta_{d},\sin\theta_{d}\right)^{T}\in{\mathbb{S}}$ be the
incident direction. By the Jacobi-Anger expansion (see e.g., [10, Formula
(3.89)]), we see

$\text{e}^{ikx\cdot
d}=\sum_{n\in\mathbb{Z}}i^{n}J_{n}\left(k|x|\right)\text{e}^{in\theta},\quad
x\in\mathbb{R}^{2},$ (5.11)

where $\theta=\theta_{x}-\theta_{d}$ denotes the angle between $\hat{x}$ and
$d$. Let $d^{\bot}=(-\sin\theta_{d},\cos\theta_{d})^{T}$ be a vector
perpendicular to $d$. Recalling the compressional part $u^{i}_{p}$ of the
incident wave $u^{i}$,

$u^{i}_{p}(x)=d\ \text{e}^{ik_{p}x\cdot d},$ (5.12)

and inserting (5.11) to (5.12), we get the form of $u^{i}_{p}$ over $\partial
B_{R}$ as

$u^{i}_{p}|_{r=R}=\sum_{n\in\mathbb{Z}}dJ_{n}(t_{p})i^{n}\text{e}^{in\theta}.$
(5.13)

Using the formulas

$\cos\theta=\frac{1}{2}\left(\text{e}^{i\theta}+\text{e}^{-i\theta}\right),\
\sin\theta=\frac{1}{2i}\left(\text{e}^{i\theta}-\text{e}^{-i\theta}\right)$
(5.14)

and the following properties of Bessel functions

$2J^{\prime}_{n}(x)=J_{n-1}(x)-J_{n+1}(x),\
J_{n-1}(x)+J_{n+1}(x)=\frac{2n}{x}J_{n}(x),$ (5.15)

we can get from (5.13) that

$\begin{split}u^{i}_{p}\cdot\nu|_{r=R}&=\frac{1}{2}\sum_{n\in\mathbb{Z}}\left(\text{e}^{i\theta}+\text{e}^{-i\theta}\right)J_{n}(t_{p})i^{n}\text{e}^{in\theta}\\\
&=\frac{1}{2}\sum_{n\in\mathbb{Z}}\left(J_{n}(t_{p})i^{n}\text{e}^{i(n+1)\theta}+J_{n}(t_{p})i^{n}\text{e}^{i(n-1)\theta}\right)\\\
&=\frac{1}{2}\sum_{n\in\mathbb{Z}}\left(J_{n}(t_{p})i^{n-1}\text{e}^{in\theta}+J_{n}(t_{p})i^{n+1}\text{e}^{in\theta}\right)\\\
&=\frac{1}{2}\sum_{n\in\mathbb{Z}}\left(J_{n-1}(t_{p})-J_{n+1}(t_{p})\right)i^{n-1}\text{e}^{in\theta}\\\
&=\sum_{n\in\mathbb{Z}}J^{\prime}_{n}(t_{p})i^{n-1}\text{e}^{in\theta}\end{split}$
(5.16)

and

$\begin{split}u^{i}_{p}\cdot\tau|_{r=R}&=\frac{1}{2i}\sum_{n\in\mathbb{Z}}\left(\text{e}^{-i\theta}-\text{e}^{i\theta}\right)J_{n}(t_{p})i^{n}\text{e}^{in\theta}\\\
&=\frac{1}{2}\sum_{n\in\mathbb{Z}}\left(J_{n}(t_{p})i^{n-1}\text{e}^{i(n-1)\theta}-J_{n}(t_{p})i^{n-1}\text{e}^{i(n+1)\theta}\right)\\\
&=\frac{1}{2}\sum_{n\in\mathbb{Z}}\left(J_{n+1}(t_{p})i^{n}\text{e}^{in\theta}+J_{n-1}(t_{p})i^{n}\text{e}^{in\theta}\right)\\\
&=\sum_{n\in\mathbb{Z}}\frac{n}{t_{p}}J_{n}(t_{p})i^{n}\text{e}^{in\theta}.\end{split}$
(5.17)

We use the notation $v^{s}$ to denote the scattered field produced by the
compressional part $u^{i}_{p}$ of the incident wave $u^{i}$. Note that $v^{s}$
is usually different from the compressional part of the scattered field for
$u^{i}$. By (5.10), we can represent $v^{s}(r,\theta_{x})$ on $|x|=R$ as the
series

$v^{s}(r,\theta_{x})|_{r=R}=\sum_{n\in\mathbb{Z}}\begin{pmatrix}\nu&\tau\end{pmatrix}\mathcal{H}_{n}Q\begin{pmatrix}A_{n,v}\\\
B_{n,v}\end{pmatrix}\frac{\text{e}^{in\theta_{x}}}{R},$ (5.18)

where the coefficients $A_{n,v},B_{n,v}\in{\mathbb{C}}$ are associated with
$v^{s}$. Making use of the Dirichlet boundary condition $u^{i}_{p}=-v^{s}$ on
$\partial B_{R}$, we have

$\left.\begin{pmatrix}u^{i}_{p}\cdot\nu\\\
u^{i}_{p}\cdot\tau\end{pmatrix}\right\rvert_{r=R}=-\left.\begin{pmatrix}v^{s}\cdot\nu\\\
v^{s}\cdot\tau\end{pmatrix}\right\rvert_{r=R},$

which together with (5.16) and (5.17) leads to

$\sum_{n\in\mathbb{Z}}\begin{pmatrix}J^{\prime}_{n}(t_{p})\\\
\frac{in}{t_{p}}J_{n}(t_{p})\end{pmatrix}i^{n-1}\text{e}^{in\theta_{x}}\text{e}^{-in\theta_{d}}=-\sum_{n\in\mathbb{Z}}\mathcal{H}_{n}Q\begin{pmatrix}A_{n,v}\\\
B_{n,v}\end{pmatrix}\frac{\text{e}^{in\theta_{x}}}{R}.$ (5.19)

By the arbitrariness of $\theta_{x}$, we can get

$\begin{pmatrix}iJ^{\prime}_{n}(t_{p})\\\
-\frac{n}{t_{p}}J_{n}(t_{p})\end{pmatrix}i^{n}\text{e}^{-in\theta_{d}}=\frac{\mathcal{H}_{n}Q}{R}\begin{pmatrix}A_{n,v}\\\
B_{n,v}\end{pmatrix},$

implying the relation

$\begin{pmatrix}A_{n,v}\\\
B_{n,v}\end{pmatrix}=Q^{-1}\mathcal{H}_{n}^{-1}\begin{pmatrix}iJ^{\prime}_{n}(t_{p})\\\
-\frac{n}{t_{p}}J_{n}(t_{p})\end{pmatrix}Ri^{n}\text{e}^{-in\theta_{d}}\,.$
(5.20)

Thus, we obtain the P-part and S-part of the far-field patterns of $v^{s}$ as
follows:

$\begin{pmatrix}u^{\infty}_{pp}(\hat{x})\\\
u^{\infty}_{sp}(\hat{x})\end{pmatrix}=\sqrt{\frac{2}{\pi}}\text{e}^{i\frac{\pi}{4}}iR\sum_{n\in\mathbb{Z}}Q^{-1}\mathcal{H}_{n}^{-1}\begin{pmatrix}J^{\prime}_{n}(t_{p})\\\
\frac{in}{t_{p}}J_{n}(t_{p})\end{pmatrix}\text{e}^{in\theta}.$ (5.21)

The far-field pattern excited by the S-part of an elastic plane wave can be
treated similarly. The shear part $u^{i}_{s}$ of the incident wave $u^{i}$
takes the form

$u^{i}_{s}(x)=d^{\bot}\text{e}^{ik_{s}x\cdot d},$ (5.22)

which together with (5.11) gives arise to

$u^{i}_{s}|_{r=R}=\sum_{n\in\mathbb{Z}}d^{\bot}J_{n}(t_{s})i^{n}\text{e}^{in\theta}.$
(5.23)

Correspondingly, we have

$u^{i}_{s}\cdot\nu|_{r=R}=-\sum_{n\in\mathbb{Z}}\frac{n}{t_{s}}J_{n}(t_{s})i^{n}\text{e}^{in\theta},\quad
u^{i}_{s}\cdot\tau|_{r=R}=\sum_{n\in\mathbb{Z}}J^{\prime}_{n}(t_{s})i^{n-1}\text{e}^{in\theta}.$

Again using (5.10), we can represent by $w^{s}$ the scattered field produced
by $u^{i}_{s}$ in the form of

$w^{s}(r,\theta_{x})|_{r=R}=\sum_{n\in\mathbb{Z}}\begin{pmatrix}\nu&\tau\end{pmatrix}\mathcal{H}_{n}Q\begin{pmatrix}A_{n,w}\\\
B_{n,w}\end{pmatrix}\frac{\text{e}^{in\theta_{x}}}{R}.$ (5.24)

Combining the proves two identities together with the boundary condition
$u^{i}_{s}=-w^{s}$ on $\partial B_{R}$, we arrive at

$\begin{pmatrix}A_{n,w}\\\
B_{n,w}\end{pmatrix}=Q^{-1}\mathcal{H}_{n}^{-1}\begin{pmatrix}\frac{n}{t_{s}}J_{n}(t_{s})\\\
iJ^{\prime}_{n}(t_{s})\end{pmatrix}Ri^{n}\text{e}^{-in\theta_{d}}.$ (5.25)

Thus, we obtain $u^{\infty}_{ps}$ and $u^{\infty}_{ss}$ as follows:

$\begin{pmatrix}u^{\infty}_{ps}(\hat{x})\\\
u^{\infty}_{ss}(\hat{x})\end{pmatrix}=\sqrt{\frac{2}{\pi}}\text{e}^{i\frac{\pi}{4}}iR\sum_{n\in\mathbb{Z}}Q^{-1}\mathcal{H}_{n}^{-1}\begin{pmatrix}-\frac{in}{t_{s}}J_{n}(t_{s})\\\
J^{\prime}_{n}(t_{s})\end{pmatrix}\text{e}^{in\theta}.$ (5.26)

This enables us to define the matrix

$U_{B_{R}}^{\infty}\coloneqq\begin{pmatrix}u^{\infty}_{pp}(\hat{x})&u^{\infty}_{ps}(\hat{x})\\\
u^{\infty}_{sp}(\hat{x})&u^{\infty}_{ss}(\hat{x})\end{pmatrix}=\sqrt{\frac{2}{\pi}}\text{e}^{i\frac{\pi}{4}}i\sum_{n\in\mathbb{Z}}Q^{-1}\mathcal{H}_{n}^{-1}\begin{pmatrix}RJ^{\prime}_{n}(t_{p})&-\frac{in}{k_{s}}J_{n}(t_{s})\\\
\frac{in}{k_{p}}J_{n}(t_{p})&RJ^{\prime}_{n}(t_{s})\end{pmatrix}\text{e}^{in\theta}.$
(5.27)

Setting

$\mathcal{J}_{n}\coloneqq\begin{pmatrix}t_{p}J^{\prime}_{n}(t_{p})&-inJ_{n}(t_{s})\\\
inJ_{n}(t_{p})&t_{s}J^{\prime}_{n}(t_{s})\end{pmatrix},$ (5.28)

we can rewrite $U_{B_{R}}^{\infty}$ as

$U_{B_{R}}^{\infty}(\hat{x})=\sqrt{\frac{2}{\pi}}\text{e}^{i\frac{\pi}{4}}i\sum_{n\in\mathbb{Z}}Q^{-1}\mathcal{H}_{n}^{-1}\mathcal{J}_{n}Q^{2}\text{e}^{in\theta}.$
(5.29)

To sum up, for the elastic plane wave $u^{i}$ of the general form (2.1), by
linear superposition its far-field pattern $u^{\infty}$ takes the form of
(2.14), where the P-part and S-part for $u^{i}_{p}$ and $u^{i}_{s}$ are given
in the matrix $U_{B_{R}}^{\infty}$ (see also (5.21) and (5.26)).

###  5.2 Spectral system of the far-field operator $F_{B_{R}}$

Now, we need to derive eigenvalues and the associated eigenfunctions of the
far-field operator $F_{B_{R}}$ defined by (e.g. (3.16))

$(F_{B_{R}}g)(\hat{x})=\text{
e}^{-i\frac{\pi}{4}}\int_{\mathtt{S}}\left\\{\sqrt{\frac{k_{p}}{\omega}}u^{\infty}_{B_{R}}(\hat{x};d,1,0)g_{p}(d)+\sqrt{\frac{k_{s}}{\omega}}u^{\infty}_{B_{R}}(\hat{x};d,0,1)g_{s}(d)\right\\}ds(d).$
(5.30)

Obviously, the spectral system of $F_{B_{R}}$ should be connected to the
spectral system of the matrix $U_{B_{R}}$. To disclose this relation, we
retain the notations from the previous subsection to define
$\widetilde{\Sigma}_{n}:=Q^{-1}\mathcal{H}_{n}^{-1}\mathcal{J}_{n}Q$.

######  Lemma 5.2.

If $(\lambda_{n},\widetilde{X}_{n})$ is the spectral pair of
$\widetilde{\Sigma}_{n}$, that is,
$\widetilde{\Sigma}_{n}\widetilde{X}_{n}=\lambda_{n}\widetilde{X}_{n},\
\widetilde{X}_{n}=(\widetilde{X}_{n}^{(1)},\widetilde{X}_{n}^{(2)})^{T}$. Then

$(F_{B_{R}}g)(\hat{x})=\sqrt{\frac{8\pi}{\omega}}i\lambda_{n}\;g(\hat{x}),\quad
g(\hat{x}):=(\widetilde{X}_{n}^{(1)}\hat{x}+\widetilde{X}_{n}^{(2)}\hat{x}^{\bot})\text{e}^{in\theta_{x}}.$
(5.31)

###### Proof.

Let $g\in\left(L^{2}({\mathbb{S}})\right)^{2}$ be given as in (5.31). It is
easy to see the P- and S-component of $g$ as
$g_{p}(d)=\widetilde{X}_{n}^{(1)}\text{e}^{in\theta_{d}}$,
$g_{s}(d)=\widetilde{X}_{n}^{(2)}\text{e}^{in\theta_{d}}$. It then follows
from the definition of $F_{B_{R}}$ in (5.30) that

$\begin{split}\left(F_{B_{R}}g\right)(\hat{x})&=\frac{1}{\sqrt{\omega}}\text{
e}^{-i\frac{\pi}{4}}\int_{\mathbb{S}}g_{p}(d)\left[\sqrt{k_{p}}u^{\infty}_{pp}(\hat{x};d)\hat{x}+\sqrt{k_{p}}u^{\infty}_{sp}(\hat{x};d)\hat{x}^{\bot}\right]\\\
&\quad+g_{s}(d)\left[\sqrt{k_{s}}u^{\infty}_{ps}(\hat{x};d)\hat{x}+\sqrt{k_{s}}u^{\infty}_{ss}(\hat{x};d)\hat{x}^{\bot}\right]ds(d)\\\
&=\frac{1}{\sqrt{\omega}}\text{
e}^{-i\frac{\pi}{4}}\int_{\mathbb{S}}\left[\widetilde{X}_{n}^{(1)}\sqrt{k_{p}}u^{\infty}_{pp}(\hat{x};d)\hat{x}+\widetilde{X}_{n}^{(1)}\sqrt{k_{p}}u^{\infty}_{sp}(\hat{x};d)\hat{x}^{\bot}\right]\text{e}^{in\theta_{d}}\\\
&\quad+\left[\widetilde{X}_{n}^{(2)}\sqrt{k_{s}}u^{\infty}_{ps}(\hat{x};d)\hat{x}+\widetilde{X}_{n}^{(2)}\sqrt{k_{s}}u^{\infty}_{ss}(\hat{x};d)\hat{x}^{\bot}\right]\text{e}^{in\theta_{d}}ds(d)\\\
&=\frac{1}{\sqrt{\omega}}\text{
e}^{-i\frac{\pi}{4}}\int_{\mathbb{S}}\left[\left(\widetilde{X}_{n}^{(1)}\sqrt{k_{p}}u^{\infty}_{pp}(\hat{x};d)+\widetilde{X}_{n}^{(2)}\sqrt{k_{s}}u^{\infty}_{ps}(\hat{x};d)\right)\hat{x}\right.\\\
&\quad+\left.\left(\widetilde{X}_{n}^{(1)}\sqrt{k_{p}}u^{\infty}_{sp}(\hat{x};d)+\widetilde{X}_{n}^{(2)}\sqrt{k_{s}}u^{\infty}_{ss}(\hat{x};d)\right)\hat{x}^{\bot}\right]\text{e}^{in\theta_{d}}ds(d)\\\
&=\frac{1}{\sqrt{\omega}}\text{
e}^{-i\frac{\pi}{4}}\int_{\mathbb{S}}\begin{pmatrix}\hat{x}&\hat{x}^{\bot}\end{pmatrix}U_{B_{R}}^{\infty}Q^{-1}\widetilde{X}_{n}\text{e}^{in\theta_{d}}ds(d)\\\
&=\sqrt{\frac{2}{\omega\pi}}i\int_{\mathbb{S}}\sum_{m\in\mathbb{Z}}\begin{pmatrix}\hat{x}&\hat{x}^{\bot}\end{pmatrix}\widetilde{\Sigma}_{m}\widetilde{X}_{n}\text{e}^{im\theta_{x}}\text{e}^{in\theta_{d}}ds(d).\end{split}$

Using the orthogonality of $\text{e}^{in\theta_{d}}$ for $n\in{\mathbb{Z}}$
and the fact that $\widetilde{\Sigma}_{n}$ and $\widetilde{X}_{n}$ are
independent of $d$, we arrive at

$\begin{split}\left(F_{B_{R}}g\right)(\hat{x})&=\sqrt{\frac{2}{\omega\pi}}i\int_{\mathbb{S}}\begin{pmatrix}\hat{x}&\hat{x}^{\bot}\end{pmatrix}\widetilde{\Sigma}_{n}\widetilde{X}_{n}\text{e}^{in\theta_{x}}ds(d)\\\
&=\sqrt{\frac{8\pi}{\omega}}i\begin{pmatrix}\hat{x}&\hat{x}^{\bot}\end{pmatrix}\widetilde{\Sigma}_{n}\widetilde{X}_{n}\text{e}^{in\theta_{x}}\\\
&=\sqrt{\frac{8\pi}{\omega}}i\begin{pmatrix}\hat{x}&\hat{x}^{\bot}\end{pmatrix}\lambda_{n}\widetilde{X}_{n}\text{e}^{in\theta_{x}}\\\
&=\sqrt{\frac{8\pi}{\omega}}i\lambda_{n}g(\hat{x}).\end{split}$

∎

As a consequence of Lemma 5.2, we obtain the spectral pair of $F_{B_{R}}$ as
follows.

######  Lemma 5.3.

The spectral pair of $F_{B_{R}}$ is given by

$(\lambda^{(n)}_{B_{R}},X^{(n)}_{B_{R}})=\left(\sqrt{\frac{8\pi}{\omega}}i\lambda_{n},\;\begin{pmatrix}\hat{x}&\hat{x}^{\bot}\end{pmatrix}Q^{-1}X_{n}\text{e}^{in\theta_{x}}\right),$

where $(\lambda_{n},X_{n})$ is the spectral pair of
$\Sigma_{n}\coloneqq\mathcal{H}_{n}^{-1}\mathcal{J}_{n}$.

###### Proof.

Suppose that $(\lambda_{n},\widetilde{X}_{n})$ is the spectral pair of
$\widetilde{\Sigma}_{n}$. Writing $X_{n}:=Q\widetilde{X}_{n}$, we have

$\lambda_{n}\widetilde{X}_{n}=\widetilde{\Sigma}_{n}\widetilde{X}_{n}=Q^{-1}\Sigma_{n}Q\widetilde{X}_{n}=Q^{-1}\Sigma_{n}X_{n}.$
(5.32)

This implies that $\lambda_{n}X_{n}=\Sigma_{n}X_{n}$. Using Lemma 5.2, we get

$\lambda^{(n)}_{B_{R}}=\sqrt{\frac{8\pi}{\omega}}i\lambda_{n},\quad
X^{(n)}_{B_{R}}(\hat{x})=\left(\hat{x}\;\;\hat{x}^{\bot}\right)Q^{-1}X_{n}\text{e}^{in\theta_{x}}.$

∎

Since the eigenvalues of $F_{B_{R}}$ have appeared in the denominator of the
indicator (4.4) with $\Omega=B_{R}$, it is necessary to show
$\lambda^{(n)}_{B_{R}}\neq 0$ for all $n\in{\mathbb{N}}$ under an additional
assumption of the frequency.

######  Lemma 5.4.

If $\omega^{2}$ is not a Dirichlet eigenvalue of $-\Delta^{\ast}$ in $B_{R}$,
then the eigenvalue $\lambda_{n}$ of the matrix $\Sigma_{n}$ cannot vanish for
any $n\in\mathbb{Z}$.

###### Proof.

Suppose that there exists $n\in\mathbb{Z}$ such that $\lambda_{n}=0$ is the
eigenvalue of the matrix $\Sigma_{n}$ and that $X_{n}\neq 0$ is the
corresponding eigenvector. Then,

$\mathcal{H}_{n}^{-1}\mathcal{J}_{n}X_{n}=\Sigma_{n}X_{n}=\lambda_{n}X_{n}=0,$

Since $\mathcal{H}_{n}^{-1}$ is invertible (see Remark 5.1), we have
$\mathcal{J}_{n}X_{n}=0$, implying that $|\mathcal{J}_{n}|=0$. Since
$X_{n}=(X_{n}^{(1)},X_{n}^{(2)})^{T}\neq 0$, we may define the non-trivial
function $u={\rm grad\,}\varphi+{\rm curl\,}\psi$ where

$\varphi=\sum_{n\in\mathbb{Z}}X_{n}^{(1)}J_{n}(k_{p}r)\text{e}^{in\theta_{x}},\quad\psi=\sum_{n\in\mathbb{Z}}X_{n}^{(2)}J_{n}(k_{s}r)\text{e}^{in\theta_{x}}.$

Then it is easy to check that

$\begin{split}&\quad u|_{\partial B_{R}}\\\
&=\sum_{n\in\mathbb{Z}}\frac{\text{e}^{in\theta_{x}}}{R}\left[\left(k_{p}RJ^{\prime}_{n}(k_{p}R)X_{n}^{(1)}-inJ_{n}(k_{s}R)X_{n}^{(2)}\right)\hat{x}+\left(inJ_{n}(k_{p}R)X_{n}^{(1)}+k_{s}RJ^{\prime}_{n}(k_{s}R)X_{n}^{(2)}\right)\hat{x}^{\bot}\right]\\\
&=\sum_{n\in\mathbb{Z}}\frac{\text{e}^{in\theta_{x}}}{R}\begin{pmatrix}\hat{x}&\hat{x}^{\bot}\end{pmatrix}\begin{pmatrix}t_{p}J^{\prime}_{n}(t_{p})&-inJ_{n}(t_{s})\\\
inJ_{n}(t_{p})&t_{s}J^{\prime}_{n}(t_{s})\end{pmatrix}\begin{pmatrix}X_{n}^{(1)}\\\
X_{n}^{(2)}\end{pmatrix}\\\
&=\sum_{n\in\mathbb{Z}}\frac{\text{e}^{in\theta_{x}}}{R}\begin{pmatrix}\hat{x}&\hat{x}^{\bot}\end{pmatrix}\mathcal{J}_{n}X_{n}\\\
&=0.\end{split}$

On the other hand, it is obvious that $u$ satisfies the Navier equation
$-\Delta^{\ast}u=\omega^{2}u$ in $B_{R}$. Hence, it is a Dirichlet
eigenfunction of $-\Delta^{*}$ over $B_{R}$, which is impossible. ∎

To calculate the spectra of $F_{B_{R}}$, by Lemma 5.3 we need to consider the
generalized eigenvalue problem

$\mathcal{J}_{n}X_{n}=\lambda_{n}\mathcal{H}_{n}X_{n},$ (5.33)

where $\lambda_{n}$ and $X_{n}$ represent eigenvalues and eigenvectors of
$\Sigma_{n}$. Recalling the Hankel functions and its derivatives,

$H^{(1)}_{n}(z)=J_{n}(z)+iY_{n}(z),\
H^{(1)\prime}_{n}(z)=J^{\prime}_{n}(z)+iY^{\prime}_{n}(z),$

and setting

$\mathcal{Y}_{n}\coloneqq\begin{pmatrix}t_{p}Y^{\prime}_{n}(t_{p})&-inY_{n}(t_{s})\\\
inY_{n}(t_{p})&t_{s}Y^{\prime}_{n}(t_{s})\end{pmatrix},$

we can rephrase the matrix $\mathcal{H}_{n}$ as

$\mathcal{H}_{n}=\mathcal{J}_{n}+i\mathcal{Y}_{n}.$ (5.34)

Below we describe an eigensystem of the generalized eigenvalue problem (5.33)
with the help of the decomposition (5.34).

######  Lemma 5.5.

A normalized eigensystem $(\lambda_{n,j},X_{n,j})$ with
$n\in{\mathbb{N}},j=1,2$ to the generalized eigenvalue problem (5.33) is given
by

$\lambda_{n,j}=\frac{t_{p}J_{n}^{\prime}(t_{p})+inJ_{n}(t_{s})\sigma_{j}^{(n)}}{t_{p}H_{n}^{(1)\prime}(t_{p})+inH_{n}^{(1)}(t_{s})\sigma_{j}^{(n)}},\quad
X_{n,j}=\frac{(1,\sigma_{j}^{(n)})^{T}}{\sqrt{1+|\sigma_{j}^{(n)}|^{2}}},$
(5.35)

with

$\begin{split}&\sigma_{1}^{(n)}=\frac{-\beta_{n}+\sqrt{\beta_{n}^{2}-4}}{2},\quad\sigma_{2}^{(n)}=\frac{-\beta_{n}-\sqrt{\beta_{n}^{2}-4}}{2},\\\
&\beta_{n}=\frac{\pi}{2in}\left[n^{2}(J_{n}(t_{s})Y_{n}(t_{p})-J_{n}(t_{p})Y_{n}(t_{s}))+t_{p}t_{s}(J_{n}^{\prime}(t_{p})Y_{n}^{\prime}(t_{s})-J_{n}^{\prime}(t_{s})Y_{n}^{\prime}(t_{p}))\right].\end{split}$

###### Proof.

Let $X_{n}=(1,\sigma^{(n)})^{T}$ be an eigenvector of the generalized
eigenvalue problem $\mathcal{J}_{n}X_{n}=\eta_{n}\mathcal{Y}_{n}X_{n}$, where
$\eta_{n}$ is the eigenvalue. Using the Wronskian

$J_{n}(t)Y_{n}^{\prime}(t)-J_{n}^{\prime}(t)Y_{n}(t)=\frac{2}{\pi t},$

simple calculations show that $\sigma^{(n)}$ should satisfy the algebraic
equation

$\sigma^{(n)2}+\beta_{n}\sigma^{(n)}+1=0,$ (5.36)

where $\beta_{n}$ is defined as in the lemma. The two roots of (5.36) are
given by

$\sigma_{1}^{(n)}=\frac{-\beta_{n}+\sqrt{\beta_{n}^{2}-4}}{2},\quad\sigma_{2}^{(n)}=\frac{-\beta_{n}-\sqrt{\beta_{n}^{2}-4}}{2}.$

On the other hand, one can also calculate the corresponding eigenvalues

$\eta_{n,j}=\frac{t_{p}J_{n}^{\prime}(t_{p})+inJ_{n}(t_{s})\sigma_{j}^{(n)}}{t_{p}Y_{n}^{\prime}(t_{p})+inY_{n}(t_{s})\sigma_{j}^{(n)}},\quad
j=1,2.$

Using the decomposition (5.34), we get

$\mathcal{H}_{n}X_{n}=(\mathcal{J}_{n}+i\mathcal{Y}_{n})X_{n}=\frac{\eta_{n}+i}{\eta_{n}}\mathcal{J}_{n}X_{n}.$

Therefore, $(\frac{\eta_{n}}{\eta_{n}+i},X_{n})$ is the eigensystem of the
generalized eigenvalue problem
$\mathcal{J}_{n}X_{n}=\lambda_{n}\mathcal{H}_{n}X_{n}$. Further, we get the
eigenvalues

$\lambda_{n,j}=\frac{\eta_{n,j}}{\eta_{n,j}+i}=\frac{t_{p}J_{n}^{\prime}(t_{p})+inJ_{n}(t_{s})\sigma_{j}^{(n)}}{t_{p}H_{n}^{(1)\prime}(t_{p})+inH_{n}^{(1)}(t_{s})\sigma_{j}^{(n)}},\
j=1,2.$ (5.37)

∎

Combining Lemma 5.3 and 5.5, we obtain an eigensystem of the far-field
operator $F_{B_{R}}$ by

$\begin{split}\lambda^{(n)}_{B_{R},j}&=\sqrt{\frac{8\pi}{\omega}}\frac{it_{p}J_{n}^{\prime}(t_{p})-nJ_{n}(t_{s})\sigma_{j}^{(n)}}{t_{p}H_{n}^{(1)\prime}(t_{p})+inH_{n}^{(1)}(t_{s})\sigma_{j}^{(n)}},\\\
X^{(n)}_{B_{R},j}&=\left(\sqrt{k_{p}}X_{n,j}^{(1)}\hat{x}+\sqrt{k_{s}}X_{n,j}^{(2)}\hat{x}^{\bot}\right)\text{e}^{in\theta_{x}}\\\
&=\left(\sqrt{k_{p}}\hat{x}+\sqrt{k_{s}}\sigma_{j}^{(n)}\hat{x}^{\bot}\right)\frac{\text{e}^{in\theta_{x}}}{\sqrt{1+|\sigma_{j}^{(n)}|^{2}}}\end{split}$
(5.38)

for $n\in{\mathbb{N}},j=1,2$.

######  Remark 5.6.

The asymptotics of $\sigma_{j}^{(n)}$ can be derived as follows. Recall the
asymptotic behavior of Bessel functions (see [10])

$\begin{split}J_{n}(z)&=\frac{z^{n}}{2^{n}n!}\left(1+\mathcal{O}(\frac{1}{n})\right),\quad
n\rightarrow+\infty,\\\
J_{n}^{\prime}(z)&=\frac{z^{n-1}}{2^{n}(n-1)!}\left(1+\mathcal{O}(\frac{1}{n})\right),\quad
n\rightarrow+\infty,\end{split}$ (5.39)

and those of Neumann functions:

$\begin{split}Y_{n}(z)&=-\frac{2^{n}(n-1)!}{\pi
z^{n}}\left(1+\mathcal{O}(\frac{1}{n})\right),\quad n\rightarrow+\infty,\\\
Y_{n}^{\prime}(z)&=\frac{2^{n}n!}{\pi
z^{n+1}}\left(1+\mathcal{O}(\frac{1}{n})\right),\quad
n\rightarrow+\infty.\end{split}$ (5.40)

Then we get from the definition of $\beta_{n}$ stated in Lemma 5.5 that

$\begin{split}\beta_{n}=i\left(\frac{t_{s}^{n}}{t_{p}^{n}}-\frac{t_{p}^{n}}{t_{s}^{n}}\right)\left(1+\mathcal{O}\left(\frac{1}{n}\right)\right),\quad\sqrt{\beta_{n}^{2}-4}=i\left(\frac{t_{s}^{n}}{t_{p}^{n}}+\frac{t_{p}^{n}}{t_{s}^{n}}\right)\left(1+\mathcal{O}\left(\frac{1}{n}\right)\right),\end{split}$

whence it follows as $n\to\infty$ that ,

$\sigma_{1}^{(n)}=i\frac{t_{p}^{n}}{t_{s}^{n}}\left(1+\mathcal{O}\left(\frac{1}{n}\right)\right),\quad\sigma_{2}^{(n)}=-i\frac{t_{s}^{n}}{t_{p}^{n}}\left(1+\mathcal{O}\left(\frac{1}{n}\right)\right).$

Now we can get the asymptotic behavior of the eigenvalues of $F_{B_{R}}$ as
$n\rightarrow\infty$. Using (5.38), Remark 5.6 and the following recurrence
relations

$\begin{split}tJ_{n}^{\prime}(t)=nJ_{n}(t)-tJ_{n+1}(t),\quad
tH_{n}^{(1)\prime}(t)=tH_{n-1}^{(1)}(t)-nH_{n}^{(1)}(t),\end{split}$

we find

$\begin{split}\lambda^{(n)}_{B_{R},1}&=\sqrt{\frac{8\pi}{\omega}}\frac{inJ_{n}(t_{p})-it_{p}J_{n+1}(t_{p})-nJ_{n}(t_{s})\sigma_{1}^{(n)}}{t_{p}H_{n}^{(1)\prime}(t_{p})+inH_{n}^{(1)}(t_{s})\sigma_{1}^{(n)}}\\\
&=-\sqrt{\frac{2\pi}{\omega}}\frac{\pi
t_{p}^{2n+2}t_{s}^{2n}}{2^{2n}(n+1)!n!(t_{p}^{2n}+t_{s}^{2n})}\left(1+\mathcal{O}(\frac{1}{n})\right),\\\
\lambda^{(n)}_{B_{R},2}&=\sqrt{\frac{8\pi}{\omega}}\frac{it_{p}J_{n}^{\prime}(t_{p})-nJ_{n}(t_{s})\sigma_{2}^{(n)}}{t_{p}H_{n-1}^{(1)}(t_{p})-nH_{n}^{(1)}(t_{p})+inH_{n}^{(1)}(t_{s})\sigma_{2}^{(n)}}\\\
&=-\sqrt{\frac{2\pi}{\omega}}\frac{\pi(t_{p}^{2n}+t_{s}^{2n})}{2^{2n-2}(n-1)!(n-2)!t_{p}^{2}}\left(1+\mathcal{O}\left(\frac{1}{n}\right)\right).\end{split}$
(5.41)

Further, from (5.35) we get the asymptotics of the eigenvectors of
$\Sigma_{n}$ as follows

$\begin{split}X_{n,1}&=(X_{n,1}^{(1)},X_{n,1}^{(2)})^{T}=\left(\frac{t_{s}^{n}}{\sqrt{t_{p}^{2n}+t_{s}^{2n}}},\frac{it_{p}^{n}}{\sqrt{t_{p}^{2n}+t_{s}^{2n}}}\right)^{T}\;\left(1+\mathcal{O}\left(\frac{1}{n}\right)\right),\\\
X_{n,2}&=(X_{n,2}^{(1)},X_{n,2}^{(2)})^{T}=\left(\frac{t_{p}^{n}}{\sqrt{t_{p}^{2n}+t_{s}^{2n}}},\frac{-it_{s}^{n}}{\sqrt{t_{p}^{2n}+t_{s}^{2n}}}\right)^{T}\;\left(1+\mathcal{O}\left(\frac{1}{n}\right)\right).\end{split}$
(5.42)

###  5.3 Proof of Corollaries 3.5 and 3.6 for testing disks

In this subsection, we will use the eigensystem
$(\lambda_{B_{R},j}^{(n)},X_{B_{R},j}^{(n)})$ for $n\in{\mathbb{N}},j=1,2$
(see (5.38)) of the far-field operator $F_{B_{R}}$ to verify Corollaries 3.5
and 3.6 with $\Omega=B_{R}$. Corollary 3.5 can be rephrased as

######  Corollary 5.7.

Let $v^{\infty}\in\left(L^{2}({\mathbb{S}})\right)^{2}$ and assume that
$\omega^{2}$ is not a Dirichlet eigenvalue of $-\Delta^{\ast}$ over $B_{R}$.
Then

$I(B_{R})=\sum_{n\in\mathbb{Z}}\sum_{j=1}^{2}\frac{\left|\left\langle
v^{\infty},X^{(n)}_{B_{R},j}\right\rangle_{\mathbb{S}}\right|^{2}}{\left|\lambda^{(n)}_{B_{R},j}\right|}<+\infty$
(5.43)

if and only if $v^{\infty}$ is the far-field pattern of some Kupradze
radiating solution $v^{s}$, where $v^{s}$ satisfies the Navier equation

$\Delta^{\ast}v^{s}+\omega^{2}v^{s}=0\qquad\text{in}\quad\mathbb{R}^{2}\backslash\overline{B_{R}},$
(5.44)

with the boundary data $v^{s}|_{\partial B_{R}}\in\left(H^{1/2}(\partial
B_{R})\right)^{2}.$

###### Proof.

Let $v^{s}$ be a Kupradze radiating solution to (5.44). By the Hodge
decomposition (2.3), we may decompose $v^{s}$ into its compressional and shear
parts by $v^{s}={\rm grad\,}\varphi+{\rm curl\,}\psi$, where
$\varphi=\sum_{n\in\mathbb{Z}}a_{n}H^{(1)}_{n}(k_{p}r)\text{e}^{in\theta_{x}}$
and
$\psi=\sum_{n\in\mathbb{Z}}b_{n}H^{(1)}_{n}(k_{s}r)\text{e}^{in\theta_{x}}$.
Straightforward calculations lead to

$\begin{split}v^{s}&=\sum_{n\in\mathbb{Z}}\frac{\text{e}^{in\theta_{x}}}{r}\left[\left(k_{p}rH^{(1)\prime}_{n}(k_{p}r)a_{n}-inH^{(1)}_{n}(k_{s}r)b_{n}\right)\hat{x}+\left(inH^{(1)}_{n}(k_{p}r)a_{n}+k_{s}rH^{(1)\prime}_{n}(k_{s}r)b_{n}\right)\hat{x}^{\bot}\right]\\\
&=\sum_{n\in\mathbb{Z}}\frac{\text{e}^{in\theta_{x}}}{r}\begin{pmatrix}\hat{x}&\hat{x}^{\bot}\end{pmatrix}\begin{pmatrix}k_{p}rH^{(1)\prime}_{n}(k_{p}r)&-inH^{(1)}_{n}(k_{s}r)\\\
inH^{(1)}_{n}(k_{p}r)&k_{s}rH^{(1)\prime}_{n}(k_{s}r)\end{pmatrix}\begin{pmatrix}a_{n}\\\
b_{n}\end{pmatrix}.\end{split}$ (5.45)

By the asymptotic behaviour of Hankel functions (see [10])

$\begin{split}H_{n}^{(1)}(z)&=\frac{2^{n}(n-1)!}{\pi
iz^{n}}\left(1+\mathcal{O}(\frac{1}{n})\right),\quad n\rightarrow+\infty,\\\
H_{n}^{(1)\prime}(z)&=-\frac{2^{n}n!}{\pi
iz^{n+1}}\left(1+\mathcal{O}(\frac{1}{n})\right),\quad
n\rightarrow+\infty,\end{split}$ (5.46)

we have

$\begin{split}v^{s}(r,\theta_{x})&=\sum_{n\in\mathbb{Z}}\frac{\text{e}^{in\theta_{x}}}{r}\begin{pmatrix}\hat{x}&\hat{x}^{\bot}\end{pmatrix}\begin{pmatrix}k_{p}r\frac{-2^{n}n!}{\pi
ik_{p}^{n+1}r^{n+1}}&-in\frac{2^{n}(n-1)!}{\pi ik_{s}^{n}r^{n}}\\\
in\frac{2^{n}(n-1)!}{\pi ik_{p}^{n}r^{n}}&k_{s}r\frac{-2^{n}n!}{\pi
ik_{s}^{n+1}r^{n+1}}\end{pmatrix}\begin{pmatrix}a_{n}\\\
b_{n}\end{pmatrix}\left(1+\mathcal{O}(\frac{1}{n})\right)\\\
&=\sum_{n\in\mathbb{Z}}\frac{\text{e}^{in\theta_{x}}}{r}\frac{2^{n}n!}{\pi
r^{n}}\begin{pmatrix}\hat{x}&\hat{x}^{\bot}\end{pmatrix}\begin{pmatrix}ik_{p}^{-n}&-k_{s}^{-n}\\\
k_{p}^{-n}&ik_{s}^{-n}\end{pmatrix}\begin{pmatrix}a_{n}\\\
b_{n}\end{pmatrix}\left(1+\mathcal{O}(\frac{1}{n})\right).\end{split}$ (5.47)

This gives the leading term of the $H^{1/2}$-norm on $\partial B_{R}$ as

$\begin{split}||v^{s}||^{2}_{\left(H^{1/2}(\partial
B_{R})\right)^{2}}&=\sum_{n\in\mathbb{Z}}(1+n^{2})^{1/2}\frac{1}{R^{2}}\frac{2^{2n}n!n!}{\pi^{2}R^{2n}}2\left|ik_{p}^{-n}a_{n}-k_{s}^{-n}b_{n}\right|^{2}\left(1+\mathcal{O}(\frac{1}{n})\right)\\\
&\sim\sum_{n\in\mathbb{Z}}\frac{1}{R^{2}}\frac{2^{2n+1}(n+1)!n!}{\pi^{2}k_{p}^{n}k_{s}^{n}R^{2n}}\left|i\left(\frac{k_{s}}{k_{p}}\right)^{n/2}a_{n}-\left(\frac{k_{p}}{k_{s}}\right)^{n/2}b_{n}\right|^{2}\left(1+\mathcal{O}(\frac{1}{n})\right)\\\
&=\sum_{n\in\mathbb{Z}}\frac{2^{2n+1}(n+1)!n!}{\pi^{2}R^{2}t_{p}^{n}t_{s}^{n}}\left|i\left(\frac{t_{s}}{t_{p}}\right)^{n/2}a_{n}-\left(\frac{t_{p}}{t_{s}}\right)^{n/2}b_{n}\right|^{2}\left(1+\mathcal{O}(\frac{1}{n})\right)\\\
&=\sum_{n\in\mathbb{Z}}C_{n}^{(1)}\frac{2^{2n+1}(n+1)!n!}{\pi^{2}R^{2}t_{p}^{n}t_{s}^{n}}\left(1+\mathcal{O}(\frac{1}{n})\right),\end{split}$
(5.48)

where

$C_{n}^{(1)}:=\left|\left(\frac{t_{s}}{t_{p}}\right)^{n/2}a_{n}+\left(\frac{t_{p}}{t_{s}}\right)^{n/2}ib_{n}\right|^{2}.$

On the other hand, the far-field pattern of $v^{s}$ can be calculated as

$v^{\infty}(\hat{x})=\sqrt{\frac{2k_{p}}{\pi}}\text{e}^{i\frac{\pi}{4}}\sum_{n\in\mathbb{Z}}i^{-n}a_{n}\text{e}^{in\theta_{x}}\hat{x}+\sqrt{\frac{2k_{s}}{\pi}}\text{e}^{i\frac{\pi}{4}}\sum_{n\in\mathbb{Z}}i^{-n}b_{n}\text{e}^{in\theta_{x}}\hat{x}^{\bot}.$
(5.49)

We proceed with the proof by computing the leading term of the the indicator
$I(B_{R})$. Using (5.38) and (5.42), the inner product over
$L^{2}({\mathbb{S}})^{2}$ can be calculated as

$\begin{split}&\quad\left|\left\langle
v^{\infty},X_{B_{R},j}^{(n)}\right\rangle_{\SS}\right|^{2}\\\
&=\left|\int_{\SS}\sqrt{\frac{2}{\pi}}k_{p}\text{e}^{i\frac{\pi}{4}}\sum_{m\in\mathbb{Z}}i^{-m}a_{m}\text{e}^{im\theta_{x}}X_{n,j}^{(1)}\text{e}^{-in\theta_{x}}+\sqrt{\frac{2}{\pi}}k_{s}\text{e}^{i\frac{\pi}{4}}\sum_{m\in\mathbb{Z}}i^{-m}b_{m}\text{e}^{im\theta_{x}}X_{n,j}^{(2)}\text{e}^{-in\theta_{x}}d\theta_{x}\right|^{2}\\\
&=\left|\sqrt{\frac{2}{\pi}}\text{e}^{i\frac{\pi}{4}}i^{-n}\int_{\SS}k_{p}a_{n}X_{n,j}^{(1)}+k_{s}b_{n}X_{n,j}^{(2)}d\theta_{x}\right|^{2}\\\
&=8\pi\left|k_{p}a_{n}X_{n,j}^{(1)}+k_{s}b_{n}X_{n,j}^{(2)}\right|^{2},\end{split}$
(5.50)

for $j=1,2$. Using asymptotic behavior shown in (5.41) and (5.42), it is easy
to check that, as $n\to\infty$,

$\left|\lambda^{(n)}_{B_{R},1}\right|\sim\left|\lambda^{(n)}_{B_{R},2}\right|\;n^{-4},\quad\left|\left\langle
v^{\infty},X_{B_{R},1}^{(n)}\right\rangle_{\SS}\right|^{2}\sim\left|\left\langle
v^{\infty},X_{B_{R},2}^{(n)}\right\rangle_{\SS}\right|^{2}.$

Thus,

$\begin{split}I(B_{R})&=\sum_{n\in\mathbb{Z}}\left(\frac{\left|\left\langle
v^{\infty},X^{(n)}_{B_{R},1}\right\rangle_{\mathbb{S}}\right|^{2}}{\left|\lambda^{(n)}_{B_{R},1}\right|}+\frac{\left|\left\langle
v^{\infty},X^{(n)}_{B_{R},2}\right\rangle_{\mathbb{S}}\right|^{2}}{\left|\lambda^{(n)}_{B_{R},2}\right|}\right)\\\
&=\sum_{n\in\mathbb{Z}}\frac{8\pi\left|k_{p}a_{n}X_{n,1}^{(1)}+k_{s}b_{n}X_{n,1}^{(2)}\right|^{2}}{\left|\lambda^{(n)}_{B_{R},1}\right|}\left(1+\mathcal{O}(\frac{1}{n})\right)\\\
&=\sum_{n\in\mathbb{Z}}C_{n}^{(2)}\sqrt{\frac{\omega}{2\pi}}\frac{2^{2n+3}(n+1)!n!}{t_{p}^{n}t_{s}^{n}R^{2}k_{p}^{2}}\left(1+\mathcal{O}(\frac{1}{n})\right),\\\
\end{split}$ (5.51)

where

$C_{n}^{(2)}:=\left|k_{p}\left(\frac{t_{s}}{t_{p}}\right)^{n/2}a_{n}+k_{s}\left(\frac{t_{p}}{t_{s}}\right)^{n/2}ib_{n}\right|^{2}.$

Noting that $\min\\{k_{p}^{2},k_{s}^{2}\\}C_{n}^{(1)}\leqslant
C_{n}^{(2)}\leqslant\max\\{k_{p}^{2},k_{s}^{2}\\}C_{n}^{(1)}$, we conclude
that the series (5.48) and (5.51) have the same convergence.

Since the boundedness of $||v^{s}||_{\left(H^{1/2}(\partial
B_{R})\right)^{2}}$ implies that $v^{s}$ of the form (5.45) is indeed a
radiating solution in $\mathbb{R}^{2}\backslash\overline{B_{R}}$ with the far-
field pattern $v^{\infty}$. This proves that $I(B_{R})<\infty$ if and only if
$v^{s}$ is a Kupradze radiating solution in
$\mathbb{R}^{2}\backslash\overline{B_{R}}$ with the far-field pattern
$v^{\infty}$ and with the $H^{1/2}$-boundary data on $\partial B_{R}$. The
proof of Corollaries 5.7 is thus complete. ∎

To prove Corollary 3.6 for testing disks, we need to consider spectral systems
of the far-field operators $F_{B_{R}}^{(p)}$ and $F_{B_{R}}^{(s)}$ . By
Definition 3.3, it follows that

$\begin{split}(F_{B_{R}}^{(p)}g_{p})(\hat{x}):=\text{e}^{-\frac{i\pi}{4}}\sqrt{\frac{k_{p}}{\omega}}\int_{{\mathbb{S}}}u^{\infty}_{B_{R},pp}(\hat{x};d)\,g_{p}(d)\,ds(d),\quad
g_{p}\in L_{p}^{2}({\mathbb{S}}),\\\
(F_{B_{R}}^{(s)}g_{s})(\hat{x}):=\text{e}^{-\frac{i\pi}{4}}\sqrt{\frac{k_{s}}{\omega}}\int_{{\mathbb{S}}}u^{\infty}_{B_{R},ss}(\hat{x};d)\,g_{s}(d)\,ds(d),\quad
g_{s}\in L_{s}^{2}({\mathbb{S}}).\end{split}$

Using (2.13) and (5.29), we see

$u^{\infty}_{B_{R},pp}(\hat{x})=\sqrt{\frac{2}{\pi
k_{p}}}\text{e}^{i\frac{\pi}{4}}i\sum_{n\in\mathbb{Z}}\Sigma_{n}(1,1)\text{e}^{in\theta},\quad
u^{\infty}_{B_{R},ss}(\hat{x})=\sqrt{\frac{2}{\pi
k_{s}}}\text{e}^{i\frac{\pi}{4}}i\sum_{n\in\mathbb{Z}}\Sigma_{n}(2,2)\text{e}^{in\theta},$

where $\Sigma_{n}(i,j)$ dentoes the $(i,j)$-th entry of the matrix
$\Sigma_{n}$. Now we can get the spectral systems of the operators
$F_{B_{R}}^{(p)}$ and $F_{B_{R}}^{(s)}$:

$\begin{split}\eta_{B_{R},p}^{(n)}&=\sqrt{\frac{2}{\pi
k_{p}}}\text{e}^{i\frac{\pi}{4}}i\Sigma_{n}(1,1),\quad\varphi_{B_{R},p}^{(n)}(\hat{x})=\text{e}^{in\theta_{x}},\\\
\eta_{B_{R},s}^{(n)}&=\sqrt{\frac{2}{\pi
k_{s}}}\text{e}^{i\frac{\pi}{4}}i\Sigma_{n}(2,2),\quad\varphi_{B_{R},s}^{(n)}(\hat{x})=\text{e}^{in\theta_{x}}.\end{split}$
(5.52)

Taking $\Omega=B_{R}$, we can rewrite Corollary 3.6 as

######  Corollary 5.8.

Let $w^{\infty}_{\alpha\alpha}\in L^{2}_{\alpha}({\mathbb{S}})$ ($\alpha=p,s$)
and assume that $\omega^{2}$ is not a Dirichlet eigenvalue of $-\Delta^{\ast}$
over $B_{R}$. Denote by
$(\lambda^{(n)}_{B_{R},\alpha},\varphi^{(n)}_{B_{R},\alpha})$ a spectral
system of the positive operator $F^{(\alpha)}_{B_{R},\\#}$. Then

$I^{(\alpha)}(B_{R})=\sum_{n\in\mathbb{Z}}\frac{\left|\left\langle
w^{\infty}_{\alpha\alpha},\varphi^{(n)}_{B_{R},\alpha}\right\rangle_{\mathbb{S}}\right|^{2}}{\left|\lambda^{(n)}_{B_{R},\alpha}\right|}<+\infty$
(5.53)

if and only if $w^{\infty}_{\alpha\alpha}$ is the far-field pattern of some
Sommerfeld radiating solution $w^{s}_{\alpha\alpha}$ which is defined in
$\mathbb{R}^{2}\backslash\overline{B_{R}}$ and
$w^{s}_{\alpha\alpha}|_{\partial B_{R}(z)}\in H^{-1/2}(\partial B_{R}).$ That
is, $w^{s}_{\alpha\alpha}$ satisfies the following boundary value problem of
the Helmholtz equation

$\Delta w^{s}_{\alpha\alpha}+k_{\alpha}^{2}w^{s}_{\alpha\alpha}=0\ \text{in}\
\mathbb{R}^{2}\backslash\overline{B_{R}},\quad w^{s}_{\alpha\alpha}|_{\partial
B_{R}}\in H^{-1/2}(\partial B_{R}).$ (5.54)

###### Proof.

Without losing generality, we only consider the case of $\alpha=p$. The case
of $\alpha=s$ can be proceeded in a similar manner. Since

$\lambda_{B_{R},p}^{(n)}=|\text{Re}(\eta_{B_{R},p}^{(n)})|+|\text{Im}(\eta_{B_{R},p}^{(n)})|,$

we deduce from (5.52) and the definition
$\Sigma_{n}:=\mathcal{H}_{n}^{-1}\mathcal{J}_{n}$ that

$\lambda_{B_{R},p}^{(n)}=\sqrt{\frac{\pi}{k_{p}}}\frac{t_{p}^{2n}}{2^{2n-1}(n-1)!(n-2)!(t_{p}^{2}+t_{s}^{2})}\left(1+\mathcal{O}(\frac{1}{n})\right),\
\varphi_{B_{R},p}^{(n)}=\text{e}^{in\theta_{x}}.$ (5.55)

By the Jacobi-Anger expansion (see e.g.,[10]), a Sommerfeld radiating solution
$w^{s}_{pp}$ to the Helmholtz equation in $|x|>R$ can be expanded into the
series

$w^{s}_{pp}(x)=\sum_{n\in\mathbb{Z}}D_{n}H_{n}^{(1)}(k_{p}|x|)\text{e}^{in\theta_{x}},\quad|x|>R,\quad
x=(|x|,\theta_{x}),$ (5.56)

with the far-field pattern given by (see [10, (3.82)])

$w^{\infty}_{pp}(\hat{x})=\sum_{n\in\mathbb{Z}}D_{n}C_{n,p}\text{e}^{in\theta_{x}},\quad
C_{n,p}:=\sqrt{\frac{2}{k_{p}\pi}}\text{e}^{-i(\frac{n\pi}{2}+\frac{\pi}{4})}.$
(5.57)

Hence,

$\begin{split}I^{(p)}(B_{R})&=\sum_{n\in\mathbb{Z}}\frac{\left|\left\langle\sum_{m\in\mathbb{Z}}D_{m}C_{m,p}\text{e}^{im\theta_{x}},\varphi^{(n)}_{B_{R},p}\right\rangle_{\mathbb{S}}\right|^{2}}{\left|\lambda^{(n)}_{B_{R},p}\right|}\\\
&=\sum_{n\in\mathbb{Z}}\frac{\left|2\pi
D_{n}C_{n,p}\right|^{2}}{\left|\lambda^{(n)}_{B_{R},p}\right|}\\\
&=\sum_{n\in\mathbb{Z}}\sqrt{\frac{\pi}{k_{p}}}\frac{2^{2n+2}(n-1)!(n-2)!(t_{p}^{2}+t_{s}^{2})}{t_{p}^{2n}}|D_{n}|^{2}\left(1+\mathcal{O}(\frac{1}{n})\right).\end{split}$
(5.58)

By the definition of $H^{-1/2}$-norm on $\partial B_{R}$, we get

$\begin{split}||w^{s}_{pp}(x)||^{2}_{H^{-1/2}(\partial
B_{R})}&=\sum_{n\in\mathbb{Z}}(1+n^{2})^{-1/2}|D_{n}H^{(1)}_{n}(t_{p})|^{2}\\\
&=\sum_{n\in\mathbb{Z}}|D_{n}|^{2}\frac{2^{2n}(n-1)!(n-2)!}{\pi^{2}t_{p}^{2n}}\left(1+\mathcal{O}(\frac{1}{n})\right).\end{split}$
(5.59)

Obviously, the series (5.58) and (5.59) have the same convergence. On the
other hand, following the proof of [10, Theorem 2.15], it is not difficulty to
prove that the boundedness of $||w^{s}_{pp}||_{H^{-1/2}(\partial B_{R})}$
implies that $w^{s}_{pp}$ is a radiating solution in $|x|>R$ with the far-
field pattern $w^{\infty}_{pp}(\hat{x})$. This proves that
$I^{(p)}(B_{R})<\infty$ if and only if $w^{s}_{pp}$ is a radiating solution to
the boundary value problem of the Helmholtz equation (5.54), with the far-
field pattern $w^{\infty}_{pp}(\hat{x})$. ∎

##  6 Imaging schemes with testing disks

Let $B_{R}(z)=z+B_{R}\coloneqq\\{y\in{\mathbb{R}}^{2}:y=z+x,x\in B_{R}\\}$ be
a rigid disk centered at $z\in{\mathbb{R}}$ with radius $R>0$. By the
translation relations (see e.g., (2.13)-(2.16),[18]), we know

$u_{B_{R}(z),\alpha\beta}^{\infty}(\hat{x})=\text{e}^{-ik_{\alpha}z\cdot\hat{x}}\text{e}^{ik_{\beta}z\cdot
d}u_{B_{R},\alpha\beta}^{\infty}(\hat{x}),$ (6.1)

where $\alpha=p,s$ and $\beta=p,s$. Define the matrices

$\begin{split}&U_{B_{R}(z)}^{\infty}(\hat{x})\coloneqq\begin{pmatrix}u^{\infty}_{B_{R}(z),pp}(\hat{x})&u^{\infty}_{B_{R}(z),ps}(\hat{x})\\\
u^{\infty}_{B_{R}(z),sp}(\hat{x})&u^{\infty}_{B_{R}(z),ss}(\hat{x})\end{pmatrix},\\\
&M_{d,z}\coloneqq\begin{pmatrix}\text{e}^{-ik_{p}z\cdot d}&0\\\
0&\text{e}^{-ik_{s}z\cdot d}\end{pmatrix},\quad
M_{\hat{x},z}\coloneqq\begin{pmatrix}\text{e}^{-ik_{p}z\cdot\hat{x}}&0\\\
0&\text{e}^{-ik_{s}z\cdot\hat{x}}\end{pmatrix}.\end{split}$

Then, it holds that

$U_{B_{R}(z)}^{\infty}=M_{d,z}^{-1}\;U^{\infty}_{B_{R}}\;M_{\hat{x},z}.$ (6.2)

Using the previous relation, we obtain spectral systems for the operators
$F_{\Omega}$, $F_{\Omega}^{(p)}$ and $F_{\Omega}^{(s)}$ with $\Omega=B_{R}(z)$
as follows.

######  Corollary 6.1.

The eigenvalues $\lambda_{B_{R}(z),j}^{(n)}$ and the associated eigenfunctions
$X^{(n)}_{B_{R}(z),j}$ of the far-field operator $F_{B_{R}(z)}$ are given by
(e.g. (5.38))

$\begin{split}&\lambda_{B_{R}(z),j}^{(n)}=\lambda_{B_{R},j}^{(n)},\\\
&X^{(n)}_{B_{R}(z),j}(\hat{x})=\frac{\left(\sqrt{k_{p}}\text{e}^{-ik_{p}z\cdot\hat{x}}\hat{x}+\sqrt{k_{s}}\sigma_{j}^{(n)}\text{e}^{-ik_{s}z\cdot\hat{x}}\hat{x}^{\bot}\right)}{\sqrt{1+|\sigma_{j}^{(n)}|^{2}}}\text{e}^{in\theta_{x}}\end{split}$

for $n\in{\mathbb{Z}},\quad j=1,2$. Moreover, the spectral systems of the
operators $F_{B_{R}(z)}^{(p)}$ and $F_{B_{R}(z)}^{(s)}$ take the form (e.g.
(5.52))

$\begin{split}\eta_{B_{R}(z),p}^{(n)}&=\sqrt{\frac{2}{\pi
k_{p}}}\text{e}^{i\frac{\pi}{4}}i\Sigma_{n}(1,1),\quad\varphi_{B_{R}(z),p}^{(n)}(\hat{x})=\text{e}^{in\theta_{x}}\text{e}^{-ik_{p}z\cdot\hat{x}},\\\
\eta_{B_{R}(z),s}^{(n)}&=\sqrt{\frac{2}{\pi
k_{s}}}\text{e}^{i\frac{\pi}{4}}i\Sigma_{n}(2,2),\quad\varphi_{B_{R}(z),s}^{(n)}(\hat{x})=\text{e}^{in\theta_{x}}\text{e}^{-ik_{s}z\cdot\hat{x}}.\end{split}$
(6.3)

Furthermore, taking $\Omega=B_{R}(z)$, we can rewrite results of Theorems 4.3
and 4.4 as

######  Theorem 6.2.

Define

$\begin{split}&W(B_{R}(z)):=\sum_{n\in\mathbb{Z}}\sum_{j=1}^{2}\frac{\left|\left\langle
u_{D}^{\infty},X^{(n)}_{B_{R}(z),j}\right\rangle_{\mathbb{S}}\right|^{2}}{\left|\lambda^{(n)}_{B_{R}(z),j}\right|},\\\
&W^{(\alpha)}(B_{R}(z)):=\sum_{n\in\mathbb{Z}}\frac{\left|\left\langle
u_{D,\alpha\alpha}^{\infty},\varphi^{(n)}_{B_{R}(z),\alpha}\right\rangle_{\mathbb{S}}\right|^{2}}{\left|\lambda^{(n)}_{B_{R}(z),\alpha}\right|},\quad\alpha=p,s\end{split}$
(6.4)

where

$\lambda^{(n)}_{B_{R}(z),\alpha}=\left|\text{\rm
Re}\left(\eta_{B_{R}(z),\alpha}^{(n)}\right)\right|+\left|\text{\rm
Im}\left(\eta_{B_{R}(z),\alpha}^{(n)}\right)\right|.$

Then $W(B_{R}(z))<\infty$ if and only if $D\subseteq B_{R}(z)$ and the same
conclusion applies to $W^{(\alpha)}(B_{R}(z))$.

Finally, we describe our imaging scheme for solving the inverse problems IP-F,
IP-P and IP-S stated at the end of Section 2. Let $D$ be a convex rigid
polygon to be recovered from far-field data. The procedure consists of the
following steps:

  * •

Suppose that $B_{R}\supset D$ for some $R>0$ and collect the measurement data
$u_{D}^{\infty}(\hat{x})$, $u_{D,pp}^{\infty}(\hat{x})$ or
$u_{D,ss}^{\infty}(\hat{x})$ for all $\hat{x}\in\mathbb{S}$. Let $Q\supset D$
be our search/computational region for imaging $D$;

  * •

Choose sampling centers $z_{n}\in\Gamma_{R}:=\\{x:\ |x|=R\\}$ for
$n=1,2,\cdots,N$ and choose sampling radii $h_{m}\in(0,2R)$ ($m=1,2,\cdots,M$)
to get the spectral systems for the operators $F_{\Omega}$, $F_{\Omega}^{(p)}$
and $F_{\Omega}^{(s)}$ with $\Omega=B_{h_{m}}(z_{n})$, $n=1,2,\cdots,N$,
$m=1,2,\cdots,M$ (see (5.38) and Corollary 6.1);

  * •

For each $z_{n}\in\Gamma_{R}$, define the function over the grid points $x\in
Q$ satsfying $h_{m+1}>|x-z_{n}|\geqslant h_{m}$ for some $m=1,2,\cdots,M$ by
(see (6.4)):

$\mathcal{I}_{n}(x)=\left\\{\begin{array}[]{lll}&[W(B_{h_{m}}(z_{n}))]^{-1}\qquad\mbox{for
the inverse problem {\bf IP-F}};\\\
&[W^{(p)}(B_{h_{m}}(z_{n}))]^{-1}\quad\mbox{for the inverse problem {\bf
IP-P}};\\\ &[W^{(s)}(B_{h_{m}}(z_{n}))]^{-1}\quad\mbox{for the inverse problem
{\bf IP-S}};\end{array}\right.$

  * •

The imaging function for recovering $D$ is defined as
$\mathcal{I}(x)=\sum_{n=1}^{N_{z}}\mathcal{I}_{n}(x)$, where $x\in Q$ are the
grid points. This can be considered as imaging function over $Q$ if the grids
are sufficiently fine.

We expect the values of the indicator function $\mathcal{I}$ for grid points
$x\in D$ should be larger than those for $x\in Q\backslash\overline{D}$,
because

$\begin{split}&[W(B_{h_{m}}(z_{n}))]^{-1}=0\quad\mbox{if}\quad
h_{m}\leq\max_{y\in\partial D}|z_{n}-y|;\\\
&[W(B_{h_{m}}(z_{n}))]^{-1}<\infty\quad\mbox{if}\quad h_{m}>\max_{y\in\partial
D}|z_{n}-y|;\end{split}$

and the same indicating behavior applies to
$[W^{(\alpha)}(B_{h_{m}}(z_{n}))]^{-1}$, $\alpha=p,s$.

######  Remark 6.3.

In implementing the above scheme, the spectral data appeared in Theorem 6.2
are all given explicitly by Corollary 6.1. For each sampling disk
$B_{h_{m}}(z_{n})$ with $n=1,2,\cdots N,m=1,2,\cdots,M$, they can be easily
calculated and stored off-line before the inversion process. This is just the
advantageous of using testing disks instead of other testing scatterers.

##  7 Acknowledgements

This work was supported by NSFC 12071236 and NSAF U1930402.

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