These authors contributed equally to this work

[2]Peijun Li

These authors contributed equally to this work

1]School of Mathematical Sciences, Zhejiang University, Hangzhou, 310027,
Zhejiang, China

[2]Department of Mathematics, Purdue University, West Lafayette, 47907,
Indiana, USA

3]School of Mathematics, Jilin University, Changchun, 130012, Jilin, China

# Convergence of the PML method for the biharmonic wave scattering problem in
periodic structures

Gang Bao<EMAIL_ADDRESS><EMAIL_ADDRESS>Xiaokai Yuan
<EMAIL_ADDRESS>[  *  [

###### Abstract

This paper investigates the scattering of biharmonic waves by a one-
dimensional periodic array of cavities embedded in an infinite elastic thin
plate. The transparent boundary conditions are introduced to formulate the
problem from an unbounded domain to a bounded one. The well-posedness of the
associated variational problem is demonstrated utilizing the Fredholm
alternative theorem. The perfectly matched layer (PML) method is employed to
reformulate the original scattering problem, transforming it from an unbounded
domain to a bounded one. The transparent boundary conditions for the PML
problem are deduced, and the well-posedness of its variational problem is
established. Moreover, exponential convergence is achieved between the
solution of the PML problem and that of the original scattering problem.

###### keywords:

Biharmonic wave equation, transparent boundary condition, perfectly matched
layer, variational problem, well-posedness, convergence analysis.

###### pacs:

[

2010]78A45, 65N30

##  1 Introduction

Scattering of flexural waves in an elastic thin plate, modeled by fourth-order
biharmonic wave equations, holds broad engineering applications. These
applications span diverse fields, including the design of ultra-broadband
elastic cloaking [1, 2, 3], platonic crystals [4, 5, 6], and the exploration
of acoustic black hole concepts [7]. Consequently, ongoing research in
theoretical analysis, numerical simulations, and industrial manufacturing
continues to draw considerable attention from both engineering and
mathematical communities.

Most works in the literature focus on the static problem in a bounded domain,
which is formulated by the bi-Laplacian equation. When addressing the fourth-
order problem using the finite element method, standard $H^{2}$-conforming
methods necessitate $C^{1}$-continuous piecewise polynomials on the mesh, a
challenge in practical implementation. Alternatively, various nonconforming
and discontinuous finite element methods have emerged, such as the weak
Galerkin finite element methods supplemented with stabilizers [8, 9, 10]; the
virtual element method, which requires no global $C^{0}$ regularity for the
numerical solution [11, 12]; and the mixed element method, effectively
reducing the fourth-order problem to coupled second-order problems [13, 14,
15, 16]. These methods have undergone comprehensive analysis.

When compared with the results concerning the bi-Laplacian equation, the
findings are relatively limited for the biharmonic wave scattering problems in
unbounded domains. In [17], the initial theoretical analysis of the boundary
integral equation method was provided for solving the biharmonic wave
equation. Through the introduction of two auxiliary variables, the biharmonic
wave equation was split into the Helmholtz and modified Helmholtz equations.
Subsequently, the Holmholtz and modified Helmholtz wave components were
represented using the double- and single-layer potentials. The well-posedness
of the coupled boundary integral system was established by applying the
Riesz–Fredholm theory. If the exterior problem is approached using the
variational approach with transparent boundary conditions (TBCs), the studies
concerning waveguide and obstacle scattering problems were presented in [18,
19] under various boundary conditions, including clamped, simply supported,
roller-supported, or free plate boundary conditions. Numerically, a mixed
element method was proposed in [20, 21] by introducing two auxiliary variables
and decomposing the biharmonic problem into the Helmholtz and modified
Helmholtz equations. Subsequently, TBCs were introduced for each equation.
Particularly, the linear finite element method, incorporating interior penalty
and boundary penalty, was proposed in [21] to effectively reduce the
oscillation of the bending moment.

The method of perfectly matched layer (PML) is a widely utilized domain
truncation technique. In contrast to the nonlocal TBC method, the PML method
generates a local boundary condition on the outer surface of the layer by
integrating an artificial absorbing region around the domain of interest. The
ease of handling the local boundary condition has contributed to the
widespread adoption of this method ever since its inception by Bérenger [22]
for solving the time-dependent Maxwell equations. It has found extensive
applications in solving various wave scattering problems, including, for
example, acoustic waves [23], electromagnetic waves [24, 25, 26], and elastic
waves [27, 28]. The PML method has also been utilized numerically in solving
biharmonic wave scattering problems [29, 30, 31], highlighting its convenience
and accuracy. However, to our knowledge, a comprehensive discussion regarding
the well-posedness of the PML method and its convergence has not been
documented in existing literature. This paper aims to address these gaps.

In this paper, we investigate the scattering of flexural waves resulting from
a plane incident wave interacting with a one-dimensional periodic array of
cavities within an infinite elastic thin plate. The wave propagation is
described by the fourth-order biharmonic wave equation. Because of the
periodic characteristics of both the incident wave and the cavities, the
solution complies with quasi-periodic conditions, allowing us to formulate the
problem within a single periodic cell. The TBCs are derived by incorporating
the bounded outgoing wave condition, utilizing the Fourier series expansion of
the solution in regions distant from the cavities. With the aid of the TBCs,
the scattering problem is equivalently transformed from an unbounded domain to
a bounded one. The corresponding variational problem is shown to satisfy
Gårding’s inequality, and its well-posedness is established through the
utilization of the Fredholm alternative theorem. To replace the nonlocal TBCs,
the PML method is adopted through the complex coordinate stretching scheme
[32]. Alternatively, the unbounded domain is truncated by imposing homogeneous
boundary conditions on the wave field and its normal derivative at the outer
boundary of the PML region. Upon studying the Fourier series expansion of the
solution to the PML problem, we deduce equivalent TBCs to reformulate the PML
problem in the domain where the original scattering problem, along with the
TBC, is imposed. The well-posedness of the PML problem is confirmed through an
examination of its variational formulation. Additionally, the PML solution
demonstrates exponential convergence concerning the thickness of the PML
regions towards the solution of the original scattering problem. For a
comprehensive account of related electromagnetic wave scattering problems in
periodic structures, we reference [33].

The paper is outlined as follows. Section 2 introduces the model equations.
The TBCs are derived in Section 3. Section 4 details the reduction of the
scattering problem to a bounded domain using the TBCs, along with the
discussion on the well-posedness of the variational problem. Section 5
addresses the PML problem, including investigations into its well-posedness
and convergence. Finally, the paper concludes with general remarks in Section
6.

##  2 Problem formulation

Let us examine the scattering phenomenon of an incident wave interacting with
a one-dimensional periodic array of cavities in an infinitely extending
elastic thin plate, which is characterized by the Kirchhoff–Love model and is
depicted in Figure 1. Assume that the alignment of the cavities coincides with
the $x_{1}$-axis, exhibiting a periodicity of $\Lambda$. Consider an incident
field represented as a time-harmonic plane wave given by

$u^{i}(x)=e^{{\rm i}(\alpha x_{1}-\beta x_{2})},\quad x\in\mathbb{R}^{2},$

where $\alpha=\kappa\sin\theta,\beta=\kappa\cos\theta$ with $\kappa>0$ and
$\theta\in\left(-\frac{\pi}{2},\frac{\pi}{2}\right)$ denoting the wavenumber
and the incident angle, respectively. It can be verified that the incident
field $u^{i}$ satisfies the biharmonic wave equation

$\Delta^{2}u^{i}-\kappa^{4}u^{i}=0\quad\text{in}~{}\mathbb{R}^{2}.$

Due to the periodic nature of the geometry, the problem can be confined to a
single periodic cell. Denote by $\Omega_{c}$ the cavity with a Lipschitz
continuous boundary $\Gamma_{c}$. Let $R$ be a rectangular domain that is
sufficiently large to enclose the region $\Omega_{c}$. Without loss of
generality, let $R$ be defined as
$R=\left\\{x\in\mathbb{R}^{2}:0<x_{1}<\Lambda,h_{2}<x_{2}<h_{1}\right\\}$,
where $h_{k},k=1,2$ are constants. Additionally, define
$\Gamma_{k}=\left\\{x\in\mathbb{R}^{2}:0<x_{1}<\Lambda,x_{2}=h_{k}\right\\}$
for $k=1,2$,
$\Gamma_{l}=\left\\{x\in\mathbb{R}^{2}:x_{1}=0,h_{2}<x_{2}<h_{1}\right\\}$,
and
$\Gamma_{r}=\left\\{x\in\mathbb{R}^{2}:x_{1}=\Lambda,h_{2}<x_{2}<h_{1}\right\\}$.
Let $\Omega=R\setminus\overline{\Omega_{c}}$. Define $\Omega_{1}$ and
$\Omega_{2}$ as the regions above and below $\Gamma_{1}$ and $\Gamma_{2}$,
respectively.

Figure 1: Schematic of the problem geometry.

The out-of-plane displacement of the plate, denoted as $u$ and referred to as
the total field, also satisfies the biharmonic wave equation

$\Delta^{2}u-\kappa^{4}u=0\quad{\rm in}~{}\Omega.$ (1)

The total field is assumed to satisfy the Dirichlet boundary condition, known
as the clamped boundary condition, on $\Gamma_{c}$:

$u=0,\quad\partial_{\nu}u=0,$ (2)

where $\nu$ denotes the unit normal vector on $\Gamma_{c}$. It is worth noting
that other types of boundary conditions, such as the Neumann boundary
condition, often known as the free plate boundary condition, can be similarly
taken into account.

Given the periodic nature of both the structure and the incident wave, the
solution to (1)–(2) demonstrates quasi-periodicity. Specifically, if $u$ is a
solution to (1)–(2), then $u(x)e^{-{\rm i}\alpha x_{1}}$ is a periodic
function of $x_{1}$ with a period of $\Lambda$. This characteristic gives rise
to the quasi-periodic boundary condition on $\Gamma_{l}$ and $\Gamma_{r}$,
i.e., $u$ satisfies $u(0,x_{2})=e^{-{\rm i}\alpha\Lambda}u(\Lambda,x_{2})$.
Furthermore, to ensure the well-posedness of the problem, it is essential to
impose a bounded outgoing wave condition on the scattered field
$u^{s}=u-u^{i}$ in $\Omega_{1}$ and the total field $u$ in $\Omega_{2}$.

We introduce notations and function spaces employed in this work. Denote by
$H^{2}(\Omega)$ the standard Sobolev space, comprising functions with square-
integrable values, as well as square-integrable first and second partial
derivatives. Let us define the quasi-periodic function space

$H_{\rm qp}^{2}(\Omega)=\left\\{u\in
H^{2}(\Omega):u(\Lambda,x_{2})=u(0,x_{2})e^{{\rm i}\alpha\Lambda}\right\\},$

along with its subspace

$H_{{\rm qp},\Gamma_{c}}^{2}(\Omega)=\left\\{u\in H^{2}_{\rm
qp}(\Omega):u=0,\partial_{\nu}u=0\text{ on }\Gamma_{c}\right\\}.$

Clearly, $H_{\rm qp}^{2}(\Omega)$ and $H_{{\rm qp},\Gamma_{c}}^{2}(\Omega)$
are subspaces of $H^{2}(\Omega)$ equipped with the standard $H^{2}$-norm.

Given a function $u\in H_{\rm qp}^{2}(\Omega)$, it allows for a Fourier
expansion on $\Gamma_{k},k=1,2:$

$u(x,h_{k})=\sum\limits_{n\in\mathbb{Z}}u^{(n)}(h_{k})e^{{\rm
i}\alpha_{n}x_{1}},$

where

$\alpha_{n}=\alpha+n\left(\frac{2\pi}{\Lambda}\right),\quad
u^{(n)}(h_{k})=\frac{1}{\Lambda}\int_{0}^{\Lambda}u(x,h_{k})e^{-{\rm
i}\alpha_{n}x_{1}}{\rm d}x_{1}.$

The trace function space $H^{s}(\Gamma_{k})$, where $s\in\mathbb{R}$, is
defined as follows:

$H^{s}(\Gamma_{k})=\left\\{u\in
L^{2}(\Gamma_{k}):\|u\|_{H^{s}(\Gamma_{k})}<\infty\right\\},$

with the norm given by

$\|u\|_{H^{s}(\Gamma_{k})}=\left(\Lambda\sum\limits_{n\in\mathbb{Z}}\left(1+\alpha_{n}^{2}\right)^{s}|u^{(n)}(h_{k})|^{2}\right)^{1/2}.$

In this paper, whenever $a\lesssim b$ is used, it denotes $a\leq Cb$, with $C$
representing a positive constant. In this context, the values of the constants
$c_{j}$ are positive and may vary in different steps of the proof. Although
the specific values of $C$ and $c_{j}$ are not explicitly stated, their
dependence should be apparent from the context.

##  3 Transparent boundary conditions

In this section, we address the challenge posed by formulating the problem in
an unbounded domain. To overcome this obstacle, we propose introducing an
equivalent transparent boundary condition (TBC) on $\Gamma_{k},k=1,2$ with the
objective of transforming the problem into the bounded domain $\Omega$.

Let $\nu=(\nu_{1},\nu_{2})$ and $\tau=(\tau_{1},\tau_{2})$ be the unit normal
and tangent vectors, respectively, to the boundary of $\Omega$. Clearly, we
have $\tau_{1}=-\nu_{2}$ and $\tau_{2}=\nu_{1}$. Define the normal and
tangential derivatives

$\partial_{\nu}:=\nu_{1}\partial_{x_{1}}+\nu_{2}\partial_{x_{2}},\quad\partial_{\tau}:=-\nu_{2}\partial_{x_{1}}+\nu_{1}\partial_{x_{2}}.$

For $0\leq\mu<1$, referred to as the Poisson ratio, define the surface
differential operators (cf. [34]):

$Mu:=\mu\Delta u+(1-\mu)M_{0}u,\quad Nu:=-\partial_{\nu}\Delta
u-(1-\mu)\partial_{\tau}N_{0}u,$ (3)

where $M_{0}$ and $N_{0}$ are explicitly given by

$\left\\{\begin{aligned} &M_{0}u:=\nu_{1}^{2}\frac{\partial^{2}u}{\partial
x_{1}^{2}}+2\nu_{1}\nu_{2}\frac{\partial^{2}u}{\partial x_{1}\partial
x_{2}}+\nu_{2}^{2}\frac{\partial^{2}u}{\partial x_{2}^{2}},\\\
&N_{0}u:=\left(\nu_{1}^{2}-\nu_{2}^{2}\right)\frac{\partial^{2}u}{\partial
x_{1}\partial x_{2}}-\nu_{1}\nu_{2}\left(\frac{\partial^{2}u}{\partial
x_{1}^{2}}-\frac{\partial^{2}u}{\partial
x_{2}^{2}}\right).\end{aligned}\right.$

First, we derive the TBC on $\Gamma_{1}$. Based on the bounded outgoing wave
condition, it is shown in [20] that the scattered field $u^{s}$ can be
represented by a Fourier series expansion in the domain $\Omega_{1}$:

$u^{s}(x_{1},x_{2})=\sum\limits_{n\in\mathbb{Z}}H_{1}^{(n)}e^{{\rm
i}\alpha_{n}x_{1}+{\rm
i}\beta_{n}(x_{2}-h_{1})}+\sum\limits_{n\in\mathbb{Z}}U_{1}^{(n)}e^{{\rm
i}\alpha_{n}x_{1}-\gamma_{n}(x_{2}-h_{1})},$ (4)

where $H_{1}^{(n)},U_{1}^{(n)}\in\mathbb{C}$ are the Fourier coefficients, and

$\beta_{n}=\left\\{\begin{aligned}
&(\kappa^{2}-\alpha_{n}^{2})^{1/2}\quad&&\text{if}~{}\kappa>|\alpha_{n}|,\\\
&{\rm
i}(\alpha_{n}^{2}-\kappa^{2})^{1/2}\quad&&\text{if}~{}\kappa<|\alpha_{n}|,\end{aligned}\right.\quad\gamma_{n}=(\kappa^{2}+\alpha_{n}^{2})^{1/2}.$
(5)

Here, we assume that $\beta_{n}\neq 0$ for all $n\in\mathbb{Z}$ to rule out
the occurrence of resonances.

Let $\left(u,\partial_{\nu}u\right)=(f_{1},g_{1})$ and
$\left(u^{s},\partial_{\nu}u^{s}\right)=(\hat{f}_{1},\hat{g}_{1})$ be the
Dirichlet data for the total and scattered fields on $\Gamma_{1}$,
respectively. It is clear to note that these data satisfy the relations

$f_{1}(x_{1})=\hat{f}_{1}(x_{1})+e^{{\rm i}\left(\alpha x_{1}-\beta
h_{1}\right)},\quad g_{1}(x_{1})=\hat{g}_{1}(x_{1})-{\rm i}\beta e^{{\rm
i}\left(\alpha x_{1}-\beta h_{1}\right)}.$

Being quasi-periodic functions, $\hat{f}_{1}$ and $\hat{g}_{1}$ admit the
Fourier series expansions

$u^{s}(x_{1},h)=\hat{f}_{1}(x_{1})=\sum_{n\in\mathbb{Z}}\hat{f}_{1}^{(n)}e^{{\rm
i}\alpha_{n}x_{1}},\quad\partial_{\nu}u^{s}(x_{1},h_{1})=\hat{g}_{1}(x_{1})=\sum_{n\in\mathbb{Z}}\hat{g}_{1}^{(n)}e^{{\rm
i}\alpha_{n}x_{1}},$

where $\hat{f}_{1}^{(n)},\hat{g}_{1}^{(n)}\in\mathbb{C}$ are the Fourier
coefficients.

On the other hand, evaluating the scattered field $u^{s}$, as defined in (4),
and its normal derivative $\partial_{x_{2}}u^{s}$ on $\Gamma_{1}$, we obtain

$\left\\{\begin{aligned}
u^{s}(x_{1},h_{1})&=\sum\limits_{n\in\mathbb{Z}}H_{1}^{(n)}e^{{\rm
i}\alpha_{n}x_{1}}+\sum\limits_{n\in\mathbb{Z}}U_{1}^{(n)}e^{{\rm
i}\alpha_{n}x_{1}},\\\
\partial_{x_{2}}u^{s}(x_{1},h_{1})&=\sum\limits_{n\in\mathbb{Z}}{\rm
i}\beta_{n}H_{1}^{(n)}e^{{\rm
i}\alpha_{n}x_{1}}-\sum\limits_{n\in\mathbb{Z}}\gamma_{n}U_{1}^{(n)}e^{{\rm
i}\alpha_{n}x_{1}}.\end{aligned}\right.$

Combining the above equations, we have from straightforward calculations that
the scattered field $u^{s}$ in domain $\Omega_{1}$ can be expressed as

$\displaystyle u^{s}(x_{1},x_{2})$
$\displaystyle=\sum\limits_{n\in\mathbb{Z}}\left(\frac{\gamma_{n}\hat{f}^{(n)}_{1}+\hat{g}^{(n)}_{1}}{\gamma_{n}+{\rm
i}\beta_{n}}\right)e^{{\rm i}\alpha_{n}x_{1}+{\rm i}\beta_{n}(x_{2}-h_{1})}$
$\displaystyle\quad+\sum\limits_{n\in\mathbb{Z}}\left(\frac{{\rm
i}\beta_{n}\hat{f}^{(n)}_{1}-\hat{g}^{(n)}_{1}}{\gamma_{n}+{\rm
i}\beta_{n}}\right)e^{{\rm i}\alpha_{n}x_{1}-\gamma_{n}(x_{2}-h_{1})}.$ (6)

On $\Gamma_{1}$, the surface differential operators $M$ and $N$ given in (3)
can be simplified to

$N_{1}u=-(2-\mu)\frac{\partial^{3}u}{\partial x_{1}^{2}\partial
x_{2}}-\frac{\partial^{3}u}{\partial x_{2}^{3}},\quad
M_{1}u=\mu\frac{\partial^{2}u}{\partial
x_{1}^{2}}+\frac{\partial^{2}u}{\partial x_{2}^{2}}.$ (7)

Substituting (3) into (7) yields the TBC of the scattered field $u^{s}$ on
$\Gamma_{1}$:

$N_{1}u^{s}=T_{11}^{(1)}\hat{f}_{1}+T_{12}^{(1)}\hat{g}_{1},\quad
M_{1}u^{s}=T_{21}^{(1)}\hat{f}_{1}+T_{22}^{(1)}\hat{g}_{1}.$

Here, the Dirichlet-to-Neumann (DtN) operators $T_{ij}^{(1)},i,j=1,2$ are
given by

$\left\\{\begin{aligned}
(T_{11}^{(1)}f)(x_{1})&=\sum\limits_{n\in\mathbb{Z}}{\rm
i}\beta_{n}\gamma_{n}\left(\gamma_{n}-{\rm i}\beta_{n}\right)f^{(n)}e^{{\rm
i}\alpha_{n}x_{1}},\\\
(T_{21}^{(1)}f)(x_{1})&=-\sum\limits_{n\in\mathbb{Z}}\left(\mu\alpha_{n}^{2}-{\rm
i}\beta_{n}\gamma_{n}\right)f^{(n)}e^{{\rm i}\alpha_{n}x_{1}},\\\
(T_{12}^{(1)}g)(x_{1})&=-\sum\limits_{n\in\mathbb{Z}}\left(\mu\alpha_{n}^{2}-{\rm
i}\beta_{n}\gamma_{n}\right)g^{(n)}e^{{\rm i}\alpha_{n}x_{1}},\\\
(T_{22}^{(1)}g)(x_{1})&=-\sum\limits_{n\in\mathbb{Z}}\left(\gamma_{n}-{\rm
i}\beta_{n}\right)g^{(n)}e^{{\rm i}\alpha_{n}x_{1}},\end{aligned}\right.$ (8)

where $f^{(n)}$ and $g^{(n)}$ are the Fourier coefficients of $f$ and $g$,
respectively. Noting $u=u^{s}+u^{i}$, we deduce the TBC for the total field
$u$ on $\Gamma_{1}$:

$N_{1}u=T_{11}^{(1)}f_{1}+T_{12}^{(1)}g_{1}+p_{1},\quad
M_{1}u=T_{21}^{(1)}f_{1}+T_{22}^{(1)}g_{1}+p_{2}.$ (9)

where

$p_{1}(x_{1})=-\left(2{\rm i}\beta\alpha^{2}+2\beta^{2}\gamma\right)e^{{\rm
i}(\alpha x_{1}-\beta h_{1})},\quad p_{2}(x_{1})=-(2\beta^{2}+2{\rm
i}\beta\gamma)e^{{\rm i}(\alpha x_{1}-\beta h_{1})}.$ (10)

Given the similarity in the derivation process of the TBC to that of
$\Gamma_{2}$, we provide a brief overview of the procedure and present the
resulting TBC. In accordance with the bounded outgoing wave condition, the
total field $u$ exhibits the Fourier series expansion in $\Omega_{2}$:

$u(x_{1},x_{2})=\sum\limits_{n\in\mathbb{Z}}H_{2}^{(n)}e^{{\rm
i}\alpha_{n}x_{1}-{\rm
i}\beta_{n}(x_{2}-h_{2})}+\sum\limits_{n\in\mathbb{Z}}U_{2}^{(n)}e^{{\rm
i}\alpha_{n}x_{1}+\gamma_{n}(x_{2}-h_{2})}.$ (11)

Evaluating (11) and its normal derivative on $\Gamma_{2}$, we obtain

$\left\\{\begin{aligned}
u(x_{1},h_{2})&=\sum\limits_{n\in\mathbb{Z}}H_{2}^{(n)}e^{{\rm
i}\alpha_{n}x_{1}}+\sum\limits_{n\in\mathbb{Z}}U_{2}^{(n)}e^{{\rm
i}\alpha_{n}x_{1}}=\sum\limits_{n\in\mathbb{Z}}f^{(n)}_{2}e^{{\rm
i}\alpha_{n}x_{1}},\\\
\partial_{\nu}u(x_{1},h_{2})&=\sum\limits_{n\in\mathbb{Z}}{\rm
i}\beta_{n}H_{2}^{(n)}e^{{\rm
i}\alpha_{n}x_{1}}-\sum\limits_{n\in\mathbb{Z}}\gamma_{n}U_{2}^{(n)}e^{{\rm
i}\alpha_{n}x_{1}}=\sum\limits_{n\in\mathbb{Z}}g^{(n)}_{2}e^{{\rm
i}\alpha_{n}x_{1}},\end{aligned}\right.$ (12)

where $(u,\partial_{\nu}u)=(f_{2},g_{2})$ are the Dirichlet data on
$\Gamma_{2}$ and have the Fourier series expansions

$f_{2}(x_{1})=\sum\limits_{n\in\mathbb{Z}}f^{(n)}_{2}e^{{\rm
i}\alpha_{n}x_{1}},\quad
g_{2}(x_{1})=\sum\limits_{n\in\mathbb{Z}}g^{(n)}_{2}e^{{\rm
i}\alpha_{n}x_{1}}.$

By solving the system (12), we deduce that the total field $u$ in $\Omega_{2}$
admits the Fourier series expansion

$\displaystyle u(x_{1},x_{2})$
$\displaystyle=\sum\limits_{n\in\mathbb{Z}}\left(\frac{\gamma_{n}f^{(n)}_{2}+g^{(n)}_{2}}{\gamma_{n}+{\rm
i}\beta_{n}}\right)e^{{\rm i}\alpha_{n}x_{1}-{\rm i}\beta_{n}(x_{2}-h_{2})}$
$\displaystyle\quad+\sum\limits_{n\in\mathbb{Z}}\left(\frac{{\rm
i}\beta_{n}f^{(n)}_{2}-g^{(n)}_{2}}{\gamma_{n}+{\rm i}\beta_{n}}\right)e^{{\rm
i}\alpha_{n}x_{1}+\gamma_{n}(x_{2}-h_{2})}.$ (13)

Noting that the surface differential operator $M$ and $N$ on $\Gamma_{2}$ can
be simplified to

$N_{2}u=(2-\mu)\frac{\partial^{3}u}{\partial x_{1}^{2}\partial
x_{2}}+\frac{\partial^{3}u}{\partial x_{2}^{3}},\quad
M_{2}u=\mu\frac{\partial^{2}u}{\partial
x_{1}^{2}}+\frac{\partial^{2}u}{\partial x_{2}^{2}},$ (14)

we substitute (3) into (14) and obtain the TBC of the total field $u$ on
$\Gamma_{2}$:

$N_{2}u=T_{11}^{(2)}f_{2}+T_{12}^{(2)}g_{2},\quad
M_{2}u=T_{21}^{(2)}f_{2}+T_{22}^{(2)}g_{2},$ (15)

where the DTN operators $T_{ij}^{(2)},i,j=1,2$ are defined as

$\left\\{\begin{aligned}
(T_{11}^{(2)}f)(x_{1})&=\sum\limits_{n\in\mathbb{Z}}{\rm
i}\beta_{n}\gamma_{n}\left(\gamma_{n}-{\rm
i}\beta_{n}\right)f^{(n)}_{2}e^{{\rm i}\alpha_{n}x_{1}},\\\
(T_{21}^{(2)}f)(x_{1})&=-\sum\limits_{n\in\mathbb{Z}}\left(\mu\alpha_{n}^{2}-{\rm
i}\beta_{n}\gamma_{n}\right)f^{(n)}_{2}e^{{\rm i}\alpha_{n}x_{1}},\\\
(T_{12}^{(2)}g)(x_{1})&=-\sum\limits_{n\in\mathbb{Z}}\left(\mu\alpha_{n}^{2}-{\rm
i}\beta_{n}\gamma_{n}\right)g^{(n)}_{2}e^{{\rm i}\alpha_{n}x_{1}},\\\
(T_{22}^{(2)}g)(x_{1})&=-\sum\limits_{n\in\mathbb{Z}}\left(\gamma_{n}-{\rm
i}\beta_{n}\right)g^{(n)}_{2}e^{{\rm i}\alpha_{n}x_{1}}.\end{aligned}\right.$

The following result concerns the properties of the DtN operators
$T^{(1)}_{ij}$ and $T^{(2)}_{ij}$, where $i,j=1,2.$

######  Lemma 1.

For $k=1,2$, the DtN operators $T^{(k)}_{11}:H^{3/2}(\Gamma_{k})\to
H^{-3/2}(\Gamma_{k})$, $T^{(k)}_{12}:H^{1/2}(\Gamma_{k})\to
H^{-3/2}(\Gamma_{k})$, $T^{(k)}_{21}:H^{3/2}(\Gamma_{k})\to
H^{-1/2}(\Gamma_{k})$, and $T^{(k)}_{22}:H^{1/2}(\Gamma_{k})\to
H^{-1/2}(\Gamma_{k})$ are bounded.

###### Proof.

We only prove the results for the operators $T_{ij}^{(1)}$, as the
corresponding properties for the operators $T_{ij}^{(2)}$ can be obtained in
the same manner. It is clear to note from (5) that

$\lim\limits_{|n|\rightarrow\infty}\frac{|\beta_{n}|}{|\alpha_{n}|}=1,\quad\lim\limits_{|n|\rightarrow\infty}\frac{|\gamma_{n}|}{|\alpha_{n}|}=1.$
(16)

For a given $f\in H^{3/2}(\Gamma_{1})$, we have from (8) and (16) that

$\displaystyle\|T^{(1)}_{11}f\|^{2}_{H^{-3/2}(\Gamma_{1})}$
$\displaystyle=\Lambda\sum\limits_{n\in\mathbb{Z}}\left(1+\alpha_{n}^{2}\right)^{-3/2}\big{|}{\rm
i}\beta_{n}\gamma_{n}\left(\gamma_{n}-{\rm
i}\beta_{n}\right)f^{(n)}\big{|}^{2}$
$\displaystyle\lesssim\sum\limits_{n\in\mathbb{Z}}(1+\alpha_{n}^{2})^{3/2}|f^{(n)}|^{2}\lesssim\|f\|^{2}_{H^{3/2}(\Gamma_{1})},$

and

$\displaystyle\|T^{(1)}_{21}f\|^{2}_{H^{-1/2}(\Gamma_{1})}$
$\displaystyle=\Lambda\sum\limits_{n\in\mathbb{Z}}\left(1+\alpha_{n}^{2}\right)^{-1/2}\big{|}\left(\mu\alpha_{n}^{2}-{\rm
i}\beta_{n}\gamma_{n}\right)f^{(n)}\big{|}^{2}$
$\displaystyle\lesssim\sum\limits_{n\in\mathbb{Z}}(1+\alpha_{n}^{2})^{3/2}|f^{(n)}|^{2}\lesssim\|f\|^{2}_{H^{3/2}(\Gamma_{1})}.$

Similarly, for any function $f\in H^{1/2}(\Gamma_{1})$, we deduce from (8) and
(16) that

$\displaystyle\|T^{(1)}_{12}f\|^{2}_{H^{-3/2}(\Gamma_{1})}$
$\displaystyle=\Lambda\sum\limits_{n\in\mathbb{Z}}\left(1+\alpha_{n}^{2}\right)^{-3/2}\big{|}\left(\mu\alpha_{n}^{2}-{\rm
i}\beta_{n}\gamma_{n}\right)g^{(n)}\big{|}^{2}$
$\displaystyle\lesssim\sum\limits_{n\in\mathbb{Z}}(1+\alpha_{n}^{2})^{1/2}|f^{(n)}|^{2}\lesssim\|f\|^{2}_{H^{1/2}(\Gamma_{1})},$

and

$\displaystyle\|T^{(1)}_{22}f\|^{2}_{H^{-1/2}(\Gamma_{1})}$
$\displaystyle=\Lambda\sum\limits_{n\in\mathbb{Z}}\left(1+\alpha_{n}^{2}\right)^{-1/2}\big{|}\left(\gamma_{n}-{\rm
i}\beta_{n}\right)f^{(n)}\big{|}^{2}$
$\displaystyle\lesssim\sum\limits_{n\in\mathbb{Z}}(1+\alpha_{n}^{2})^{1/2}|f^{(n)}|^{2}\lesssim\|f\|^{2}_{H^{1/2}(\Gamma_{1})},$

thus completing the proof. ∎

######  Lemma 2.

If $|n|$ is sufficiently large, then the following inequality holds for any
complex values $f$ and $g$:

$\displaystyle\Re\big{\\{}-{\rm i}\beta_{n}\gamma_{n}\left(\gamma_{n}-{\rm
i}\beta_{n}\right)|f|^{2}+\left(\mu\alpha_{n}^{2}-{\rm
i}\beta_{n}\gamma_{n}\right)g\bar{f}$
$\displaystyle+\left(\mu\alpha_{n}^{2}-{\rm
i}\beta_{n}\gamma_{n}\right)f\bar{g}+\left(\gamma_{n}-{\rm
i}\beta_{n}\right)|g|^{2}\big{\\}}\geq 0.$

###### Proof.

By the definitions of $\gamma_{n}$ and $\beta_{n}$ given in (5), a
straightforward calculation shows that for sufficiently large $|n|$

$-{\rm i}\beta_{n}\gamma_{n}\left(\gamma_{n}-{\rm
i}\beta_{n}\right)>0,\quad\mu\alpha_{n}^{2}-{\rm
i}\beta_{n}\gamma_{n}>0,\quad\gamma_{n}-{\rm i}\beta_{n}>0.$

It suffices to demonstrate for sufficiently large $|n|$ that

$-{\rm i}\beta_{n}\gamma_{n}\left(\gamma_{n}-{\rm
i}\beta_{n}\right)|f|^{2}-2\left(\mu\alpha_{n}^{2}-{\rm
i}\beta_{n}\gamma_{n}\right)|f||g|+\left(\gamma_{n}-{\rm
i}\beta_{n}\right)|g|^{2}\geq 0.$

Utilizing the Cauchy inequality, we deduce from a simple calculation that

$\displaystyle\big{(}(-{\rm i}\beta_{n}\gamma_{n})^{1/2}\left(\gamma_{n}-{\rm
i}\beta_{n}\right)\big{)}^{2}-\big{(}\mu\alpha_{n}^{2}-{\rm
i}\beta_{n}\gamma_{n}\big{)}^{2}$
$\displaystyle=-\gamma_{n}^{2}\beta_{n}^{2}-\mu\alpha_{n}^{4}-2{\rm
i}(1-\mu)\gamma_{n}\beta_{n}\alpha_{n}^{2}$
$\displaystyle=\alpha_{n}^{4}-\kappa^{4}-\mu\alpha_{n}^{4}-2{\rm
i}(1-\mu)\gamma_{n}\beta_{n}\alpha_{n}^{2}$
$\displaystyle=(1-\mu)\left(\alpha_{n}^{4}-2{\rm
i}\gamma_{n}\beta_{n}\alpha_{n}^{2}\right)-\kappa^{4},$

which is positive for sufficiently large $|n|$ by noting that $\mu<1$ and
$\alpha_{n}^{4}-2{\rm i}\gamma_{n}\beta_{n}\alpha_{n}^{2}\to\infty$ as
$|n|\to\infty$. ∎

######  Lemma 3.

For any $f\in H^{3/2}(\Gamma_{k})$ and $g\in H^{1/2}(\Gamma_{k})$ with
$k=1,2$, there exists a positive constant $C$ such that

$\displaystyle-\Re\left\\{\int_{\Gamma_{k}}\Big{(}T_{11}^{(k)}|f|^{2}+T_{12}^{(k)}g\bar{f}+T_{21}^{(k)}f\bar{g}+T_{22}^{(k)}|g|^{2}\Big{)}{\rm
d}s\right\\}$
$\displaystyle\geq-C\Big{(}\|f\|^{2}_{L^{2}(\Gamma_{k})}+\|g\|^{2}_{L^{2}(\Gamma_{k})}\Big{)}.$

###### Proof.

We prove the result for $k=1$, as the result for $k=2$ can be obtained
similarly. By the definitions of $T^{(1)}_{ij}$ and Lemma 2, we have

$\displaystyle-\Re\int_{\Gamma_{1}}\left(T_{11}^{(1)}|f|^{2}+T_{12}^{(1)}g\bar{f}+T_{21}^{(1)}f\bar{g}+T_{22}^{(1)}|g|^{2}\right){\rm
d}s$ $\displaystyle=\Lambda\sum\limits_{|n|\leq n_{0}}\Re\Big{\\{}-{\rm
i}\beta_{n}\gamma_{n}\left(\gamma_{n}-{\rm
i}\beta_{n}\right)|f^{(n)}|^{2}+\left(\mu\alpha_{n}^{2}-{\rm
i}\beta_{n}\gamma_{n}\right)g^{(n)}\overline{f^{(n)}}$
$\displaystyle\quad+\left(\mu\alpha_{n}^{2}-{\rm
i}\beta_{n}\gamma_{n}\right)f^{(n)}\overline{g^{(n)}}+\left(\gamma_{n}-{\rm
i}\beta_{n}\right)|g^{(n)}|^{2}\Big{\\}}$
$\displaystyle\quad+\Lambda\sum\limits_{|n|>n_{0}}\Re\Big{\\{}-{\rm
i}\beta_{n}\gamma_{n}\left(\gamma_{n}-{\rm i}\beta_{n}\right)|f^{(n)}|^{2}$
$\displaystyle\quad+\left(\mu\alpha_{n}^{2}-{\rm
i}\beta_{n}\gamma_{n}\right)g^{(n)}\overline{f^{(n)}}+\left(\mu\alpha_{n}^{2}-{\rm
i}\beta_{n}\gamma_{n}\right)f^{(n)}\overline{g^{(n)}}+\left(\gamma_{n}-{\rm
i}\beta_{n}\right)|g^{(n)}|^{2}\Big{\\}}$
$\displaystyle\geq\Lambda\sum\limits_{|n|\leq n_{0}}\Re\Big{\\{}-{\rm
i}\beta_{n}\gamma_{n}\left(\gamma_{n}-{\rm
i}\beta_{n}\right)|f^{(n)}|^{2}+\left(\mu\alpha_{n}^{2}-{\rm
i}\beta_{n}\gamma_{n}\right)g^{(n)}\overline{f^{(n)}}$
$\displaystyle\quad+\left(\mu\alpha_{n}^{2}-{\rm
i}\beta_{n}\gamma_{n}\right)f^{(n)}\overline{g^{(n)}}+\left(\gamma_{n}-{\rm
i}\beta_{n}\right)|g^{(n)}|^{2}\Big{\\}}$
$\displaystyle\geq-C\left(\|f\|^{2}_{L^{2}(\Gamma_{1})}+\|g\|^{2}_{L^{2}(\Gamma_{1})}\right),$

where $n_{0}$ is the smallest integer such that Lemma 2 holds. ∎

Utilizing the TBCs given by (9) and (15), we transform the original problem
(1)–(2) from an unbounded domain into the bounded domain $\Omega$, which is to
find a quasi-periodic function $u$ satisfying

$\left\\{\begin{aligned} &\Delta^{2}u-\kappa^{4}u=0&&\text{in}~{}\Omega,\\\
&u=0,\quad\partial_{\nu}u=0&&\text{on}~{}\Gamma_{c},\\\
&N_{1}u=T_{11}^{(1)}f_{1}+T_{12}^{(1)}g_{1}+p_{1}&&\text{on}~{}\Gamma_{1},\\\
&M_{1}u=T_{21}^{(1)}f_{1}+T_{22}^{(1)}g_{1}+p_{2}&&\text{on}~{}\Gamma_{1},\\\
&N_{2}u=T_{11}^{(2)}f_{2}+T_{12}^{(2)}g_{2}&&\text{on}~{}\Gamma_{2},\\\
&M_{2}u=T_{21}^{(2)}f_{2}+T_{22}^{(2)}g_{2}&&\text{on}~{}\Gamma_{2},\end{aligned}\right.$
(17)

where $p_{1}$ and $p_{2}$ are defined in (10), and $(f_{k},g_{k})$ are the
Dirichlet data of the total field $u$ on $\Gamma_{k}$ for $k=1,2.$ The
objective of this study is to examine the PML formulation applied to the
boundary value problem (17) and to establish the convergence of the PML
solution.

##  4 The variational problem

In this section, we present a variational formulation for the boundary value
problem (17) and examine its well-posedness.

Observing that the bi-Laplacian can be expressed in terms of the Poisson
ratio, as demonstrated in [34], we have

$\displaystyle\Delta^{2}u-\kappa^{4}u$
$\displaystyle=\frac{\partial^{2}}{\partial
x_{1}^{2}}\left(\frac{\partial^{2}u}{\partial
x_{1}^{2}}+\mu\frac{\partial^{2}u}{\partial
x_{2}^{2}}\right)+2(1-\mu)\frac{\partial^{2}}{\partial x_{1}\partial
x_{2}}\left(\frac{\partial^{2}u}{\partial x_{1}\partial x_{2}}\right)$
$\displaystyle\quad+\frac{\partial^{2}}{\partial
x_{2}^{2}}\left(\frac{\partial^{2}u}{\partial
x_{2}^{2}}+\mu\frac{\partial^{2}u}{\partial x_{1}^{2}}\right)-\kappa^{4}u.$

Multipling both sides of the above equation with a test function $v\in H_{{\rm
qp},\Gamma_{c}}^{2}(\Omega)$, integrating across the domain $\Omega$, and
applying integration by parts, we obtain

$\displaystyle\int_{\Omega}\bigg{[}\mu\Delta
u\Delta\bar{v}+(1-\mu)\sum\limits_{i,j=1}^{2}\frac{\partial^{2}u}{\partial
x_{i}\partial x_{j}}\frac{\partial^{2}\bar{v}}{\partial x_{i}\partial
x_{j}}-\kappa^{4}u\bar{v}\bigg{]}{\rm d}x$
$\displaystyle-\sum_{k=1}^{2}\int_{\Gamma_{k}}\left(\bar{v}N_{k}u+\partial_{\nu}\bar{v}M_{k}u\right){\rm
d}s=0.$ (18)

Substituting the TBCs on $\Gamma_{k},k=1,2$ into (4), we arrive at the
variational problem: find $u\in H_{{\rm qp},\Gamma_{c}}^{2}(\Omega)$ such that

$a(u,v)=\int_{\Gamma_{1}}\left(p_{1}\bar{v}+p_{2}\partial_{\nu}\bar{v}\right){\rm
d}s\quad\forall\,v\in H_{{\rm qp},\Gamma_{c}}^{2}(\Omega),$ (19)

where the sesquilinear form $a(u,v):H_{{\rm qp},\Gamma_{c}}^{2}(\Omega)\times
H_{{\rm qp},\Gamma_{c}}^{2}(\Omega)\rightarrow\mathbb{C}$ is defined as

$\displaystyle a(u,v)=\int_{\Omega}\bigg{[}\mu\Delta
u\Delta\bar{v}+(1-\mu)\sum\limits_{i,j=1}^{2}\frac{\partial^{2}u}{\partial
x_{i}\partial x_{j}}\frac{\partial^{2}\bar{v}}{\partial x_{i}\partial
x_{j}}-\kappa^{4}u\bar{v}\bigg{]}{\rm d}x$
$\displaystyle-\sum_{k=1}^{2}\int_{\Gamma_{k}}(\mathbb{T}^{(k)}\boldsymbol{u})\cdot\overline{\boldsymbol{v}}{\rm
d}s$ (20)

with $\boldsymbol{u},\boldsymbol{v}$, and $\mathbb{T}^{(k)}$ given by

$\boldsymbol{u}=\begin{bmatrix}u\\\
\partial_{\nu}u\end{bmatrix},\quad\boldsymbol{v}=\begin{bmatrix}v\\\
\partial_{\nu}v\end{bmatrix},\quad\mathbb{T}^{(k)}=\begin{bmatrix}T_{11}^{(k)}&T_{12}^{(k)}\\\\[5.0pt]
T_{21}^{(k)}&T_{22}^{(k)}\end{bmatrix}.$

The following trace theorem can be found in [35, Theorem 1.1.6].

######  Lemma 4.

Let $\Omega$ be a Lipschitz domain. Then, there is a positive constant $C$ for
which

$\|u\|_{L^{2}(\partial\Omega)}\leq
C\|u\|_{L^{2}(\Omega)}^{1/2}\|u\|_{H^{1}(\Omega)}^{1/2}\quad\forall\,u\in
H^{1}(\Omega).$

######  Theorem 1.

The variational problem (19) has a unique weak solution $u\in H^{2}_{{\rm
qp},\Gamma_{c}}(\Omega)$ except for a discrete set of wavenumbers $\kappa$.

###### Proof.

It follows from Lemma 1 and the trace theorem (cf. [36, Lemmas 2.2 and 2.3])
that the continuity of the sesquilinear form (4) is evident. It can be shown
from [18] that there exist positive constants $c_{1}$ and $c_{2}$ such that

$\int_{\Omega}\bigg{[}\mu|\Delta
u|^{2}+(1-\mu)\sum\limits_{i,j=1}^{2}\bigg{|}\frac{\partial^{2}u}{\partial
x_{i}\partial x_{j}}\bigg{|}^{2}-\kappa^{4}|u|^{2}\bigg{]}{\rm d}x\geq
c_{1}\|u\|_{H^{2}(\Omega)}^{2}-c_{2}\|u\|_{L^{2}(\Omega)}^{2}.$ (21)

By combining Lemmas 1, 3, and 4 with the Cauchy inequality, we deduce

$\displaystyle-\Re\sum_{k=1}^{2}\int_{\Gamma_{k}}(\mathbb{T}^{(k)}\boldsymbol{u})\cdot\overline{\boldsymbol{u}}{\rm
d}s$ $\displaystyle\geq-
c_{3}\big{(}\|u\|^{2}_{L^{2}(\Gamma_{1}\cup\Gamma_{2})}+\|\partial_{x_{2}}u\|^{2}_{L^{2}(\Gamma_{1}\cup\Gamma_{2})}\big{)}$
$\displaystyle\geq-
c_{4}\big{(}\|u\|_{L^{2}(\Omega)}\|u\|_{H^{1}(\Omega)}+\|\nabla
u\|_{L^{2}(\Omega)}\|\nabla u\|_{H^{1}(\Omega)}\big{)}$ $\displaystyle\geq-
c_{5}\|u\|^{2}_{H^{1}(\Omega)}-c_{6}\|u\|^{2}_{H^{2}(\Omega)},$

where $c_{6}>0$ is sufficiently small. Combining the above inequality with
(21), we verify that the sesquilinear form (4) satisfies the Gårding
inequality

$\Re a(u,u)\geq c_{7}\|u\|_{H^{2}(\Omega)}-c_{8}\|u\|_{H^{1}(\Omega)},$

which completes the proof by applying the Fredholm alternative theorem. ∎

##  5 The PML problem

This section focuses on the PML problem, aiming to establish its well-
posedness while also providing a convergence analysis of the PML solution.

###  5.1 The PML formulation

Denote by $\Omega_{k}^{\rm PML}$, for $k=1,2$, the PML regions above and below
the interfaces $\Gamma_{k}$, respectively. Assume that the thickness of each
region is $\delta_{k}$ and denote the outer boundary of $\Omega_{k}^{\rm PML}$
by $\Gamma_{k}^{\rm PML}$. Let $\Omega^{\rm PML}=\Omega\cup\Omega_{1}^{\rm
PML}\cup\Omega_{2}^{\rm PML}$ be the domain in which the PML problem is
formulated. The schematic of the PML problem is depicted in Figure 2.

Figure 2: The schematic of the PML problem.

The PML parameters in $\Omega_{k}^{\rm PML}$ are introduced by the complex
coordinate stretching (cf. [32]):

$\tilde{x}_{2}=\varphi(x_{2})=\int_{0}^{x_{2}}s(t){\rm d}t,\quad
s(t)=\left\\{\begin{aligned} &1&&\text{if}~{}h_{2}\leq t\leq h_{1},\\\ &1+{\rm
i}\sigma_{0}\left(\frac{t-h_{1}}{\delta_{1}}\right)^{m}&&\text{if}~{}t>h_{1},\\\
&1+{\rm
i}\sigma_{0}\left(\frac{h_{2}-t}{\delta_{2}}\right)^{m}&&\text{if}~{}t<h_{2},\end{aligned}\right.$

where $\sigma_{0}$ is a positive constant, and $m\geq 2$ is an integer.

Let $\tilde{u}$ be the solution of the total field for the PML problem in the
complex coordinates $\tilde{x}=(x_{1},\tilde{x}_{2})$. It satisfies

$\left\\{\begin{aligned}
&\tilde{\Delta}^{2}\tilde{u}-\kappa^{4}\tilde{u}=w&&{\rm in}~{}\Omega^{\rm
PML},\\\
&\tilde{u}=u^{i},\quad\partial_{\nu}\tilde{u}=\partial_{\nu}u^{i}&&{\rm
on}~{}\Gamma_{1}^{\rm PML},\\\
&\tilde{u}=0,\quad\partial_{\nu}\tilde{u}=0&&{\rm on}~{}\Gamma_{2}^{\rm
PML},\\\ \end{aligned}\right.$ (22)

where $\tilde{\Delta}$ is the Laplacian operator in the complex coordinates
and is given by

$\tilde{\Delta}\tilde{u}=\frac{\partial^{2}\tilde{u}}{\partial
x_{1}^{2}}+\frac{\partial^{2}\tilde{u}}{\partial\tilde{x}_{2}^{2}},$

and

$w(\tilde{x})=\left\\{\begin{aligned}
&\tilde{\Delta}^{2}u^{i}-\kappa^{4}u^{i}&&\text{in}~{}\Omega_{1}^{\rm PML},\\\
&0&&\text{in}~{}\Omega\setminus\overline{\Omega_{1}^{\rm
PML}}.\end{aligned}\right.$

Here, the complex coordinate variable $\tilde{x}=(x_{1},\tilde{x}_{2})$ is
considered to be in $\Omega^{\rm PML}_{k}$ or on $\Gamma_{k}^{\rm PML}$ if
$x=(x_{1},\psi(\tilde{x}_{2}))$ is in $\Omega^{\rm PML}_{k}$ or on
$\Gamma_{k}^{\rm PML}$, where $\psi$ is the inverse function of $\varphi$,
i.e., $x_{2}=\psi(\tilde{x}_{2})=\varphi^{-1}(\tilde{x}_{2})$.

###  5.2 TBC for the PML problem

To equivalently formulate the PML problem from the domain $\Omega^{\rm PML}$
to the domain $\Omega$, we investigate the TBCs of the PML problem (22) on the
interfaces $\Gamma_{k}$.

First, we deduce the TBC on $\Gamma_{1}$. Consider the scattered field

$\tilde{u}^{s}(x_{1},\varphi(x_{2}))=\tilde{u}(x_{1},\varphi(x_{2}))-u^{i}(x_{1},x_{2}).$

It follows from (22) that the scattered field $\tilde{u}^{s}$ satisfies

$\left\\{\begin{aligned}
&\tilde{\Delta}\tilde{u}^{s}-\kappa^{4}\tilde{u}^{s}=0&&{\rm
in}~{}\Omega_{1}^{\rm PML},\\\
&\tilde{u}^{s}=\hat{f}_{1},\quad\partial_{\nu}\tilde{u}^{s}=\hat{g}_{1}&&{\rm
on}~{}\Gamma_{1},\\\ &\tilde{u}^{s}=0,\quad\partial_{\nu}\tilde{u}^{s}=0&&{\rm
on}~{}\Gamma_{1}^{\rm PML},\\\ \end{aligned}\right.$ (23)

where $(\tilde{u}^{s},\partial_{\nu}\tilde{u}^{s})=(\hat{f}_{1},\hat{g}_{1})$
are the Dirichlet data for the scattered field $\tilde{u}^{s}$ on
$\Gamma_{1}$. As $\hat{f}_{1}$ and $\hat{g}_{1}$ are quasi-periodic functions
of $x_{1}$ with a period $\Lambda$, they have the Fourier series expansions

$\hat{f}_{1}(x_{1})=\sum\limits_{n\in\mathbb{Z}}\hat{f}_{1}^{(n)}e^{{\rm
i}\alpha_{n}x_{1}},\quad\hat{g}_{1}(x_{1})=\sum\limits_{n\in\mathbb{Z}}\hat{g}_{1}^{(n)}e^{{\rm
i}\alpha_{n}x_{1}}.$

Since the scattered field $\tilde{u}^{s}$ satisfies
$\tilde{\Delta}\tilde{u}^{s}-\kappa^{4}\tilde{u}^{s}=0$ in $\Omega_{1}^{\rm
PML}$, we can verify that $\tilde{u}^{s}$ admits the following analytical
expression in $\Omega_{1}^{\rm PML}$:

$\displaystyle\tilde{u}^{s}(x_{1},\tilde{x}_{2})$
$\displaystyle=\sum\limits_{n\in\mathbb{Z}}\Big{[}W_{1}^{(n)}e^{-{\rm
i}\beta_{n}(\tilde{x}_{2}-h_{1})}+V_{1}^{(n)}e^{{\rm
i}\beta_{n}(\tilde{x}_{2}-h_{1})}$
$\displaystyle\quad+X_{1}^{(n)}e^{\gamma_{n}(\tilde{x}_{2}-h_{1})}+Y_{1}^{(n)}e^{-\gamma_{n}(\tilde{x}_{2}-h_{1})}\Big{]}e^{{\rm
i}\alpha_{n}x_{1}}.$ (24)

Substituting (24) into the boundary conditions in (23), we obtain a linear
system of algebraic equations for the Fourier coefficients
$W_{1}^{(n)},V_{1}^{(n)},X_{1}^{(n)}$, and $Y_{1}^{(n)}$:

$\left\\{\begin{aligned}
&W_{1}^{(n)}+V_{1}^{(n)}+X_{1}^{(n)}+Y_{1}^{(n)}=\hat{f}_{1}^{(n)},\\\ &-{\rm
i}\beta_{n}W_{1}^{(n)}+{\rm
i}\beta_{n}V_{1}^{(n)}+\gamma_{n}X_{1}^{(n)}-\gamma_{n}Y_{1}^{(n)}=\hat{g}_{1}^{(n)},\\\
&W_{1}^{(n)}e^{-{\rm i}\beta_{n}\tilde{h}_{1}}+V_{1}^{(n)}e^{{\rm
i}\beta_{n}\tilde{h}_{1}}+X_{1}^{(n)}e^{\gamma_{n}\tilde{h}_{1}}+Y_{1}^{(n)}e^{-\gamma_{n}\tilde{h}_{1}}=0,\\\
&-{\rm i}\beta_{n}W_{1}^{(n)}e^{-{\rm i}\beta_{n}\tilde{h}_{1}}+{\rm
i}\beta_{n}V_{1}^{(n)}e^{{\rm
i}\beta_{n}\tilde{h}_{1}}+\gamma_{n}X_{1}^{(n)}e^{\gamma_{n}\tilde{h}_{1}}-\gamma_{n}Y_{1}^{(n)}e^{-\gamma_{n}\tilde{h}_{1}}=0,\end{aligned}\right.$
(25)

where

$\tilde{h}_{1}=\varphi(h_{1}+\delta_{1})-h_{1}=\int_{0}^{h_{1}+\delta_{1}}s(t){\rm
d}t-h_{1}=\delta_{1}\Big{(}1+\frac{{\rm i}\sigma_{0}}{m+1}\Big{)}.$ (26)

Through tedious yet straightforward calculations, we solve the linear system
(25) and obtain the solutions for the Fourier coefficients:

$\displaystyle W_{1}^{(n)}$
$\displaystyle=\frac{1}{D^{(n)}_{1}}\bigg{\\{}\hat{f}_{1}^{(n)}\Big{[}\left(\gamma_{n}\beta_{n}-{\rm
i}\gamma_{n}^{2}\right)e^{2{\rm
i}\beta_{n}\tilde{h}_{1}}-2\beta_{n}\gamma_{n}e^{({\rm
i}\beta_{n}+\gamma_{n})\tilde{h}_{1}}$
$\displaystyle\quad+\left(\gamma_{n}\beta_{n}+{\rm
i}\gamma_{n}^{2}\right)e^{2({\rm
i}\beta_{n}+\gamma_{n})\tilde{h}_{1}}\Big{]}-\hat{g}_{1}^{(n)}\Big{[}\left(\beta_{n}-{\rm
i}\gamma_{n}\right)e^{2{\rm i}\beta_{n}\tilde{h}_{1}}$
$\displaystyle\quad+2{\rm i}\gamma_{n}e^{({\rm
i}\beta_{n}+\gamma_{n})\tilde{h}_{1}}-\left({\rm
i}\gamma_{n}+\beta_{n}\right)e^{2({\rm
i}\beta_{n}+\gamma_{n})\tilde{h}_{1}}\Big{]}\bigg{\\}},$ $\displaystyle
V_{1}^{(n)}$
$\displaystyle=\frac{1}{D^{(n)}_{1}}\bigg{\\{}\hat{f}_{1}^{(n)}\left[\left(\beta_{n}\gamma_{n}+{\rm
i}\gamma_{n}^{2}\right)+\left(\gamma_{n}\beta_{n}-{\rm
i}\gamma_{n}^{2}\right)e^{2\gamma_{n}\tilde{h}_{1}}-2\beta_{n}\gamma_{n}e^{({\rm
i}\beta_{n}+\gamma_{n})\tilde{h}_{1}}\right]$
$\displaystyle\quad-\hat{g}_{1}^{(n)}\left[\left(\beta_{n}+{\rm
i}\gamma_{n}\right)-\left(\beta_{n}-{\rm
i}\gamma_{n}\right)e^{2\gamma_{n}\tilde{h}_{1}}-2{\rm i}\gamma_{n}e^{({\rm
i}\beta_{n}+\gamma_{n})\tilde{h}_{1}}\right]\bigg{\\}},$ $\displaystyle
X_{1}^{(n)}$
$\displaystyle=\frac{1}{D^{(n)}_{1}}\bigg{\\{}\hat{f}_{1}^{(n)}\left[(\beta_{n}\gamma_{n}-{\rm
i}\beta_{n}^{2})+\left({\rm i}\beta_{n}^{2}+\beta_{n}\gamma_{n}\right)e^{2{\rm
i}\beta_{n}\tilde{h}_{1}}-2\beta_{n}\gamma_{n}e^{({\rm
i}\beta_{n}+\gamma_{n})\tilde{h}_{1}}\right]$
$\displaystyle\quad-\hat{g}_{1}^{(n)}\left[-\left(\beta_{n}+{\rm
i}\gamma_{n}\right)-\left(\beta_{n}-{\rm i}\gamma_{n}\right)e^{2{\rm
i}\beta_{n}\tilde{h}_{1}}+2\beta_{n}e^{({\rm
i}\beta_{n}+\gamma_{n})\tilde{h}_{1}}\right]\bigg{\\}},$ $\displaystyle
Y_{1}^{(n)}$
$\displaystyle=\frac{1}{D^{(n)}_{1}}\bigg{\\{}\hat{f}_{1}^{(n)}\Big{[}\left({\rm
i}\beta_{n}^{2}+\beta_{n}\gamma_{n}\right)e^{2\gamma_{n}\tilde{h}_{1}}-2\beta_{n}\gamma_{n}e^{({\rm
i}\beta_{n}+\gamma_{n})\tilde{h}_{1}}$
$\displaystyle\quad+\left(\beta_{n}\gamma_{n}-{\rm
i}\beta_{n}^{2}\right)e^{2({\rm
i}\beta_{n}+\gamma_{n})\tilde{h}_{1}}\Big{]}-\hat{g}_{1}^{(n)}\Big{[}\left(\beta_{n}-{\rm
i}\gamma_{n}\right)e^{2\gamma_{n}\tilde{h}_{1}}$
$\displaystyle\quad-2\beta_{n}e^{({\rm
i}\beta_{n}+\gamma_{n})\tilde{h}_{1}}+\left({\rm
i}\gamma_{n}+\beta_{n}\right)e^{2({\rm
i}\beta_{n}+\gamma_{n})\tilde{h}_{1}}\Big{]}\bigg{\\}},$

where the denominator $D^{(n)}_{1}$ is defined as

$\displaystyle D^{(n)}_{1}$ $\displaystyle=-8\beta_{n}\gamma_{n}e^{({\rm
i}\beta_{n}+\gamma_{n})\tilde{h}_{1}}+\left(-{\rm i}\beta_{n}^{2}+{\rm
i}\gamma_{n}^{2}+2\beta_{n}\gamma_{n}\right)\big{(}1+e^{2({\rm
i}\beta_{n}+\gamma_{n})\tilde{h}_{1}}\big{)}$ $\displaystyle\quad+\left({\rm
i}\beta_{n}^{2}-{\rm
i}\gamma_{n}^{2}+2\beta_{n}\gamma_{n}\right)\big{(}e^{2{\rm
i}\beta_{n}\tilde{h}_{1}}+e^{2\gamma_{n}\tilde{h}_{1}}\big{)}.$ (27)

It is clear to note from (26) that both of the real and imaginary parts of
$\tilde{h}_{1}$ are positive. If the thickness of the layer $\delta_{1}$ is
sufficiently large, i.e., $\Re\tilde{h}_{1}=\delta_{1}$ is sufficiently large,
then the leading term in (5.2) is the one containing
$e^{2\gamma_{n}\tilde{h}_{1}}$. A simple calculations yields

$\left|{\rm i}\beta_{n}^{2}-{\rm
i}\gamma_{n}^{2}+2\beta_{n}\gamma_{n}\right|^{2}=|\gamma_{n}+{\rm
i}\beta_{n}|^{4}\neq 0\quad\forall\,n\in\mathbb{Z},$

which ensures that $D^{(n)}_{1}$ is non-zero for sufficiently large
$\delta_{1}$.

Substituting (24) into (7), we obtain the TBC of the scattered field
$\tilde{u}^{s}$ for the PML problem on $\Gamma_{1}$:

$N_{1}\tilde{u}^{s}=\hat{T}_{11}^{(1)}\hat{f}_{1}+\hat{T}_{12}^{(1)}\hat{g}_{1},\quad
M_{1}\tilde{u}^{s}=\hat{T}_{21}^{(1)}\hat{f}_{1}+\hat{T}_{22}^{(1)}\hat{g}_{1},$
(28)

where the DtN operators $\hat{T}_{ij}^{(1)},i,j=1,2$ are given by

$\displaystyle(\hat{T}^{(1)}_{11}\hat{f}_{1})(x_{1})$
$\displaystyle=-\sum\limits_{n\in\mathbb{Z}}\frac{e^{{\rm
i}\alpha_{n}x_{1}}}{D^{(n)}_{1}}\hat{f}_{1}^{(n)}\Big{\\{}-\left({\rm
i}\beta_{n}^{4}\gamma_{n}-\beta_{n}^{3}\gamma_{n}^{2}+{\rm
i}\beta_{n}^{2}\gamma_{n}^{3}-\beta_{n}\gamma_{n}^{4}\right)$
$\displaystyle\quad+\left({\rm
i}\beta_{n}^{4}\gamma_{n}+\beta_{n}^{3}\gamma_{n}^{2}+{\rm
i}\beta_{n}^{2}\gamma_{n}^{3}+\beta_{n}\gamma_{n}^{4}\right)e^{2{\rm
i}\beta_{n}\tilde{h}_{1}}$ $\displaystyle\quad-\left({\rm
i}\beta_{n}^{4}\gamma_{n}+\beta_{n}^{3}\gamma_{n}^{2}+{\rm
i}\beta_{n}^{2}\gamma_{n}^{3}\beta_{n}\gamma_{n}^{4}\right)e^{2\gamma_{n}\tilde{h}_{1}}$
$\displaystyle\quad+\left({\rm
i}\beta_{n}^{4}\gamma_{n}-\beta_{n}^{3}\gamma_{n}^{2}+{\rm
i}\beta_{n}^{2}\gamma_{n}^{3}-\beta_{n}\gamma_{n}^{4}\right)e^{2({\rm
i}\beta_{n}+\gamma_{n})\tilde{h}_{1}}\Big{\\}},$
$\displaystyle(\hat{T}^{(1)}_{12}\hat{g}_{1})(x_{1})$
$\displaystyle=-\sum\limits_{n\in\mathbb{Z}}\frac{e^{{\rm
i}\alpha_{n}x_{1}}}{D^{(n)}_{1}}\hat{g}_{1}^{(n)}\bigg{\\{}\Big{(}\frac{\rm
i}{2}\mu\beta_{n}^{4}+(1-\mu)\beta_{n}^{3}\gamma_{n}+(2-\mu){\rm
i}\beta_{n}^{2}\gamma_{n}^{2}$
$\displaystyle\quad-(1-\mu)\beta_{n}\gamma_{n}^{3}+\frac{\mu}{2}{\rm
i}\gamma_{n}^{4}\Big{)}+\Big{(}-\frac{\rm
i}{2}\mu\beta_{n}^{4}+(1-\mu)\beta_{n}^{3}\gamma_{n}-(2-\mu){\rm
i}\beta_{n}^{2}\gamma_{n}^{2}$
$\displaystyle\quad-(1-\mu)\beta_{n}\gamma_{n}^{3}-\frac{\rm
i}{2}\mu\gamma_{n}^{4}\Big{)}e^{2{\rm
i}\beta_{n}\tilde{h}_{1}}+\Big{(}-\frac{\rm
i}{2}\mu\beta_{n}^{4}+(1-\mu)\beta_{n}^{3}\gamma_{n}$
$\displaystyle\quad-(2-\mu){\rm
i}\beta_{n}^{2}\gamma_{n}^{2}-(1-\mu)\beta_{n}\gamma_{n}^{3}-\frac{\rm
i}{2}\mu\gamma_{n}^{4}\Big{)}e^{2\gamma_{n}\tilde{h}_{1}}$
$\displaystyle\quad+(4-4\mu)\left(\beta_{n}\gamma_{n}^{3}-\beta_{n}^{3}\gamma_{n}\right)e^{({\rm
i}\beta_{n}+\gamma_{n})\tilde{h}_{1}}+\Big{(}\frac{\rm
i}{2}\mu\beta_{n}^{4}+(1-\mu)\beta_{n}^{3}\gamma_{n}$
$\displaystyle\quad+(2-\mu){\rm
i}\beta_{n}^{2}\gamma_{n}^{2}-(1-\mu)\beta_{n}\gamma_{n}^{3}+\frac{\mu}{2}{\rm
i}\gamma_{n}^{4}\Big{)}e^{2({\rm
i}\beta_{n}+\gamma_{n})\tilde{h}_{1}}\bigg{\\}},$
$\displaystyle(\hat{T}_{21}^{(1)}\hat{f}_{1})(x_{1})$
$\displaystyle=-\sum\limits_{n\in\mathbb{Z}}\frac{e^{{\rm
i}\alpha_{n}x_{1}}}{D^{(n)}_{1}}\hat{f}_{1}^{(n)}\bigg{\\{}\Big{(}\frac{\rm
i}{2}\mu\beta_{n}^{4}+(1-\mu)\beta_{n}^{3}\gamma_{n}+(2-\mu){\rm
i}\beta_{n}^{2}\gamma_{n}^{2}$
$\displaystyle\quad-(1-\mu)\beta_{n}\gamma_{n}^{3}+\frac{\rm
i}{2}\mu\gamma_{n}^{4}\Big{)}+\Big{(}-\frac{\rm
i}{2}\mu\beta_{n}^{4}+(1-\mu)\beta_{n}^{3}\gamma_{n}-(2-\mu){\rm
i}\beta_{n}^{2}\gamma_{n}^{2}$
$\displaystyle\quad-(1-\mu)\beta_{n}\gamma_{n}^{3}-\frac{\rm
i}{2}\mu\gamma_{n}^{4}\Big{)}e^{2{\rm
i}\beta_{n}\tilde{h}_{1}}+\Big{(}-\frac{\rm
i}{2}\mu\beta_{n}^{4}+(1-\mu)\beta_{n}^{3}\gamma_{n}$
$\displaystyle\quad-(2-\mu){\rm
i}\beta_{n}^{2}\gamma_{n}^{2}-(1-\mu)\beta_{n}\gamma_{n}^{3}-\frac{\rm
i}{2}\mu\gamma_{n}^{4}\Big{)}e^{2\gamma_{n}\tilde{h}_{1}}$
$\displaystyle\quad+(4-4\mu)\left(\gamma_{n}^{3}\beta_{n}-\beta_{n}^{3}\gamma_{n}\right)e^{({\rm
i}\beta_{n}+\gamma_{n})\tilde{h}_{1}}+\Big{(}\frac{\rm
i}{2}\mu\beta_{n}^{4}+(1-\mu)\beta_{n}^{3}\gamma_{n}$
$\displaystyle\quad+(2-\mu){\rm
i}\beta_{n}^{2}\gamma_{n}^{2}-(1-\mu)\beta_{n}\gamma_{n}^{3}+\frac{\rm
i}{2}\mu\gamma_{n}^{4}\Big{)}e^{2({\rm
i}\beta_{n}+\gamma_{n})\tilde{h}_{1}}\bigg{\\}},$

and

$\displaystyle(\hat{T}_{22}^{(1)}\hat{g}_{1})(x_{1})$
$\displaystyle=-\sum\limits_{n\in\mathbb{Z}}\frac{e^{{\rm
i}\alpha_{n}x_{1}}}{D^{(n)}_{1}}\hat{g}_{1}^{(n)}\Big{\\{}-\left(\beta_{n}^{3}+{\rm
i}\beta_{n}^{2}\gamma_{n}+\beta_{n}\gamma_{n}^{2}+{\rm
i}\gamma_{n}^{3}\right)$ $\displaystyle\quad-\left(\beta_{n}^{3}-{\rm
i}\beta_{n}^{2}\gamma_{n}+\beta_{n}\gamma_{n}^{2}-{\rm
i}\gamma_{n}^{3}\right)e^{2{\rm i}\beta_{n}\tilde{h}_{1}}$
$\displaystyle\quad+\left(\beta_{n}^{3}-{\rm
i}\beta_{n}^{2}\gamma_{n}+\beta_{n}\gamma_{n}^{2}-{\rm
i}\gamma_{n}^{3}\right)e^{2\gamma_{n}\tilde{h}_{1}}$
$\displaystyle\quad+\left(\beta_{n}^{3}+{\rm
i}\beta_{n}^{2}\gamma_{n}+\beta_{n}\gamma_{n}^{2}+{\rm
i}\gamma_{n}^{3}\right)e^{2({\rm
i}\beta_{n}+\gamma_{n})\tilde{h}_{1}}\Big{\\}}.$

Let $(\tilde{u},\partial_{\nu}\tilde{u})=(f_{1},g_{1})$ be the Dirichlet data
of the total field $\tilde{u}$ on $\Gamma_{1}$. Utilizing (28) and noting
$f_{1}=\hat{f}_{1}+u^{i}$ and $g_{1}=\hat{g}_{1}+\partial_{x_{2}}u^{i}$, the
TBC for the total field $\tilde{u}$ on $\Gamma_{1}$ can be formulated as

$N_{1}\tilde{u}=\hat{T}_{11}^{(1)}f_{1}+\hat{T}_{12}^{(1)}g_{1}+\hat{p}_{1},\quad
M_{1}\tilde{u}=\hat{T}_{21}^{(1)}f_{1}+\hat{T}_{22}^{(1)}g_{1}+\hat{p}_{2},$
(29)

where

$\hat{p}_{1}(x_{1})=N_{1}u^{i}-\hat{T}_{11}^{(1)}u^{i}-\hat{T}_{12}^{(1)}\partial_{x_{2}}u^{i},\quad\hat{p}_{2}(x_{1})=M_{1}u^{i}-\hat{T}_{21}^{(1)}u^{i}-\hat{T}_{22}^{(1)}\partial_{x_{2}}u^{i}.$
(30)

The TBC can be similarly deduced on $\Gamma_{2}$. Derived from the biharmonic
wave equation in (22), the PML solution $\tilde{u}$ exhibits the following
analytical expansion in $\Omega_{2}^{\rm PML}$:

$\displaystyle\tilde{u}(x_{1},\tilde{x}_{2})$
$\displaystyle=\sum\limits_{n\in\mathbb{Z}}\Big{[}W_{2}^{(n)}e^{-{\rm
i}\beta_{n}(\tilde{x}_{2}-h_{2})}+V_{2}^{(n)}e^{{\rm
i}\beta_{n}(\tilde{x}_{2}-h_{2})}$
$\displaystyle\quad+X_{2}^{(n)}e^{\gamma_{n}(\tilde{x}_{2}-h_{2})}+Y_{2}^{(n)}e^{-\gamma_{n}(\tilde{x}_{2}-h_{2})}\Big{]}e^{{\rm
i}\alpha_{n}x_{1}}.$ (31)

Substituting (5.2) into the boundary conditions in (22), we obtain a linear
system for the Fourier coefficients $W_{2}^{(n)},V_{2}^{(n)},X_{2}^{(n)}$, and
$Y_{2}^{(n)}$:

$\left\\{\begin{aligned}
&W_{2}^{(n)}+V_{2}^{(n)}+X_{2}^{(n)}+Y_{2}^{(n)}=f_{2}^{(n)},\\\ &-{\rm
i}\beta_{n}W_{2}^{(n)}+{\rm
i}\beta_{n}V_{2}^{(n)}+\gamma_{n}X_{2}^{(n)}-\gamma_{n}Y_{2}^{(n)}=-g_{2}^{(n)},\\\
&W_{2}^{(n)}e^{-{\rm i}\beta_{n}\tilde{h}_{2}}+V_{2}^{(n)}e^{{\rm
i}\beta_{n}\tilde{h}_{2}}+X_{2}^{(n)}e^{\gamma_{n}\tilde{h}_{2}}+Y_{2}^{(n)}e^{-\gamma_{n}\tilde{h}_{2}}=0,\\\
&-{\rm i}\beta_{n}W_{2}^{(n)}e^{-{\rm i}\beta_{n}\tilde{h}_{2}}+{\rm
i}\beta_{n}V_{2}^{(n)}e^{{\rm
i}\beta_{n}\tilde{h}_{2}}+\gamma_{n}X_{2}^{(n)}e^{\gamma_{n}\tilde{h}_{2}}-\gamma_{n}Y_{2}^{(n)}e^{-\gamma_{n}\tilde{h}_{2}}=0,\end{aligned}\right.$
(32)

where $(\tilde{u},\partial_{\nu}\tilde{u})=(f_{2},g_{2})$ represent the
Dirichlet data of the total field on $\Gamma_{2}$ and have the Fourier series
expansions

$f_{2}=\sum\limits_{n\in\mathbb{Z}}f_{2}^{(n)}e^{{\rm i}\alpha_{n}x_{1}},\quad
g_{2}=\sum\limits_{n\in\mathbb{Z}}g_{2}^{(n)}e^{{\rm i}\alpha_{n}x_{1}},$

and

$\tilde{h}_{2}:=\varphi(h_{2}-\delta_{2})-h_{2}=\int_{0}^{h_{2}-\delta_{2}}s(t){\rm
d}t-h_{2}=-\delta_{2}\left(1+\frac{{\rm i}\sigma_{0}}{m+1}\right).$ (33)

Upon solving the linear system (32), we have

$\displaystyle W_{2}^{(n)}$
$\displaystyle=\frac{1}{D^{(n)}_{2}}\bigg{\\{}f_{2}^{(n)}\Big{[}\left(\gamma_{n}\beta_{n}-{\rm
i}\gamma_{n}^{2}\right)e^{2{\rm
i}\beta_{n}\tilde{h}_{2}}-2\beta_{n}\gamma_{n}e^{({\rm
i}\beta_{n}+\gamma_{n})\tilde{h}_{2}}$
$\displaystyle\quad+\left(\gamma_{n}\beta_{n}+{\rm
i}\gamma_{n}^{2}\right)e^{2({\rm
i}\beta_{n}+\gamma_{n})\tilde{h}_{2}}\Big{]}+g_{2}^{(n)}\Big{[}\left(\beta_{n}-{\rm
i}\gamma_{n}\right)e^{2{\rm i}\beta_{n}\tilde{h}_{2}}$
$\displaystyle\quad+2{\rm i}\gamma_{n}e^{({\rm
i}\beta_{n}+\gamma_{n})\tilde{h}_{2}}-\left({\rm
i}\gamma_{n}+\beta_{n}\right)e^{2({\rm
i}\beta_{n}+\gamma_{n})\tilde{h}_{2}}\Big{]}\bigg{\\}},$ $\displaystyle
V_{2}^{(n)}$
$\displaystyle=\frac{1}{D^{(n)}_{2}}\bigg{\\{}f_{2}^{(n)}\left[\left(\beta_{n}\gamma_{n}+{\rm
i}\gamma_{n}^{2}\right)+\left(\gamma_{n}\beta_{n}-{\rm
i}\gamma_{n}^{2}\right)e^{2\gamma_{n}\tilde{h}_{2}}-2\beta_{n}\gamma_{n}e^{({\rm
i}\beta_{n}+\gamma_{n})\tilde{h}_{2}}\right]$
$\displaystyle\quad+g_{2}^{(n)}\left[\left(\beta_{n}+{\rm
i}\gamma_{n}\right)-\left(\beta_{n}-{\rm
i}\gamma_{n}\right)e^{2\gamma_{n}\tilde{h}_{2}}-2{\rm i}\gamma_{n}e^{({\rm
i}\beta_{n}+\gamma_{n})\tilde{h}_{2}}\right]\bigg{\\}},$ $\displaystyle
X_{2}^{(n)}$
$\displaystyle=\frac{1}{D^{(n)}_{2}}\bigg{\\{}f_{2}^{(n)}\left[\left(\beta_{n}\gamma_{n}-{\rm
i}\beta_{n}^{2}\right)+\left({\rm
i}\beta_{n}^{2}+\beta_{n}\gamma_{n}\right)e^{2{\rm
i}\beta_{n}\tilde{h}_{2}}-2\beta_{n}\gamma_{n}e^{({\rm
i}\beta_{n}+\gamma_{n})\tilde{h}_{2}}\right]$
$\displaystyle\quad+g_{2}^{(n)}\left[-\left(\beta_{n}+{\rm
i}\gamma_{n}\right)-\left(\beta_{n}-{\rm i}\gamma_{n}\right)e^{2{\rm
i}\beta_{n}\tilde{h}_{2}}+2\beta_{n}e^{({\rm
i}\beta_{n}+\gamma_{n})\tilde{h}_{2}}\right]\bigg{\\}},$ $\displaystyle
Y_{2}^{(n)}$
$\displaystyle=\frac{1}{D^{(n)}_{2}}\bigg{\\{}f_{2}^{(n)}\Big{[}\left({\rm
i}\beta_{n}^{2}+\beta_{n}\gamma_{n}\right)e^{2\gamma_{n}\tilde{h}_{2}}-2\beta_{n}\gamma_{n}e^{({\rm
i}\beta_{n}+\gamma_{n})\tilde{h}_{2}}$
$\displaystyle\quad+\left(\beta_{n}\gamma_{n}-{\rm
i}\beta_{n}^{2}\right)e^{2({\rm
i}\beta_{n}+\gamma_{n})\tilde{h}_{2}}\Big{]}+g_{2}^{(n)}\Big{[}\left(\beta_{n}-{\rm
i}\gamma_{n}\right)e^{2\gamma_{n}\tilde{h}_{2}}$
$\displaystyle\quad-2\beta_{n}e^{({\rm
i}\beta_{n}+\gamma_{n})\tilde{h}_{2}}+\left({\rm
i}\gamma_{n}+\beta_{n}\right)e^{2({\rm
i}\beta_{n}+\gamma_{n})\tilde{h}_{2}}\Big{]}\bigg{\\}},$

where the denominator $D^{(n)}_{2}$ is given by

$\displaystyle D^{(n)}_{2}$ $\displaystyle=-8\beta_{n}\gamma_{n}e^{({\rm
i}\beta_{n}+\gamma_{n})\tilde{h}_{2}}+\left(-{\rm i}\beta_{n}^{2}+{\rm
i}\gamma_{n}^{2}+2\beta_{n}\gamma_{n}\right)\big{(}1+e^{2({\rm
i}\beta_{n}+\gamma_{n})\tilde{h}_{2}}\big{)}$ (34)
$\displaystyle\quad+\left({\rm i}\beta_{n}^{2}-{\rm
i}\gamma_{n}^{2}+2\beta_{n}\gamma_{n}\right)\big{(}e^{2{\rm
i}\beta_{n}\tilde{h}_{2}}+e^{2\gamma_{n}\tilde{h}_{2}}\big{)}.$

By (33), both of the real and imaginary parts of $\tilde{h}_{2}$ are negative.
If the thickness of the layer $\delta_{2}$ is sufficiently large, i.e.,
$-\Re\tilde{h}_{2}=\delta_{2}$ is sufficiently large, then the leading term in
(34) is the one containing $e^{2{\rm i}\beta_{n}\tilde{h}_{2}}$. Observing
$|{\rm i}\beta_{n}^{2}-{\rm
i}\gamma_{n}^{2}+2\beta_{n}\gamma_{n}|^{2}=|\gamma_{n}+{\rm
i}\beta_{n}|^{4}\neq 0$ for any $n\in\mathbb{Z}$, we deduce that the
denominator $D^{(2)}_{n}$ in non-zero for sufficiently large $\delta_{2}$.

Substituting (5.2) into (14), we obtain the TBC for the PML problem on
$\Gamma_{2}$:

$N_{2}\tilde{u}=\hat{T}_{11}^{(2)}f_{2}+\hat{T}_{12}^{(2)}g_{2},\quad
M_{2}\tilde{u}=\hat{T}_{21}^{(2)}f_{2}+\hat{T}_{22}^{(2)}g_{2},$ (35)

where the DtN operators $\hat{T}_{ij}^{(2)},i,j=1,2$ are defined as

$\displaystyle(\hat{T}^{(2)}_{11}f_{2})(x_{1})$
$\displaystyle=-\sum\limits_{n\in\mathbb{Z}}\frac{e^{{\rm
i}\alpha_{n}x_{1}}}{D^{(2)}_{n}}f_{2}^{(n)}\Big{\\{}\left({\rm
i}\beta_{n}^{4}\gamma_{n}-\beta_{n}^{3}\gamma_{n}^{2}+{\rm
i}\beta_{n}^{2}\gamma_{n}^{3}-\beta_{n}\gamma_{n}^{4}\right)$
$\displaystyle\quad-\left({\rm
i}\beta_{n}^{4}\gamma_{n}+\beta_{n}^{3}\gamma_{n}^{2}+{\rm
i}\beta_{n}^{2}\gamma_{n}^{3}+\beta_{n}\gamma_{n}^{4}\right)e^{2{\rm
i}\beta_{n}\tilde{h}_{2}}$ $\displaystyle\quad+\left({\rm
i}\beta_{n}^{4}\gamma_{n}+\beta_{n}^{3}\gamma_{n}^{2}+{\rm
i}\beta_{n}^{2}\gamma_{n}^{3}+\beta_{n}\gamma_{n}^{4}\right)e^{2\gamma_{n}\tilde{h}_{2}}$
$\displaystyle\quad-\left({\rm
i}\beta_{n}^{4}\gamma_{n}-\beta_{n}^{3}\gamma_{n}^{2}+{\rm
i}\beta_{n}^{2}\gamma_{n}^{3}-\beta_{n}\gamma_{n}^{4}\right)e^{2({\rm
i}\beta_{n}+\gamma_{n})\tilde{h}_{2}}\Big{\\}},$
$\displaystyle(\hat{T}^{(2)}_{12}g_{2})(x_{1})$
$\displaystyle=-\sum\limits_{n\in\mathbb{Z}}\frac{e^{{\rm
i}\alpha_{n}x_{1}}}{D^{(2)}_{n}}g_{2}^{(n)}\bigg{\\{}\Big{(}\frac{\rm
i}{2}\mu\beta_{n}^{4}+(1-\mu)\beta_{n}^{3}\gamma_{n}+(2-\mu){\rm
i}\beta_{n}^{2}\gamma_{n}^{2}$
$\displaystyle\quad-(1-\mu)\beta_{n}\gamma_{n}^{3}+\frac{\mu}{2}{\rm
i}\gamma_{n}^{4}\Big{)}+\Big{(}-\frac{\rm
i}{2}\mu\beta_{n}^{4}+(1-\mu)\beta_{n}^{3}\gamma_{n}-(2-\mu){\rm
i}\beta_{n}^{2}\gamma_{n}^{2}$
$\displaystyle\quad-(1-\mu)\beta_{n}\gamma_{n}^{3}-\frac{\rm
i}{2}\mu\gamma_{n}^{4}\Big{)}e^{2{\rm
i}\beta_{n}\tilde{h}_{2}}+\Big{(}-\frac{\rm
i}{2}\mu\beta_{n}^{4}+(1-\mu)\beta_{n}^{3}\gamma_{n}$
$\displaystyle\quad-(2-\mu){\rm
i}\beta_{n}^{2}\gamma_{n}^{2}-(1-\mu)\beta_{n}\gamma_{n}^{3}-\frac{\rm
i}{2}\mu\gamma_{n}^{4}\Big{)}e^{2\gamma_{n}\tilde{h}_{2}}$
$\displaystyle\quad+(4-4\mu)\left(\beta_{n}\gamma_{n}^{3}-\beta_{n}^{3}\gamma_{n}\right)e^{({\rm
i}\beta_{n}+\gamma_{n})\tilde{h}_{2}}+\Big{(}\frac{\rm
i}{2}\mu\beta_{n}^{4}+(1-\mu)\beta_{n}^{3}\gamma_{n}$
$\displaystyle\quad+(2-\mu){\rm
i}\beta_{n}^{2}\gamma_{n}^{2}-(1-\mu)\beta_{n}\gamma_{n}^{3}+\frac{\mu}{2}{\rm
i}\gamma_{n}^{4}\Big{)}e^{2({\rm
i}\beta_{n}+\gamma_{n})\tilde{h}_{2}}\bigg{\\}},$
$\displaystyle(\hat{T}_{21}^{(2)}(n)f_{2})(x_{1})$
$\displaystyle=-\sum\limits_{n\in\mathbb{Z}}\frac{e^{{\rm
i}\alpha_{n}x_{1}}}{D^{(2)}_{n}}f_{2}^{(n)}\bigg{\\{}\Big{(}\frac{\rm
i}{2}\mu\beta_{n}^{4}+(1-\mu)\beta_{n}^{3}\gamma_{n}+(2-\mu){\rm
i}\beta_{n}^{2}\gamma_{n}^{2}$
$\displaystyle\quad-(1-\mu)\beta_{n}\gamma_{n}^{3}+\frac{\rm
i}{2}\mu\gamma_{n}^{4}\Big{)}+\Big{(}-\frac{\rm
i}{2}\mu\beta_{n}^{4}+(1-\mu)\beta_{n}^{3}\gamma_{n}-(2-\mu){\rm
i}\beta_{n}^{2}\gamma_{n}^{2}$
$\displaystyle\quad-(1-\mu)\beta_{n}\gamma_{n}^{3}-\frac{\rm
i}{2}\mu\gamma_{n}^{4}\Big{)}e^{2{\rm
i}\beta_{n}\tilde{h}_{2}}+\Big{(}-\frac{\rm
i}{2}\mu\beta_{n}^{4}+(1-\mu)\beta_{n}^{3}\gamma_{n}$
$\displaystyle\quad-(2-\mu){\rm
i}\beta_{n}^{2}\gamma_{n}^{2}-(1-\mu)\beta_{n}\gamma_{n}^{3}-\frac{\rm
i}{2}\mu\gamma_{n}^{4}\Big{)}e^{2\gamma_{n}\tilde{h}_{2}}$
$\displaystyle\quad+(4-4\mu)\left(\gamma_{n}^{3}\beta_{n}-\beta_{n}^{3}\gamma_{n}\right)e^{({\rm
i}\beta_{n}+\gamma_{n})\tilde{h}_{2}}+\Big{(}\frac{\rm
i}{2}\mu\beta_{n}^{4}+(1-\mu)\beta_{n}^{3}\gamma_{n}$
$\displaystyle\quad+(2-\mu){\rm
i}\beta_{n}^{2}\gamma_{n}^{2}-(1-\mu)\beta_{n}\gamma_{n}^{3}+\frac{\rm
i}{2}\mu\gamma_{n}^{4}\Big{)}e^{2({\rm
i}\beta_{n}+\gamma_{n})\tilde{h}_{2}}\bigg{\\}},$

and

$\displaystyle(\hat{T}_{22}^{(2)}(n)g_{2})(x_{1})$
$\displaystyle=-\sum\limits_{n\in\mathbb{Z}}\frac{e^{{\rm
i}\alpha_{n}x_{1}}}{D^{(2)}_{n}}g_{2}^{(n)}\Big{\\{}\left(\beta_{n}^{3}+{\rm
i}\beta_{n}^{2}\gamma_{n}+\beta_{n}\gamma_{n}^{2}+{\rm
i}\gamma_{n}^{3}\right)$ $\displaystyle\quad+\left(\beta_{n}^{3}-{\rm
i}\beta_{n}^{2}\gamma_{n}+\beta_{n}\gamma_{n}^{2}-{\rm
i}\gamma_{n}^{3}\right)e^{2{\rm i}\beta_{n}\tilde{h}_{2}}$
$\displaystyle\quad-\left(\beta_{n}^{3}-{\rm
i}\beta_{n}^{2}\gamma_{n}+\beta_{n}\gamma_{n}^{2}-{\rm
i}\gamma_{n}^{3}\right)e^{2\gamma_{n}\tilde{h}_{2}}$
$\displaystyle\quad-\left(\beta_{n}^{3}+{\rm
i}\beta_{n}^{2}\gamma_{n}+\beta_{n}\gamma_{n}^{2}+{\rm
i}\gamma_{n}^{3}\right)e^{2({\rm
i}\beta_{n}+\gamma_{n})\tilde{h}_{2}}\Big{\\}}.$

######  Lemma 5.

Assuming that the thickness of the PML layers $\delta_{k}$ for $k=1,2$ are
sufficiently large, the DtN operators
$\hat{T}^{(k)}_{11}:H^{3/2}(\Gamma_{k})\to H^{-3/2}(\Gamma_{k})$,
$\hat{T}^{(k)}_{12}:H^{1/2}(\Gamma_{k})\to H^{-3/2}(\Gamma_{k})$,
$\hat{T}^{(k)}_{21}:H^{3/2}(\Gamma_{k})\to H^{-1/2}(\Gamma_{k})$, and
$\hat{T}^{(k)}_{22}:H^{1/2}(\Gamma_{k})\to H^{-1/2}(\Gamma_{k})$ are bounded.

###### Proof.

It is sufficient to present the results for the DtN operators on $\Gamma_{1}$,
as the corresponding outcomes can be similarly established for the DtN
operators on $\Gamma_{2}$.

As demonstrated in the derivation of the TBC on $\Gamma_{1}$, when the
thickness of the PML region is sufficiently large, the term containing
$e^{2\gamma_{n}\tilde{h}_{1}}$ is dominant in both the denominator and the
numerator. Noting (16), i.e., $\beta_{n}\sim\alpha_{n}$ and
$\gamma_{n}\sim\alpha_{n}$ as $n\to\infty$, we assert that for any function
$f\in H^{3/2}(\Gamma_{1})$

$\displaystyle\|\hat{T}_{11}^{(1)}f\|^{2}_{H^{-3/2}(\Gamma_{1})}$
$\displaystyle\lesssim\sum\limits_{n\in\mathbb{Z}}(1+\alpha_{n}^{2})^{-3/2}\left|\frac{f^{(n)}}{D_{1}^{(n)}}\left({\rm
i}\beta_{n}^{4}\gamma_{n}+\beta_{n}^{3}\gamma_{n}^{2}+{\rm
i}\beta_{n}^{2}\gamma_{n}^{3}+\beta_{n}\gamma_{n}^{4}\right)e^{2\gamma_{n}\tilde{h}_{1}}\right|^{2}$
$\displaystyle\lesssim\sum\limits_{n\in\mathbb{Z}}(1+\alpha_{n}^{2})^{-3/2}\left|\frac{\left({\rm
i}\beta_{n}^{4}\gamma_{n}+\beta_{n}^{3}\gamma_{n}^{2}+{\rm
i}\beta_{n}^{2}\gamma_{n}^{3}+\beta_{n}\gamma_{n}^{4}\right)e^{2\gamma_{n}\tilde{h}_{1}}}{\left({\rm
i}\beta_{n}^{2}-{\rm
i}\gamma_{n}^{2}+2\beta_{n}\gamma_{n}\right)e^{2\gamma_{n}\tilde{h}_{1}}}\right|^{2}|f^{(n)}|^{2}$
$\displaystyle\lesssim\sum\limits_{n\in\mathbb{Z}}(1+\alpha_{n}^{2})^{3/2}|f^{(n)}|^{2}=\|f\|^{2}_{H^{3/2}(\Gamma_{1})},$

and

$\displaystyle\|\hat{T}_{21}^{(1)}f\|^{2}_{H^{-1/2}(\Gamma_{1})}$
$\displaystyle\lesssim\sum\limits_{n\in\mathbb{Z}}(1+\alpha_{n}^{2})^{-1/2}$
$\displaystyle\times\left|\frac{f^{(n)}}{D_{1}^{(n)}}\Big{(}-\frac{\rm
i}{2}\mu\beta_{n}^{4}+(1-\mu)\beta_{n}^{3}\gamma_{n}-(2-\mu){\rm
i}\beta_{n}^{2}\gamma_{n}^{2}-(1-\mu)\beta_{n}\gamma_{n}^{3}-\frac{\rm
i}{2}\mu\gamma_{n}^{4}\Big{)}e^{2\gamma_{n}\tilde{h}_{1}}\right|^{2}$
$\displaystyle\lesssim\sum\limits_{n\in\mathbb{Z}}(1+\alpha_{n}^{2})^{-1/2}$
$\displaystyle\times\left|\frac{-\frac{\rm
i}{2}\mu\beta_{n}^{4}+(1-\mu)\beta_{n}^{3}\gamma_{n}-(2-\mu){\rm
i}\beta_{n}^{2}\gamma_{n}^{2}-(1-\mu)\beta_{n}\gamma_{n}^{3}-\frac{\rm
i}{2}\mu\gamma_{n}^{4}}{{\rm i}\beta_{n}^{2}-{\rm
i}\gamma_{n}^{2}+2\beta_{n}\gamma_{n}}\right|^{2}|f^{(n)}|^{2}$
$\displaystyle\lesssim\sum\limits_{n\in\mathbb{Z}}(1+\alpha_{n}^{2})^{3/2}|f^{(n)}|^{2}=\|f\|^{2}_{H^{3/2}(\Gamma_{1})}.$

Similarly, we have that for any function $f\in H^{1/2}(\Gamma_{1})$

$\displaystyle\|\hat{T}_{12}^{(1)}f\|^{2}_{H^{-3/2}(\Gamma_{1})}$
$\displaystyle\lesssim\sum\limits_{n\in\mathbb{Z}}(1+\alpha_{n}^{2})^{-3/2}$
$\displaystyle\times\left|\frac{f^{(n)}}{D_{1}^{(n)}}\Big{(}-\frac{\rm
i}{2}\mu\beta_{n}^{4}+(1-\mu)\beta_{n}^{3}\gamma_{n}-(2-\mu){\rm
i}\beta_{n}^{2}\gamma_{n}^{2}-(1-\mu)\beta_{n}\gamma_{n}^{3}-\frac{\rm
i}{2}\mu\gamma_{n}^{4}\Big{)}e^{2\gamma_{n}\tilde{h}_{1}}\right|^{2}$
$\displaystyle\lesssim\sum\limits_{n\in\mathbb{Z}}(1+\alpha_{n}^{2})^{-3/2}$
$\displaystyle\times\left|\frac{-\frac{\rm
i}{2}\mu\beta_{n}^{4}+(1-\mu)\beta_{n}^{3}\gamma_{n}-(2-\mu){\rm
i}\beta_{n}^{2}\gamma_{n}^{2}-(1-\mu)\beta_{n}\gamma_{n}^{3}-\frac{\rm
i}{2}\mu\gamma_{n}^{4}}{{\rm i}\beta_{n}^{2}-{\rm
i}\gamma_{n}^{2}+2\beta_{n}\gamma_{n}}\right|^{2}|f^{(n)}|^{2}$
$\displaystyle\lesssim\sum\limits_{n\in\mathbb{Z}}(1+\alpha_{n}^{2})^{1/2}|f^{(n)}|^{2}=\|f\|^{2}_{H^{1/2}(\Gamma_{1})},$

and

$\displaystyle\|\hat{T}_{22}^{(1)}f\|^{2}_{H^{-1/2}(\Gamma_{1})}$
$\displaystyle\lesssim\sum\limits_{n\in\mathbb{Z}}(1+\alpha_{n}^{2})^{-1/2}\left|\frac{f^{(n)}}{D_{1}^{(n)}}\left(\beta_{n}^{3}-{\rm
i}\beta_{n}^{2}\gamma_{n}+\beta_{n}\gamma_{n}^{2}-{\rm
i}\gamma_{n}^{3}\right)e^{2\gamma_{n}\tilde{h}_{1}}\right|^{2}$
$\displaystyle\lesssim\sum\limits_{n\in\mathbb{Z}}(1+\alpha_{n}^{2})^{-1/2}\left|\frac{\left(\beta_{n}^{3}-{\rm
i}\beta_{n}^{2}\gamma_{n}+\beta_{n}\gamma_{n}^{2}-{\rm
i}\gamma_{n}^{3}\right)e^{2\gamma_{n}\tilde{h}_{1}}}{\left({\rm
i}\beta_{n}^{2}-{\rm
i}\gamma_{n}^{2}+2\beta_{n}\gamma_{n}\right)e^{2\gamma_{n}\tilde{h}_{1}}}\right|^{2}|f^{(n)}|^{2}$
$\displaystyle\lesssim\sum\limits_{n\in\mathbb{Z}}(1+\alpha_{n}^{2})^{1/2}|f^{(n)}|^{2}=\|f\|^{2}_{H^{1/2}(\Gamma_{1})},$

which complete the proof. ∎

The following lemma addresses the error estimates of the DtN operators between
the PML problem and the original scattering problem.

######  Lemma 6.

Let

$\Delta^{-}=\min_{n\in\mathbb{Z}}\left\\{\Re(\beta_{n})>0\right\\},\quad\Delta^{+}=\min_{n\in\mathbb{Z}}\left\\{\Im(\beta_{n})>0\right\\}.$

For $k=1,2$, if the thickness of the PML region $\delta_{k}$ is sufficiently
large, then the following estimates hold for any $f\in H^{3/2}(\Gamma_{k})$:

$\displaystyle\|(\hat{T}_{11}^{(k)}-T_{11}^{(k)})f\|_{H^{-3/2}(\Gamma_{k})}$
$\displaystyle\lesssim\Theta\|f\|_{H^{3/2}(\Gamma_{k})},$
$\displaystyle\|(\hat{T}_{21}^{(k)}-T_{21}^{(k)})f\|_{H^{-1/2}(\Gamma_{k})}$
$\displaystyle\lesssim\Theta\|f\|_{H^{3/2}(\Gamma_{k})},$

and the following estimates hold for any $f\in H^{1/2}(\Gamma_{k})$:

$\displaystyle\|(\hat{T}_{12}^{(k)}-T_{12}^{(k)})f\|_{H^{-3/2}(\Gamma_{k})}$
$\displaystyle\lesssim\Theta\|f\|_{H^{1/2}(\Gamma_{k})},$
$\displaystyle\|(\hat{T}_{22}^{(k)}-T_{22}^{(k)})f\|_{H^{-1/2}(\Gamma_{k})}$
$\displaystyle\lesssim\Theta\|f\|_{H^{1/2}(\Gamma_{k})},$

where

$\Theta=\max\Big{\\{}e^{-\frac{2\delta_{k}\sigma_{0}}{m+1}\Delta^{-}},e^{-2\delta_{k}(\kappa^{2}+\alpha^{2})^{1/2}},e^{-2\delta_{k}\Delta^{+}}\Big{\\}}.$
(36)

###### Proof.

Given the similarity in proof, we only present the details of the error
estimate between $\hat{T}_{11}^{(1)}$ and $T_{11}^{(1)}$, with the
understanding that the results for the other operators can be obtained in the
same manner.

It follows from (28) and (8) that

$\displaystyle(T_{11}^{(1)}-\hat{T}_{11}^{(1)})f=\sum\limits_{n\in\mathbb{Z}}\frac{e^{{\rm
i}\alpha_{n}x}}{D^{(n)}_{1}}f_{1}^{(n)}\Big{\\{}\left(-2{\rm
i}\beta_{n}^{4}\gamma_{n}+4\beta_{n}^{3}\gamma_{n}^{2}+2{\rm
i}\beta_{n}^{2}\gamma_{n}^{3}\right)$ $\displaystyle\quad+\left(2{\rm
i}\beta_{n}^{4}\gamma_{n}+2\beta_{n}^{3}\gamma_{n}^{2}+2{\rm
i}\beta_{n}^{2}\gamma_{n}^{3}+2\beta_{n}\gamma_{n}^{4}\right)e^{2{\rm
i}\beta_{n}\tilde{h}_{1}}-\left(8{\rm
i}\beta_{n}^{2}\gamma_{n}^{3}+\beta_{n}^{3}\gamma_{n}^{2}\right)e^{({\rm
i}\beta_{n}+\gamma_{n})\tilde{h}_{1}}$
$\displaystyle\quad+\left(2\beta_{n}^{3}\gamma_{n}^{2}+4{\rm
i}\beta_{n}^{2}\gamma_{n}^{3}-2\beta_{n}\gamma_{n}^{4}\right)e^{2({\rm
i}\beta_{n}+\gamma_{n})\tilde{h}_{1}}\Big{\\}}.$

When $\delta_{1}$ is sufficiently large, the dominant term in the denominator
is the one involving $e^{2\gamma_{n}\tilde{h}_{1}}$, which has a coefficient
of ${\rm i}\beta_{n}^{2}-{\rm i}\gamma_{n}^{2}+2\beta_{n}\gamma_{n}$. The
dominant terms in the numerator are either the constant term or the one that
contains $e^{2({\rm i}\beta_{n}+\gamma_{n})\tilde{h}_{1}}$. By the choice of
PML parameters given in (26), a straightforward calculation yields

$\displaystyle 2\gamma_{h}\tilde{h}_{1}=2\delta_{1}\left(1+\frac{{\rm
i}\sigma_{0}}{m+1}\right)(\kappa^{2}+\alpha_{n}^{2})^{1/2}=2\delta_{1}(\kappa^{2}+\alpha_{n}^{2})^{1/2}+\frac{2{\rm
i}\delta_{1}\sigma_{0}}{m+1}(\kappa^{2}+\alpha_{n}^{2})^{1/2},$

and

$\displaystyle 2{\rm i}\beta_{n}\tilde{h}_{1}=\left\\{\begin{aligned} &2{\rm
i}\delta_{1}\left(1+\frac{{\rm
i}\sigma_{0}}{m+1}\right)(\kappa^{2}-\alpha_{n}^{2})^{1/2}\\\ &\hskip
28.45274pt=-\frac{2\delta_{1}\sigma_{0}}{m+1}(\kappa^{2}-\alpha_{n}^{2})^{1/2}+2{\rm
i}\delta_{1}(\kappa^{2}-\alpha_{n}^{2})^{1/2}\quad\text{if}~{}|\alpha_{n}|<\kappa,\\\
&-2\delta_{1}\left(1+\frac{{\rm
i}\sigma_{0}}{m+1}\right)(\alpha_{n}^{2}-\kappa^{2})^{1/2}\\\ &\hskip
28.45274pt=-2\delta_{1}(\alpha_{n}^{2}-\kappa^{2})^{1/2}-\frac{2{\rm
i}\delta_{1}\sigma_{0}}{m+1}(\alpha_{n}^{2}-\kappa^{2})^{1/2}\quad\text{if}~{}|\alpha_{n}|>\kappa.\end{aligned}\right.$

Then we obtain

$\displaystyle\|(T_{11}^{(1)}-\hat{T}_{11}^{(1)})f\|^{2}_{H^{-3/2}(\Gamma_{1})}\lesssim\sum\limits_{n\in\mathbb{Z}}(1+\alpha_{n}^{2})^{-3/2}\Bigg{(}\bigg{|}\frac{-2{\rm
i}\beta_{n}^{4}\gamma_{n}+4\beta_{n}^{3}\gamma_{n}^{2}+2{\rm
i}\beta_{n}^{2}\gamma_{n}^{3}}{\left({\rm i}\beta_{n}^{2}-{\rm
i}\gamma_{n}^{2}+2\beta_{n}\gamma_{n}\right)e^{2\gamma_{n}\tilde{h}_{1}}}\bigg{|}^{2}$
$\displaystyle\qquad+\bigg{|}\frac{\left(2\beta_{n}^{3}\gamma_{n}^{2}+4{\rm
i}\beta_{n}^{2}\gamma_{n}^{3}-2\beta_{n}\gamma_{n}^{4}\right)e^{2({\rm
i}\beta_{n}+\gamma_{n})\tilde{h}_{1}}}{\left({\rm i}\beta_{n}^{2}-{\rm
i}\gamma_{n}^{2}+2\beta_{n}\gamma_{n}\right)e^{2\gamma_{n}\tilde{h}_{1}}}\bigg{|}^{2}\Bigg{)}|f_{1}^{(n)}|^{2}$
$\displaystyle\lesssim\sum\limits_{n\in\mathbb{Z}}(1+\alpha_{n}^{2})^{3/2}\left(e^{-4\delta_{1}(\kappa^{2}+\alpha_{n}^{2})^{1/2}}+e^{-\frac{4\delta_{1}\sigma_{0}}{m+1}|\kappa^{2}-\alpha_{n}^{2}|^{1/2}}+e^{-4\delta_{1}|\alpha_{n}^{2}-\kappa^{2}|^{1/2}}\right)|f_{1}^{(n)}|^{2}$
$\displaystyle\lesssim\Theta\|f\|_{H^{3/2}(\Gamma_{1})},$

which completes the proof. ∎

It is evident from Lemma 6 that the DtN operators of the PML problem exhibit
exponential convergence in the operator norm to the DtN operators of the
original scattering problem. This convergence is a crucial factor contributing
to the exponential convergence of the PML solution towards the solution of the
original scattering problem.

###  5.3 Convergence analysis

By employing the TBCs for the PML problem as given in (29) and (35), the PML
problem (22) can be reformulated equivalently in the domain $\Omega$:

$\left\\{\begin{aligned} &\Delta^{2}u^{\rm PML}-\kappa^{4}u^{\rm PML}=0&&{\rm
in}~{}\Omega,\\\ &u^{\rm PML}=0,\quad\partial_{\nu}u^{\rm PML}=0&&{\rm
on}~{}\Gamma_{c},\\\ &N_{1}u^{\rm
PML}=\hat{T}_{11}^{(1)}f_{1}+\hat{T}_{12}^{(1)}g_{1}+\hat{p}_{1}&&{\rm
on}~{}\Gamma_{1},\\\ &M_{1}u^{\rm
PML}=\hat{T}_{21}^{(1)}f_{1}+\hat{T}_{22}^{(1)}g_{1}+\hat{p}_{2}&&{\rm
on}~{}\Gamma_{1},\\\ &N_{2}u^{\rm
PML}=\hat{T}_{11}^{(2)}f_{2}+\hat{T}_{12}^{(2)}g_{2}&&{\rm
on}~{}\Gamma_{2},\\\ &M_{2}u^{\rm
PML}=\hat{T}_{21}^{(2)}f_{2}+\hat{T}_{22}^{(2)}g_{2}&&{\rm
on}~{}\Gamma_{2},\end{aligned}\right.$ (37)

where $\hat{p}_{1}$ and $\hat{p}_{2}$ are given in (30), $(f_{k},g_{k})$
represent the Dirichlet data of the total field $u^{\rm PML}$ on $\Gamma_{k}$
for $k=1,2$.

The variational problem of (37) is to find $u^{\rm PML}\in H_{{\rm
qp},\Gamma_{c}}^{2}(\Omega)$ such that

$a^{\rm PML}(u^{\rm
PML},v)=\int_{\Gamma_{1}}\left(\hat{p}_{1}\bar{v}+\hat{p}_{2}\partial_{\nu}\bar{v}\right){\rm
d}s\quad\forall\,v\in H_{{\rm qp},\Gamma_{c}}^{2}(\Omega),$ (38)

where the sesquilinear form $a^{\rm PML}(u,v):H_{{\rm
qp},\Gamma_{c}}^{2}(\Omega)\times H_{{\rm
qp},\Gamma_{c}}^{2}(\Omega)\rightarrow\mathbb{C}$ is defined as

$\displaystyle a^{\rm PML}(u,v)=\int_{\Omega}\bigg{[}\mu\Delta
u\Delta\bar{v}+(1-\mu)\sum\limits_{i,j=1}^{2}\frac{\partial^{2}u}{\partial
x_{i}\partial x_{j}}\frac{\partial^{2}\bar{v}}{\partial x_{i}\partial
x_{j}}-\kappa^{4}u\bar{v}\bigg{]}{\rm d}x$
$\displaystyle-\sum_{k=1}^{2}\int_{\Gamma_{k}}(\mathbb{\hat{T}}^{(k)}\boldsymbol{u})\cdot\overline{\boldsymbol{v}}{\rm
d}s.$ (39)

with $\boldsymbol{u},\boldsymbol{v}$, and $\mathbb{\hat{T}}^{(k)}$ given by

$\boldsymbol{u}=\begin{bmatrix}u\\\
\partial_{\nu}u\end{bmatrix},\quad\boldsymbol{v}=\begin{bmatrix}v\\\
\partial_{\nu}v\end{bmatrix},\quad\mathbb{\hat{T}}^{(k)}=\begin{bmatrix}{\hat{T}}_{11}^{(k)}&{\hat{T}}_{12}^{(k)}\\\\[5.0pt]
{\hat{T}}_{21}^{(k)}&{\hat{T}}_{22}^{(k)}\end{bmatrix}.$

######  Theorem 2.

Assuming that the thickness of the PML regions is sufficiently large, the
variational problem (38) has a unique weak solution $u^{\rm PML}\in
H^{2}_{{\rm qp},\Gamma_{c}}(\Omega)$ except for a discrete set of wavenumbers
$\kappa$. Moreover, the solution satisfies the error estimate

$\displaystyle\|u-u^{\rm
PML}\|_{H^{2}(\Omega)}\lesssim\Theta\|u^{i}\|_{H^{2}(\Omega)},$ (40)

where $u^{i}$ is the incident field and $u$ is the solution of the variational
problem (19).

###### Proof.

First, we demonstrate that the sesquilinear form (5.3) satisfies the Gårding
inequality. It follows from Lemmas 3, 4, 5–6 and the trace theorem that

$\displaystyle\Re a^{\rm PML}(u,u)$ $\displaystyle=\Re
a(u,u)+\Re\left\\{\sum_{k=1}^{2}\int_{\Gamma_{k}}\big{(}({\mathbb{T}}^{(k)}-\hat{\mathbb{T}}^{(k)})\boldsymbol{u}\big{)}\cdot\overline{\boldsymbol{u}}{\rm
d}s\right\\}$ $\displaystyle\geq
c_{1}\|u\|^{2}_{H^{2}(\Omega)}-c_{2}\|u\|^{2}_{H^{1}(\Omega)}-\Theta\left(c_{3}\|u\|^{2}_{H^{3/2}(\Gamma_{1}\cup\Gamma_{2})}+c_{4}\|u\|^{2}_{H^{1/2}(\Gamma_{1}\cup\Gamma_{2})}\right)$
$\displaystyle\geq
c_{1}\|u\|^{2}_{H^{2}(\Omega)}-c_{2}\|u\|^{2}_{H^{1}(\Omega)}-c_{5}\Theta\|u\|^{2}_{H^{2}(\Omega)}.$

Given the exponential decay of $\Theta$ concerning $\delta_{k}$, we can choose
$\delta_{k}$ to be sufficiently large to ensure $c_{1}-c_{5}\Theta>0$. For all
but a possibly discrete set of wavenumbers $\kappa$, it follows from the
Fredholm alternative theorem that the variational problem is well-posed.
Consequently, there is a positive constant $\gamma$ for which the subsequent
inf-sup condition is satisfied:

$\sup\limits_{0\neq v\in H^{2}_{{\rm qp},\Gamma_{c}}(\Omega)}\frac{|a^{\rm
PML}(u,v)|}{\|v\|_{H^{2}(\Omega)}}\geq\gamma\|u\|_{H^{2}(\Omega)}\quad\forall\,u\in
H_{{\rm qp},\Gamma_{c}}^{2}(\Omega).$

Moreover, the PML solution $u^{\rm PML}$ satisfies the stability estimate

$\|u^{\rm PML}\|_{H^{2}(\Omega)}\lesssim\|u^{\rm inc}\|_{H^{2}(\Omega)}.$ (41)

It remains to prove the error estimate (40). Denote by $e=u^{\rm PML}-u$ the
error between the PML solution and the solution to the original scattering
problem. Upon a straightforward calculation, we obtain

$\displaystyle\int_{\Gamma_{1}}\left(\left(\hat{p}_{1}-p_{1}\right)\bar{v}+\left(\hat{p}_{2}-p_{2}\right)\partial_{\nu}\bar{v}\right){\rm
d}s$ $\displaystyle=a^{\rm PML}(u^{\rm PML},v)-a(u,v)$
$\displaystyle=a(e,v)+\sum_{k=1}^{2}\int_{\Gamma_{k}}\big{(}({\mathbb{T}}^{(k)}-\hat{\mathbb{T}}^{(k)}){\boldsymbol{u}}^{\rm
PML}\big{)}\cdot\overline{\boldsymbol{v}}{\rm d}s,$

where ${\boldsymbol{u}}^{\rm PML}=[u^{\rm PML},\partial_{\nu}u^{\rm
PML}]^{\top}$. Hence we have

$\displaystyle a^{\rm PML}(e,v)$
$\displaystyle=-\sum_{k=1}^{2}\int_{\Gamma_{k}}\big{(}({\mathbb{T}}^{(k)}-\hat{\mathbb{T}}^{(k)}){\boldsymbol{u}}^{\rm
PML}\big{)}\cdot\overline{\boldsymbol{v}}{\rm
d}s-\int_{\Gamma_{1}}\big{(}({\mathbb{T}}^{(1)}-\hat{\mathbb{T}}^{(1)}){\boldsymbol{u}}^{i}\big{)}\cdot\overline{\boldsymbol{v}}{\rm
d}s,$

where ${\boldsymbol{u}}^{i}=[u^{i},\partial_{\nu}u^{i}]^{\top}$. By utilizing
the continuity of the sesquilinear form (4), the stability estimate (41), and
Lemma 6, we obtain

$\displaystyle\|e\|_{H^{2}(\Omega)}\lesssim\sup\limits_{0\neq v\in H^{2}_{{\rm
qp},\Gamma_{c}}(\Omega)}\frac{|a^{\rm
PML}(e,v)|}{\|v\|_{H^{2}(\Omega)}}\lesssim\Theta\left(\|u^{\rm
PML}\|_{H^{2}(\Omega)}+\|u^{i}\|_{H^{2}(\Omega)}\right)\lesssim\Theta\|u^{i}\|_{H^{2}(\Omega)},$

which completes the proof. ∎

It is apparent from Theorem 2 and the definition of $\Theta$ in (36) that as
the thickness of the PML regions increases, the PML solution $u^{\rm PML}$
exhibits exponential convergence towards the solution of the original
scattering problem $u$.

##  6 Conclusion

In this paper, we have investigated the scattering of flexural waves by a one-
dimensional periodic array of cavities embedded in an infinite elastic thin
plate. The problem is formulated using the biharmonic wave equation in an
unbounded domain. Initially, TBCs are introduced to reduce the scattering
problem into a bounded domain, and the well-posedness of the associated
variational problem is examined. Subsequently, the PML method is employed to
transform the problem from an unbounded domain to a bounded one. The
corresponding TBCs are derived, and the well-posedness of the PML problem is
established. Notably, exponential convergence is achieved between the PML
solution and the solution to the original scattering problem.

This work is centered on formulating and analyzing the biharmonic wave
scattering problem in one-dimensional periodic structures. Currently, we are
developing numerical methods, including the finite element method, to solve
the PML problem. The progress and results of this ongoing development will be
detailed in a forthcoming publication.

Acknowledgments

The first author is supported partially by National Natural Science Foundation
of China (U21A20425) and a Key Laboratory of Zhejiang Province. The second
author is supported in part by the NSF grant DMS-2208256. The third author is
supported by the NSFC grants 12201245 and 12171017.

## Declarations

  * •

Conflict of interest: On behalf of all authors, the corresponding author
states that there is no conflict of interest.

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