Inverse Acoustic Scattering from a Bounded Homogeneous Penetrable Obstacle

Abstract

We consider time domain acoustic scattering from a bounded homogeneous penetrable obstacle. The problem is reduced to a system of time-dependent integral equations on the boundary of the scatter. Our aim is to determine the location and shape of the obstacle. Using convolution quadrature in time combined with the nonlinear integral equation method, we obtain the Helmholtz equation with complex wave numbers. Then the iterative scheme is presented to solve the nonlinear boundary integral equations, and the fully discrete system is given. Numerical experiments are presented to show the effectiveness of the proposed method.

1. Introduction

Acoustic scattering problems have received increasing attention in diverse scientific areas such as medical diagnostics, ultrasound tomography, geophysical exploration, and nondestructive testing. However, compared with the frequency domain data, the time domain data is easier to get and contains more information at discrete frequencies. Thus, it is natural to consider the time domain inverse scattering problems. Although time domain inversion algorithms are more challenging due to the time dependence, different methods have been used to solve this problem either theoretically or numerically. Such as the enclosure method, linear sampling method, factorization method, Laguerre transform method, etc. For the framework of the enclosure method with dynamical data and its applications, we refer to [1]. For numerical examples of using the enclosure method for finding the convex hull of polygonal cavities, we refer to [2], and in the case of electromagnetic waves in the time domain, we refer to [3]. The advantage of this method is that one can calculate the support function of an unknown polygonal obstacle from only one measurement in a theoretical sense. However, the formula is ensured to work for estimating the value of the support function at only ‘regular’ directions. Using the enclosure method for the heat equation, we refer to [4], and for more details, we refer to [5,6]. The linear sampling method [7] is also a feasible tool in inverse acoustic scattering problems for the wave equation with Dirichlet, Robin, and Neumann boundary conditions from time domain measurements of scattered waves. In the article [8], the author determined the support of the inhomogeneity from measurements of causal scattered waves by using the linear sampling method in the time domain, and the first examples of using this method in three dimensions are included. The linear sampling method for sparse small aperture data we refer to [9]. Refer to [10,11,12,13,14,15] for more details of the linear sampling method. The factorization method [16] is used to characterize the Dirichlet scattering object from measurements of time-dependent causal scattered waves in the far field regime, and the method for obstacles with impedance boundary conditions in the time domain we refer to [17]. The Laguerre transform (LT) method is another useful tool for the acoustic scattering problem in the time domain. Compared with the Fourier–Laplace integral transform, the inverse transform of the LT method is to find the sum of corresponding Fourier–Laguerre series, which is proved to be more constructive. For more detail and applications of the LT method, we refer to [18,19,20].

In this paper, the inverse scattering problem we considered is the determination of the shape and location of an unknown homogeneous penetrable acoustic scatterer from the measurement of the scattered field in the time domain. Approaches based on retarded potentials are used since they reduce the time-dependent problem in the unbounded domain to an integral equation on the bounded surface. A particular class of methods for the discretization in time domain boundary integral equations (TDBIEs) for the wave equation are the so-called convolution quadratures (CQs); see Lunbich [21,22], in which Christian Lubich introduced this method by extending classical ideas of discrete operational calculus. This method is based on sustainable methods for ordinary differential equations and is more efficient than Galerkin or collocation time approximations. Refer to [23]; the author considered theoretically acoustic scattering problems with respect to various kinds of situations, such as scattering from sound soft, sound hard, non-penetrable, and penetrable with homogeneous or non-homogeneous media. For more details and applications of the CQ method, we refer to [24,25,26,27,28,29,30].

In this work, we propose the convolution quadrature method combined with the nonlinear integral equation method for the inverse problem. We note that this method is also used in [31] for the sound-soft obstacle. Specifically, the author uses the convolution quadrature method (CQ) for time discretization, and then a system of nonlinear integral equations-based iterative reconstruction method is developed to reconstruct both the location and shape of the obstacle. To our best knowledge, here the author first employs it in an inverse problem combined with a nonlinear integral equation method, which belongs to a simplified Newton method [32,33]. In this work, we also use the convolution quadrature method combined with the nonlinear integral equation method for the model of the homogeneous penetrable obstacle (see Equation (2) below). Given time-dependent scattering data, the Fourier transform of time-domain scattered field data is employed in the classic method, and then single frequency reconstruction methods are applied at several frequencies. Using the convolution quadrature method, we obtain a system of boundary integral equations for the Helmholtz equation with time-dependent complex wave numbers. Then, frequency-domain boundary integral operators are linearized with respect to the two-dimensional boundary’s parametrization to construct an iterative domain reconstruction method.

The outline of the paper is as follows. In Section 2, we formulate the direct and inverse scattering problem for the wave equation with a homogeneous penetrable obstacle. In Section 3, we obtain the boundary integral equations in the time domain and Laplace domain. In Section 4, we parameterize the curve and demonstrate that the regularized boundary integral equation has at least one solution in the frequency domain for the case of a ball-shaped obstacle. In Section 5, we introduce the convolution quadrature method for time discretization. In Section 6, an iterative scheme is presented, and the nonlinear boundary integral equation is linearized. Full discretization is given. In Section 7, numerical experiments are presented to demonstrate the effectiveness of the proposed methods. In Section 8, the conclusion of this paper with some future work. Detailed proof of eigenvalues of single-layer operator for unit circle is in Appendix A.

2. Problem Formulation

Let ${\Omega }_{-}$ be a bounded open set in ${\mathbb{R}}^2$, and ${\Omega }_{+}={\mathbb{R}}^2\setminus {\overline{\Omega }}_{-}$ denote the complement of the scatter ${\Omega }_{-}$, with the common interface denoted by $\Gamma$. Let $\nu$ denote the unit outward normal to ${\Omega }_{-}$. Given an incident wave field $u^{id}$ in free space, the scattering produced by the obstacle can be computed as follows [23]:

$$\left\{\begin{array}{cc}{\nabla }^{2}u(\mathit{x},t)-u_{tt}(\mathit{x},t)=0,\hfill & \mathrm{in}{\Omega }_{+}\times (0,+\infty )\hfill \\ \nabla ·({\kappa }_0\nabla v(\mathit{x},t))-{\kappa }_1{v}_{tt}(\mathit{x},t)=0,\hfill & \mathrm{in}{\Omega }_{-}\times (0,+\infty )\hfill \\ v(\mathit{x},t)-u(\mathit{x},t)=u^{id}(\mathit{x},t),\hfill & \mathrm{on}\Gamma \times (0,+\infty )\hfill \\ {\kappa }_0{\partial }_{\nu }v(\mathit{x},t)-{\partial }_{\nu }u(\mathit{x},t)={\partial }_{\nu }{u}^{id}(\mathit{x},t),\hfill & \mathrm{on}\Gamma \times (0,+\infty ).\hfill \end{array}\right.$$

(1)

We assume the support of $u^{id}$ at time $t=0$ shall not intersect with the boundary of the obstacle $\Gamma$. This assumption guarantees vanishing initial conditions for the scattered field $u(\mathit{x},t)$ and the total field inside the obstacle $v(\mathit{x},t)$ with their derivatives, respectively:

$$u(·,0)=u_{\nu }(·,0)=0;v(·,0)=v_{\nu }(·,0)=0.$$

The physical parameters ${\kappa }_0$ and ${\kappa }_1$ are defined, respectively, by $1/\rho$ and $1/c^2\rho$, where $\rho$ is the density and c is the speed of propagation of sound. In this paper we only consider the situation where the obstacle is homogeneous and penetrable such that the parameters ${\kappa }_0$ and ${\kappa }_1$ are set to be constants. Thus, in the interior domain, the total field satisfies the equation:

$${\nabla }^{2}v(\mathit{x},t)-c^{-2}{v}_{tt}(\mathit{x},t)=0\mathrm{in}{\Omega }_{-}\times (0,+\infty ).$$

Consequently, Equation (1) will be reduced to a system of equations as [23]

$$\left\{\begin{array}{cc}{\nabla }^{2}u(\mathit{x},t)-u_{tt}(\mathit{x},t)=0,\hfill & \mathrm{in}{\Omega }_{+}\times (0,+\infty )\hfill \\ {\nabla }^{2}v(\mathit{x},t)-c^{-2}{v}_{tt}(\mathit{x},t)=0,\hfill & \mathrm{in}{\Omega }_{-}\times (0,+\infty )\hfill \\ v(\mathit{x},t)-u(\mathit{x},t)=u^{id}(\mathit{x},t),\hfill & \mathrm{on}\Gamma \times (0,+\infty )\hfill \\ {\kappa }_0{\partial }_{\nu }v(\mathit{x},t)-{\partial }_{\nu }u(\mathit{x},t)={\partial }_{\nu }{u}^{id}(\mathit{x},t),\hfill & \mathrm{on}\Gamma \times (0,+\infty )\hfill \end{array}\right.$$

(2)

for which the incident wave satisfies

$$\left\{\begin{array}{cc}{u}_{tt}^{id}-{\nabla }^2{u}^{id}=f,\hfill & \mathrm{in}{\mathbb{R}}^2\times (0,\infty )\hfill \\ u^{id}(·,0)={\partial }_{\nu }{u}^{id}(·,0)=0,\hfill & \mathrm{in}{\mathbb{R}}^2\hfill \end{array}\right.$$

where the source term $f(·,t)$ should not intersect with the obstacle.

The Inverse Problem. For a suitable constant $R>0$, we define $B_R=\{\mathit{x}\in {\mathbb{R}}^2:|\mathit{x}|<R\}$ such that ${\overline{\Omega }}_{-}\subset B_R$. Let the boundary $\partial B_R$ be the observation curve. Using the incident wave emitted from a launch position ${\mathit{x}}_{\mathbf{0}}$ and the scattered field received at observation curve $u(\mathit{x},t),\mathit{x}\in \partial B_R,t\in [0,T]$, our aim is to locate the obstacle and predict the shape of it.

3. From Time Domain to Laplace Domain

The retarded single-layer potential with density $g(\mathit{x},t)$ in two dimensions is defined as follows [31]:

$$\begin{array}{c}\hfill (\mathcal{S}\ast g)(\mathit{x},t;c):={\int }_0^t{\int }_{\Gamma }k(\mathit{x}-\mathit{y},t-\tau ;c)g(\mathit{y},\tau )\mathrm{d}s(\mathit{y})\mathrm{d}\tau \\ \hfill t\in (0,\infty ),\mathit{x}\in {\mathbb{R}}^2\setminus \Gamma \end{array}$$

(3)

where

$$k(t,r;c)={\displaystyle \frac{H(t-c^{-1}r)}{2\pi \sqrt{t^2-c^{-2}{r}^2}}}$$

and H is the Heaviside function. The trace of a function u to the boundary $\Gamma$ from the interior and exterior of $\Gamma$ will be denoted ${\gamma }^{-}u$ and ${\gamma }^{+}u$, respectively. The normal derivatives from inside and outside are ${\partial }_{\nu }^{-}u$ and ${\partial }_{\nu }^{+}u$. Averages are denoted as follows [34]:

$$\{\{\gamma u\}\}:={\displaystyle \frac{1}{2}}({\gamma }^{-}u+{\gamma }^{+}u);\{\{{\partial }_{\nu }u\}\}:={\displaystyle \frac{1}{2}}({\partial }_{\nu }^{-}u+{\partial }_{\nu }^{+}u).$$

On the boundary, we define the following two operators:

$$\begin{array}{cc}\hfill & \mathcal{V}\ast g:=\{\{\gamma (\mathcal{S}\ast g)\}\}={\gamma }^{-}(\mathcal{S}\ast g)={\gamma }^{+}(\mathcal{S}\ast g)\hfill \\ \hfill & {\mathcal{K}}^t\ast g:=\{\{{\partial }_{\nu }(\mathcal{S}\ast g)\}\}.\hfill \end{array}$$

The definitions above imply the jump relation of potentials [34]:

$${\partial }_{\nu }^{\pm }(\mathcal{S}\ast g)=\mp {\displaystyle \frac{1}{2}}g+{\mathcal{K}}^t\ast g.$$

In this work, the scattered field $u(\mathit{x},t)$ and the interior total field $v(\mathit{x},t)$ in Equation (2) can be represented by a single layer potential as follows, with densities $\varphi$ and $\lambda$, respectively.

$$\begin{array}{cc}\hfill & u(\mathit{x},t)=({\mathcal{S}}_1\ast \varphi )(\mathit{x},t;1),t\in (0,\infty ),\mathit{x}\in {\Omega }_{+}\hfill \\ \hfill & v(\mathit{x},t)=({\mathcal{S}}_2\ast \lambda )(\mathit{x},t;c),t\in (0,\infty ),\mathit{x}\in {\Omega }_{-}.\hfill \end{array}$$

Thus we have

$$\begin{array}{cc}\hfill & {\mathcal{V}}_1\ast \varphi =\{\{\gamma ({\mathcal{S}}_1\ast \varphi )\}\}{\mathcal{V}}_2\ast \lambda =\{\{\gamma ({\mathcal{S}}_2\ast \lambda )\}\}\hfill \\ \hfill & {\mathcal{K}}_1^t\ast \varphi =\{\{{\partial }_{\nu }({\mathcal{S}}_1\ast \varphi )\}\}{\mathcal{K}}_2^t\ast \lambda =\{\{{\partial }_{\nu }({\mathcal{S}}_2\ast \lambda )\}\}.\hfill \end{array}$$

From the boundary conditions and jump relations [34], we obtain the time domain boundary integral equations:

$$\left(\begin{array}{cc}{\mathcal{V}}_2& -{\mathcal{V}}_1\\ {\kappa }_0({\displaystyle \frac{1}{2}}I+{\mathcal{K}}_2^t)& {\displaystyle \frac{1}{2}}I-{\mathcal{K}}_1^t\end{array}\right)\ast \left(\begin{array}{c}\lambda \\ \varphi \end{array}\right)=\left(\begin{array}{c}\gamma u^{id}\\ {\partial }_{\upsilon }{u}^{id}\end{array}\right).$$

(4)

Now, we consider the corresponding Laplace-domain problem as follows:

$$\left\{\begin{array}{cc}{\nabla }^2\widehat{u}(s)-s^2\widehat{u}(s)=0,\hfill & \mathrm{in}{\Omega }_{+}\hfill \\ {\nabla }^2\widehat{v}(s)-{({\displaystyle \frac{s}{c}})}^2\widehat{v}(s)=0,\hfill & \mathrm{in}{\Omega }_{-}\hfill \\ {\gamma }^{-}\widehat{v}(s)-{\gamma }^{+}\widehat{u}(s)=\gamma {\widehat{u}}^{id},\hfill & \mathrm{on}\Gamma \hfill \\ {\kappa }_0{\partial }_{\nu }^{-}\widehat{v}(s)-{\partial }_{\nu }^{+}\widehat{u}(s)={\partial }_{\nu }{\widehat{u}}^{id},\hfill & \mathrm{on}\Gamma \hfill \end{array}\right.$$

(5)

$\widehat{u}$ and $\widehat{v}$ are the Laplace transforms of $u(\mathit{x},t)$ and $v(\mathit{x},t)$, respectively, while ${\widehat{u}}^{id}$ is the Laplace transform of the income wave $u^{id}(\mathit{x},t)$. We first recall the fundamental solution for the operator ${\nabla }^2-s^2$ in two dimensions [34]:

$$E(\mathit{x},\mathit{y};s)={\displaystyle \frac{\mathrm{i}}{4}}{H}_0^{(1)}(\mathrm{i}s|\mathit{x}-\mathit{y}|)$$

where $H_0^{(1)}$ denotes the zero-order Hankel function of the first kind.

Consider the layer potential and integral operators associated with ${\nabla }^2-s^2$ for $s\in {\mathbb{C}}_{+}$. Given the density function $\psi$, for arbitrary $\mathit{x}\in {\mathbb{R}}^2\setminus \Gamma$, the single layer potential has the expression in the frequency domain [34]:

$$(\mathrm{S}\psi )(\mathit{x}):={\int }_{\Gamma }E(\mathit{x},\mathit{y};s)\psi (\mathit{y})\mathrm{d}{s}_{\mathit{y}},\mathit{x}\in {\mathbb{R}}^2\setminus \Gamma .$$

Consequently, on the boundary, we can define two boundary integral operators [34]:

$$(\mathrm{V}\psi )(\mathit{x}):={\int }_{\Gamma }E(\mathit{x},\mathit{y};s)\psi (\mathit{y})\mathrm{d}{s}_{\mathit{y}},\mathit{x}\in \Gamma$$

$$({\mathrm{K}}^t\psi )(\mathit{x}):={\int }_{\Gamma }{\partial }_{\nu (x)}E(\mathit{x},\mathit{y};s)\psi (\mathit{y})\mathrm{d}{s}_{\mathit{y}},\mathit{x}\in \Gamma .$$

Using single layer potential for the problem (5), we assume that the exterior and interior solutions can be represented by

$$\begin{array}{ccc}\hfill & \widehat{u}(s)={\int }_{\Gamma }E(\mathit{x},\mathit{y};s)\widehat{\varphi }(\mathit{y})\mathrm{d}{s}_{\mathit{y}},\hfill & \hfill \mathit{x}\in {\Omega }_{+}\\ \hfill & \widehat{v}(s)={\int }_{\Gamma }E(\mathit{x},\mathit{y};{\displaystyle \frac{s}{c}})\widehat{\lambda }(\mathit{y})\mathrm{d}{s}_{\mathit{y}},\hfill & \hfill \mathit{x}\in {\Omega }_{-}\end{array}$$

with densities $\widehat{\varphi }$ and $\widehat{\lambda }$, respectively. Thus, on the boundary, we can define

$$\begin{array}{cc}\hfill & ({\mathrm{V}}_1\widehat{\varphi })(\mathit{x}):={\int }_{\Gamma }E(\mathit{x},\mathit{y};s)\widehat{\varphi }(\mathit{y})\mathrm{d}{s}_{\mathit{y}}={\displaystyle \frac{\mathrm{i}}{4}}{\int }_{\Gamma }{H}_0^{(1)}(\mathrm{i}s|\mathit{x}-\mathit{y}|)\widehat{\varphi }(\mathit{y})\mathrm{d}\mathit{y},\mathit{x}\in \Gamma \hfill \\ \hfill & ({\mathrm{V}}_2\widehat{\lambda })(\mathit{x}):={\int }_{\Gamma }E(\mathit{x},\mathit{y};s/c)\widehat{\lambda }(\mathit{y})\mathrm{d}{s}_{\mathit{y}}={\displaystyle \frac{\mathrm{i}}{4}}{\int }_{\Gamma }{H}_0^{(1)}(\mathrm{i}{\displaystyle \frac{s}{c}}|\mathit{x}-\mathit{y}|)\widehat{\lambda }(\mathit{y})\mathrm{d}\mathit{y},\mathit{x}\in \Gamma \hfill \end{array}$$

and,

$$\begin{array}{cc}\hfill ({\mathrm{K}}_1^t\widehat{\varphi })(\mathit{x})& :={\int }_{\Gamma }{\partial }_{\nu (\mathit{x})}E(\mathit{x},\mathit{y};s)\widehat{\varphi }(\mathit{y})\mathrm{d}{s}_{\mathit{y}}\hfill \\ \hfill & ={\displaystyle \frac{s}{4}}{\int }_{\Gamma }{H}_1^{(1)}(\mathrm{i}s|\mathit{x}-\mathit{y}|){\displaystyle \frac{(\mathit{x}-\mathit{y})·\nu (\mathit{x})}{|\mathit{x}-\mathit{y}|}}\widehat{\varphi }(\mathit{y})\mathrm{d}\mathit{y},\hfill & \hfill \mathit{x}\in \Gamma \end{array}$$

$$\begin{array}{cc}\hfill ({\mathrm{K}}_2^t\widehat{\lambda })(\mathit{x})& :={\int }_{\Gamma }{\partial }_{\nu (\mathit{x})}E(\mathit{x},\mathit{y};s/c)\widehat{\lambda }(\mathit{y})\mathrm{d}{s}_{\mathit{y}}\hfill \\ \hfill & ={\displaystyle \frac{s/c}{4}}{\int }_{\Gamma }{H}_1^{(1)}(\mathrm{i}{\displaystyle \frac{s}{c}}|\mathit{x}-\mathit{y}|){\displaystyle \frac{(\mathit{x}-\mathit{y})·\nu (\mathit{x})}{|\mathit{x}-\mathit{y}|}}\widehat{\lambda }(\mathit{y})\mathrm{d}\mathit{y},\hfill & \hfill \mathit{x}\in \Gamma \end{array}$$

where $H_1^{(1)}$ is the Hankel function of the first kind of order one.

In the end, from the boundary conditions and jump relations [23], we obtain the Laplace domain boundary integral equations as follows:

$$\left[\begin{array}{cc}{\mathrm{V}}_2& -{\mathrm{V}}_1\\ {\kappa }_0({\displaystyle \frac{\mathrm{I}}{2}}+{\mathrm{K}}_2^t)& {\displaystyle \frac{\mathrm{I}}{2}}-{\mathrm{K}}_1^t\end{array}\right]\left[\begin{array}{c}\widehat{\lambda }\\ \widehat{\varphi }\end{array}\right]=\left[\begin{array}{c}\gamma {\widehat{u}}^{id}\\ {\partial }_{\nu }{\widehat{u}}^{id}\end{array}\right].$$

(6)

4. Parameterization of the Curve

For brevity, let $\sigma =1or2$; if correspondingly, ${\kappa }_{\sigma }=sor{\displaystyle \frac{s}{c}}$. Now we can define new operators for some density $\phi$:

$$\begin{array}{ccc}\hfill & ({\mathrm{V}}_{\sigma }\phi )(\mathit{x})={\displaystyle \frac{\mathrm{i}}{4}}{\int }_{\Gamma }{H}_0^{(1)}(\mathrm{i}{\kappa }_{\sigma }|\mathit{x}-\mathit{y}|)\phi (\mathit{y})\mathrm{d}\mathit{y},\hfill & \hfill \mathit{x}\in \Gamma \\ \hfill & ({\mathrm{K}}_{\sigma }^t\phi )(\mathit{x})={\displaystyle \frac{{\kappa }_{\sigma }}{4}}{\int }_{\Gamma }{H}_1^{(1)}(\mathrm{i}{\kappa }_{\sigma }|\mathit{x}-\mathit{y}|){\displaystyle \frac{(\mathit{x}-\mathit{y})·\nu (\mathit{x})}{|\mathit{x}-\mathit{y}|}}\phi (\mathit{y})\mathrm{d}\mathit{y},\hfill & \hfill \mathit{x}\in \Gamma .\end{array}$$

We assume that the boundary $\Gamma$ is an analytic curve with the parametric form [35,36]

$$\Gamma =\{z(t)|(z_1(t),z_2(t)):0\le t<2\pi \}$$

where $z:\mathbb{R}\to {\mathbb{R}}^2$ is analytic and $2\pi$ period with $|z^{\prime }(t)|>0$ for all t. The normal derivative is represented by $n(t)=(z_2^{\prime }(t),-z_1^{\prime }(t))$, while the outward unit normal derivative is denoted by $\nu (t)={\displaystyle \frac{n(t)}{|z^{\prime }(t)|}}$. The parameterized integral operators are denoted by ${\mathrm{V}}_{\sigma }$ and ${\mathrm{K}}_{\sigma }$, i.e.,

$$({\mathrm{V}}_{\sigma }\phi )(t)={\int }_0^{2\pi }{h}_{\sigma }(t,\tau )\phi (\tau )\mathrm{d}\tau$$

$$({\mathrm{K}}_{\sigma }^t\phi )(t)={\int }_0^{2\pi }{k}_{\sigma }(t,\tau )\phi (\tau )\mathrm{d}\tau$$

where

$$\begin{array}{cc}\hfill & h_{\sigma }(t,\tau )={\displaystyle \frac{\mathrm{i}}{4}}{H}_0^{(1)}(\mathrm{i}{\kappa }_{\sigma }|z(t)-z(\tau )|)|z^{\prime }(\tau )|\hfill \\ \hfill & k_{\sigma }(t,\tau )={\displaystyle \frac{{\kappa }_{\sigma }}{4}}{H}_1^{(1)}(\mathrm{i}{\kappa }_{\sigma }|z(t)-z(\tau )|){\displaystyle \frac{(z(t)-z(\tau ))·\nu (t)}{|z(t)-z(\tau )|}}|z^{\prime }(\tau )|.\hfill \end{array}$$

Firstly, we split the kernel $k_{\sigma }(t,\tau )$ into [36]

$$k_{\sigma }(t,\tau )=k_{\sigma }^1(t,\tau )ln(4{sin}^2{\displaystyle \frac{t-\tau }{2}})+k_{\sigma }^2(t,\tau )$$

with

$$k_{\sigma }^1(t,\tau )={\displaystyle \frac{\mathrm{i}{\kappa }_{\sigma }}{4\pi }}|z^{\prime }(\tau )|{\displaystyle \frac{(z(t)-z(\tau ))·\nu (t)}{|z(t)-z(\tau )|}}{J}_1(\mathrm{i}{\kappa }_{\sigma }|z(t)-z(\tau )|).$$

Accordingly,

$$\begin{array}{cc}\hfill ({\mathrm{K}}_{\sigma }^t\phi )(t)& =({\mathrm{K}}_{\sigma }^{t,1}\phi )(t)+({\mathrm{K}}_{\sigma }^{t,2}\phi )(t)\hfill \\ \hfill & :={\int }_0^{2\pi }{k}_{\sigma }^1(t,\tau )ln(4{sin}^2{\displaystyle \frac{t-\tau }{2}})\phi (\tau )\mathrm{d}\tau +{\int }_0^{2\pi }{k}_{\sigma }^2(t,\tau )\phi (\tau )\mathrm{d}\tau .\hfill \end{array}$$

(7)

Similarly, the kernel $h_{\sigma }(t,\tau )$ can be written in the form [36]

$$h_{\sigma }(t,\tau )=h_{\sigma }^1(t,\tau )ln(4{sin}^2{\displaystyle \frac{t-\tau }{2}})+h_{\sigma }^2(t,\tau )$$

with analytic function

$$h_{\sigma }^1(t,\tau )=-{\displaystyle \frac{1}{4\pi }}|z^{\prime }(\tau )|J_0(\mathrm{i}{\kappa }_{\sigma }|z(t)-z(\tau )|)$$

• where $J_0$ is known as Bessel function of order zero.
• Correspondingly,

$$\begin{array}{cc}\hfill ({\mathrm{V}}_{\sigma }\phi )(t)& =({\mathrm{V}}_{\sigma }^1\phi )(t)+({\mathrm{V}}_{\sigma }^2\phi )(t)\hfill \\ \hfill & :={\int }_0^{2\pi }{h}_{\sigma }^1(t,\tau )ln(4{sin}^2{\displaystyle \frac{t-\tau }{2}})\phi (\tau )\mathrm{d}\tau +{\int }_0^{2\pi }{h}_{\sigma }^2(t,\tau )\phi (\tau )\mathrm{d}\tau .\hfill \end{array}$$

(8)

Let ${\mathrm{H}}^p[0,2\pi ],p\ge 0$ denote the space of $2\pi -$ periodic functions $u:\mathbb{R}\to \mathbb{C}$ equipped with the norm [36]

$${\Vert u\Vert }_p^2:=\sum _{m-\infty }^{\infty }{(1+m^2)}^p{|{\widehat{u}}_m|}^2<\infty$$

where

$${\widehat{u}}_m={\displaystyle \frac{1}{2\pi }}{\int }_0^{2\pi }u(t)e^{\mathrm{i}mt}\mathrm{d}t,m=0,\pm 1,\pm 2,\cdots$$

are the Fourier coefficients of u.

Theorem 1. 

The operators ${\mathrm{K}}_{\sigma }^t$ and ${\mathrm{V}}_{\sigma }$ are both compact operators from ${\mathrm{H}}^p[0,2\pi ]$ to ${\mathrm{H}}^p[0,2\pi ]$.

Proof of Theorem 1. 

Using (Theorem 12.15 [36]) and referring to [35], we can deduce that operators ${\mathrm{K}}_{\sigma }^{t,1}$ and ${\mathrm{V}}_{\sigma }^1$ are bounded from ${\mathrm{H}}^p[0,2\pi ]$ to ${\mathrm{H}}^{p+1}[0,2\pi ]$ for all $p\ge 0$. Consequently, both operators are compact from ${\mathrm{H}}^p[0,2\pi ]$ to ${\mathrm{H}}^p[0,2\pi ]$.

Since the kernel functions $k_{\sigma }^2(t,r)$ and $h_{\sigma }^2(t,r)$ are analytic, it follows from (Theorem 8.13 [36]) and (Theorem A.45 [37]) that the operator ${\mathrm{K}}_{\sigma }^{t,2}$ and ${\mathrm{V}}_{\sigma }^2$ from ${\mathrm{H}}^p[0,2\pi ]$ to ${\mathrm{H}}^{p+r}[0,2\pi ]$ are bounded for all integers $r\ge 0$ and arbitrary $p\ge 0$. Thus, we can deduce that the operator ${\mathrm{K}}_{\sigma }^t$ is compact from ${\mathrm{H}}^p[0,2\pi ]$ to ${\mathrm{H}}^p[0,2\pi ]$.

Since the operator ${\mathrm{V}}_{\sigma }$ is bounded from ${\mathrm{H}}^p[0,2\pi ]$ to ${\mathrm{H}}^{p+1}[0,2\pi ]$, the operator ${\mathrm{V}}_{\sigma }$ is also compact from ${\mathrm{H}}^p[0,2\pi ]$ to ${\mathrm{H}}^p[0,2\pi ]$. □

Let $\mathrm{B}={\kappa }_0({\displaystyle \frac{\mathrm{I}}{2}}+{\mathrm{K}}_2^t)$; assuming $-{\displaystyle \frac{1}{2}}$ is not the eigenvalue of operator ${\mathrm{K}}_2^t$, then the operator $\mathrm{B}$ has a bounded inverse ${\mathrm{B}}^{-1}$. We can then turn the equation:

$$\left[\begin{array}{cc}{\mathrm{V}}_2& -{\mathrm{V}}_1\\ {\kappa }_0({\displaystyle \frac{\mathrm{I}}{2}}+{\mathrm{K}}_2^t)& {\displaystyle \frac{\mathrm{I}}{2}}-{\mathrm{K}}_1^t\end{array}\right]\left[\begin{array}{c}\widehat{\lambda }\\ \widehat{\varphi }\end{array}\right]=\left[\begin{array}{c}\gamma {\widehat{u}}^{id}\\ {\partial }_{\nu }{\widehat{u}}^{id}\end{array}\right]$$

(9)

into the form

$$[{\mathrm{V}}_1+{\mathrm{V}}_2{\mathrm{B}}^{-1}({\displaystyle \frac{\mathrm{I}}{2}}-{\mathrm{K}}_1^t)]\widehat{\lambda }={\mathrm{V}}_2{\mathrm{B}}^{-1}{\partial }_{\nu }{\widehat{u}}^{id}-\gamma {\widehat{u}}^{id}$$

(10)

Let $\mathcal{A}:=[{\mathrm{V}}_1+{\mathrm{V}}_2{\mathrm{B}}^{-1}({\displaystyle \frac{\mathrm{I}}{2}}-{\mathrm{K}}_1^t)]$, $f:={\mathrm{V}}_2{\mathrm{B}}^{-1}{\partial }_{\nu }{\widehat{u}}^{id}-\gamma {\widehat{u}}^{id}$. We can reformulate the above equation as

$$\mathcal{A}\widehat{\lambda }=f.$$

(11)

Since ${\mathrm{V}}_{\sigma }$ is a compact operator, Equation (11) is an integral equation of the first kind. For simplicity here, we only prove the existence of the solution of the regularized equation for the unit circle.

Let $z(t)=(cost,sint)$, $z(\tau )=(cos\tau ,sin\tau )$ and $f_m(t):=e^{imt}$, $(t\in \mathrm{R}$ and $m\in \mathrm{Z})$. We use the incident waves (40) to ensure that ${\partial }_{\nu }{\widehat{u}}^{id}$, $\gamma {\widehat{u}}^{id}$ both belong to ${\mathrm{H}}^p[0,2\pi ]$ for some $p>0$; thus, we have the following theorem:

Theorem 2. 

For the trigonometric monomials $f_m(t):=e^{\mathrm{i}mt},m=0,\pm 1,\pm 2,\cdots$, the eigenvalues ${\lambda }_{\pm m}(m=0,1,2,\cdots )$ of operator ${\mathrm{V}}_{\sigma }$ are tende to zero as $|m|\to +\infty$. Let ${\mathrm{H}}_{\mathrm{M}}$ denote the linear subspace of ${\mathrm{H}}^p[0,2\pi ]$ generated by $\{e^{\pm \mathrm{i}mt},m=0,\pm 1\cdots ,\pm \mathrm{M}\}$ and ${\mathrm{P}}_{\mathrm{M}}$ denote the regularizer operator from ${\mathrm{H}}^p[0,2\pi ]$ to ${\mathrm{H}}_M$. Then, for Equation (11), there exists a solution ${\widehat{\lambda }}_{\mathrm{M}}$ satisfies the regularized equation: ${\mathrm{P}}_{\mathrm{M}}\mathcal{A}\widehat{\lambda }={\mathrm{P}}_{\mathrm{M}}f$.

Remark 1. 

Detailed proof of these eigenvalues; refer to Appendix A.

Proof of Theorem 2. 

With the eigenfunctions $f_m(t):=e^{\mathrm{i}mt}$, the eigenvalues are as follows:

for $m=0$,

$$\begin{array}{cc}\hfill {\lambda }_0=& \sum _{p=1}^{\infty }{(-1)}^{p+1}{\displaystyle \frac{k_{\sigma }^{2p}}{4^p{(p!)}^2}}\sum _{j=0}^{p-1}{C}_{2p}^j{\displaystyle \frac{{(-1)}^j}{p-j}}\hfill \\ \hfill & +{\displaystyle \frac{1}{4}}\{{\displaystyle \frac{2\mathrm{i}}{\pi }}ln{\displaystyle \frac{\mathrm{i}{k}_{\sigma }}{2}}+{\displaystyle \frac{2\mathrm{i}}{\pi }}C+1\}\sum _{p=1}^{\infty }{\displaystyle \frac{k_{\sigma }^{2p}}{4^p{(p!)}^2}}{\displaystyle \frac{(2p)!}{p!}}\hfill \\ \hfill & +{\displaystyle \frac{-\mathrm{i}}{2\pi }}\sum _{p=1}^{\infty }{\displaystyle \frac{k_{\sigma }^{2p}}{4^p{(p!)}^2}}{\displaystyle \frac{(2p)!}{p!}}\Psi (p)+\{-ln{\displaystyle \frac{\mathrm{i}{k}_{\sigma }}{2}}-E_C+{\displaystyle \frac{\mathrm{i}\pi }{2}}\}.\hfill \end{array}$$

• Here, we define $\Psi (0):=0$, $\Psi (p):={\displaystyle \sum _{m=1}^p}{\displaystyle \frac{1}{m}},p=1,2,\cdots$, $E_C$ denote Euler’s constant.
• For $m=\pm 1,\pm 2,\cdots$,

$$\begin{array}{cc}\hfill {\lambda }_{(\pm m)}& ={\displaystyle \frac{1}{2m}}\hfill \\ \hfill & +{\displaystyle \frac{{(-1)}^{p+1}}{2}}\sum _{p=m+1}^{\infty }{\displaystyle \frac{k_{\sigma }^{2p}}{4^p{(p!)}^2}}[\sum _{j=0}^{p-m-1}{C}_{2p}^j{\displaystyle \frac{{(-1)}^j}{2p}}{\displaystyle \frac{{(-1)}^j}{p-m-j}}+\sum _{j=0}^{p+m-1}{C}_{2p}^j{\displaystyle \frac{{(-1)}^j}{p+m-j}}]\hfill \\ \hfill & +\{{\displaystyle \frac{-\mathrm{i}}{2\pi }}ln{\displaystyle \frac{\mathrm{i}{k}_{\sigma }}{2}}-{\displaystyle \frac{\mathrm{i}}{2\pi }}{E}_C-{\displaystyle \frac{1}{4}}\}\sum _{p=m}^{\infty }{(-1)}^{m+1}{\displaystyle \frac{k_{\sigma }^{2p}}{4^p{(p!)}^2}}{\displaystyle \frac{(2p)!}{(p+m)!}}\hfill \\ \hfill & +{\displaystyle \frac{\mathrm{i}}{2\pi }}\sum _{p=m}^{\infty }{(-1)}^{m+1}{\displaystyle \frac{k_{\sigma }^{2p}}{4^p{(p!)}^2}}{\displaystyle \frac{(2p)!}{(p+m)!}}\Psi (p)\hfill \end{array}$$

For ${\lambda }_0$, we first consider the series

$$\sum _{p=1}^{\infty }{(-1)}^{p+1}{\displaystyle \frac{k_{\sigma }^{2p}}{4^p{(p!)}^2}}\sum _{j=0}^{p-1}{C}_{2p}^j{\displaystyle \frac{{(-1)}^j}{p-j}}$$

It is easy to show that

$$\begin{array}{c}\hfill |\sum _{p=1}^{\infty }{(-1)}^{p+1}{\displaystyle \frac{k_{\sigma }^{2p}}{4^p{(p!)}^2}}\sum _{j=0}^{p-1}{C}_{2p}^j{\displaystyle \frac{{(-1)}^j}{p-j}}|\le \sum _{p=1}^{\infty }{\displaystyle \frac{p|k_{\sigma }{|}^{2p}}{4^p{(p!)}^2}}{C}_{2p}^p\end{array}$$

where the series ${\displaystyle \sum _{p=1}^{\infty }}{\displaystyle \frac{p|k_{\sigma }{|}^{2p}}{4^p{(p!)}^2}}{C}_{2p}^p$ is absolutely convergent. Similarly, for the second and the third part of ${\lambda }_0$, the series

$$\sum _{p=1}^{\infty }{\displaystyle \frac{k_{\sigma }^{2p}}{4^p{(p!)}^2}}{\displaystyle \frac{(2p)!}{p!}};\sum _{p=1}^{\infty }{\displaystyle \frac{k_{\sigma }^{2p}}{4^p{(p!)}^2}}{\displaystyle \frac{(2p)!}{p!}}\Psi (p)$$

• are both absolutely convergent.
• For ${\lambda }_{\pm m},m=\pm 1,\pm 2,\cdots ,$ we consider the second part of ${\lambda }_{\pm m}$:

$$\begin{array}{cc}\hfill \sum _{p=m+1}^{\infty }{\mathcal{U}}_p:={\displaystyle \frac{{(-1)}^{p+1}}{2}}\sum _{p=m+1}^{\infty }{\displaystyle \frac{k_{\sigma }^{2p}}{4^p{(p!)}^2}}& [\sum _{j=0}^{p-m-1}{C}_{2p}^j{\displaystyle \frac{{(-1)}^j}{2p}}{\displaystyle \frac{{(-1)}^j}{p-m-j}}+\hfill \\ \hfill & \sum _{j=0}^{p+m-1}{C}_{2p}^j{\displaystyle \frac{{(-1)}^j}{p+m-j}}].\hfill \end{array}$$

If we assume that ${\mathcal{U}}_p=0,1\le p\le m$, it is easy to compute that the series ${\displaystyle \sum _{p=1}^{\infty }}{\mathcal{U}}_p$ is absolutely convergent:

$$\begin{array}{c}\hfill \sum _{p=1}^{\infty }|{\mathcal{U}}_p|\le {\displaystyle \frac{1}{2}}\sum _{p=m+1}^{\infty }{\displaystyle \frac{|k_{\sigma }{|}^{2p}}{4^p{(p!)}^2}}[(p-m)C_{2p}^p+(p+m)C_{2p}^p]\\ \hfill \le \sum _{p=m+1}^{\infty }{\displaystyle \frac{|k_{\sigma }{|}^{2p}}{4^p{(p!)}^2}}(p+m)C_{2p}^p\end{array}$$

Similarly, for the second and third parts of ${\lambda }_{\pm m}$, it is easy to show that the series

$$\begin{array}{c}\hfill \sum _{p=m}^{\infty }{\mathcal{V}}_p:=\{{\displaystyle \frac{-\mathrm{i}}{2\pi }}ln{\displaystyle \frac{\mathrm{i}{k}_{\sigma }}{2}}-{\displaystyle \frac{\mathrm{i}}{2\pi }}{E}_C-{\displaystyle \frac{1}{4}}\}\sum _{p=m}^{\infty }{(-1)}^{m+1}{\displaystyle \frac{k_{\sigma }^{2p}}{4^p{(p!)}^2}}{\displaystyle \frac{(2p)!}{(p+m)!}}\end{array}$$

and

$$\begin{array}{c}\hfill \sum _{p=m}^{\infty }{\mathcal{W}}_p:={\displaystyle \frac{\mathrm{i}}{2\pi }}\sum _{p=m}^{\infty }{(-1)}^{m+1}{\displaystyle \frac{k_{\sigma }^{2p}}{4^p{(p!)}^2}}{\displaystyle \frac{(2p)!}{(p+m)!}}\Psi (p)\end{array}$$

satisfy the inequality (assuming ${\mathcal{V}}_p=0$, $p=1,\cdots m-1$; ${\mathcal{W}}_p=0$, $p=1\cdots ,m-1$):

$$\sum _{p=1}^{\infty }|{\mathcal{V}}_p|\le \mathfrak{C}\sum _{p=1}^{\infty }{\displaystyle \frac{|k_{\sigma }{|}^{2p}}{4^p{(p!)}^2}}{\displaystyle \frac{(2p)!}{(p+m)!}}$$

$$\sum _{p=1}^{\infty }|{\mathcal{W}}_p|\le {\displaystyle \frac{1}{2\pi }}\sum _{p=1}^{\infty }{\displaystyle \frac{|k_{\sigma }{|}^{2p}}{4^p{(p!)}^2}}{\displaystyle \frac{(2p)!}{(p+m)!}}\Psi (p)$$

for some constant $\mathfrak{C}\in \mathbb{R}$. It is clear that the series ${\displaystyle \sum _{p=1}^{\infty }}{\mathcal{V}}_p$ and ${\displaystyle \sum _{p=1}^{\infty }}{\mathcal{W}}_p$ are both absolutely convergent. Accordingly, ${\lambda }_{\pm m}$ tends to zero, as $m\to \infty$. □

5. Time Discretization

We use the convolution quadrature method [29,31] to handle the time discretization of the boundary integral Equation (4). Briefly, the right-hand side of expression (3) can be written as

$$A({\partial }_{\mathrm{t}})g:={\int }_0^{t}w(t-\tau ;c)g(\tau )\mathrm{d}\tau :=(w\ast g)(t),$$

(12)

where w is a parameter-dependent integral operator:

$$(w(t-\tau ;c)g(\tau ))(\mathit{x}):={\int }_{\Gamma }k(t-\tau ,\mathit{x}-\mathit{y};c)g(\mathit{y},\tau )\mathrm{d}s(\mathit{y}),\mathit{x}\in \Gamma .$$

(13)

The Laplace transform of w is defined by

$$\mathrm{W}(s;c)={\int }_0^{\infty }w(t)e^{-st}\mathrm{d}t.$$

Its specific expression has the form

$$(\mathrm{W}(s;c)g)(\mathit{x})={\int }_{\Gamma }K(|\mathit{x}-\mathit{y}|,{\displaystyle \frac{s}{c}})g(\mathit{y})\mathrm{d}{s}_{\mathit{y}},\mathit{x}\in \Gamma ,$$

(14)

where $K(|\mathit{x}-\mathit{y}|,s/c)={\displaystyle \frac{\mathrm{i}}{4}}{H}_0^{(1)}({\displaystyle \frac{\mathrm{i}s}{c}}|\mathit{x}-\mathit{y}|)$.

In order to discretize the time convolution, we split the time interval $[0,T]$ into $N+1$ time steps of equal length $\Delta t=T/N$ with $t_n=n\Delta t$,$n=0,1,\cdots ,N$. Then, using the convolution quadrature method [29,31], we replace the continuous convolution operator $A({\partial }_{\mathrm{t}})$ at the discrete times $t_n$ by the discrete convolution operator, for $n=0,1,\cdots ,N$,

$$(A({\partial }_{\mathrm{t}}^{\Delta t})g^{\Delta t})(\mathit{x},t_n):=(w\ast g)(t_n)=\sum _{j=0}^n{w}_{n-j}^{\Delta t}(\mathrm{W})g^{\Delta t}(t_j).$$

(15)

The convolution weights $w_n^{\Delta t}(\mathrm{W})$ are defined below:

$$\mathrm{W}({\displaystyle \frac{{\rm Y}(\zeta )}{\Delta t}})=\sum _{n=0}^{\infty }{w}_n^{\Delta t}{\zeta }^n,|\zeta |<1$$

(16)

where ${\rm Y}(\zeta )={\displaystyle \frac{1}{2}}({\zeta }^2-4\zeta +3)$ is the quotient of the generating polynomials of the BDF2 scheme. As recommended in [31], the convolution weights $w_n^{\Delta t}(\mathrm{W})$ have their representation as a contour integral

$$w_j^{\Delta t}(\mathrm{W})={\displaystyle \frac{1}{2\pi i}}{\mathit{\oint }}_C{\displaystyle \frac{\mathrm{W}({\rm Y}(\zeta )/\Delta t)}{{\zeta }^{j+1}}}\mathrm{d}\zeta ,$$

(17)

where C can be chosen as a circle centered at the origin of radius $\varsigma <1$. Using the trapezoidal rule, the approximate convolution weights are given by

$$w_j^{\Delta t,\varsigma }(\mathrm{W}):={\displaystyle \frac{{\varsigma }^{-j}}{N+1}}\sum _{l=0}^N\mathrm{W}(s_l;c){\zeta }_{N+1}^{lj},$$

(18)

where ${\zeta }_{N+1}=e^{{\displaystyle \frac{2\pi \mathrm{i}}{N+1}}}$, $s_l={\displaystyle \frac{{\rm Y}(\varsigma {\zeta }_{N+1}^{-l})}{\Delta t}}$. Substituting the definition of $w_j^{\Delta t,\varsigma }$ in (15), we obtain the expression as follows:

$$(A({\partial }_{\mathrm{t}}^{\Delta t})g^{\Delta t,\varsigma })(\mathit{x},t_n):={\displaystyle \frac{{\varsigma }^{-n}}{N+1}}\sum _{l=0}^N(\mathrm{W}(s_l;c){\widehat{g}}_l)(\mathit{x}){\zeta }_{N+1}^{nl},n=0,1,\cdots ,N$$

(19)

where

$${\widehat{g}}_l:=\sum _{j=0}^N{\varsigma }^j{g}_j{\zeta }_{N+1}^{-lj}.$$

(20)

Using the above results, for the time domain boundary integral to Equation (4), we can define new discrete convolution quadrature operators as follows:

$$\left\{\begin{array}{c}(A({\mathrm{V}}_2^{\Delta t}){\lambda }^{\Delta t,\varsigma })(\mathit{x},t_n):={\displaystyle \frac{{\varsigma }^{-n}}{N+1}}\sum _{l=0}^N({\mathrm{V}}_2(s_l;c){\widehat{\lambda }}_l)(\mathit{x}){\zeta }_{N+1}^{nl};\hfill \\ (A({\mathrm{V}}_1^{\Delta t}){\varphi }^{\Delta t,\varsigma })(\mathit{x},t_n):={\displaystyle \frac{{\varsigma }^{-n}}{N+1}}\sum _{l=0}^N({\mathrm{V}}_1(s_l;1){\widehat{\varphi }}_l)(\mathit{x}){\zeta }_{N+1}^{nl};\hfill \\ (A({\mathrm{K}}_2^{\Delta t}){\lambda }^{\Delta t,\varsigma })(\mathit{x},t_n):={\displaystyle \frac{{\varsigma }^{-n}}{N+1}}\sum _{l=0}^N({\partial }_{\nu (x)}{\mathrm{K}}_2^t(s_l;c){\widehat{\lambda }}_l)(\mathit{x}){\zeta }_{N+1}^{nl};\hfill \\ (A({\mathrm{K}}_1^{\Delta t}){\varphi }^{\Delta t,\varsigma })(\mathit{x},t_n):={\displaystyle \frac{{\varsigma }^{-n}}{N+1}}\sum _{l=0}^N({\partial }_{\nu (x)}{\mathrm{K}}_1^t(s_l;1){\widehat{\varphi }}_l)(\mathit{x}){\zeta }_{N+1}^{nl}.\hfill \end{array}\right.$$

(21)

The time-discrete problem is given as follows: find ${\varphi }_j={\varphi }^{\Delta t,\varsigma }(·,t_j)$, ${\lambda }_j={\lambda }^{\Delta t,\varsigma }(·,t_j)$ such that

$$\left\{\begin{array}{c}(A({\mathrm{V}}_2^{\Delta t}){\lambda }^{\Delta t,\varsigma })(\mathit{x})-(A({\mathrm{V}}_1^{\Delta t}){\varphi }^{\Delta t,\varsigma })(\mathit{x})=u_n^{id}(\mathit{x})\hfill \\ {\displaystyle \frac{1}{2}}({\kappa }_0{\lambda }_n(\mathit{x})+{\varphi }_n(\mathit{x}))+{\kappa }_0(A({\mathrm{K}}_2^{\Delta t}){\lambda }^{\Delta t,\varsigma })(\mathit{x})-(A({\mathrm{K}}_1^{\Delta t}){\varphi }^{\Delta t,\varsigma })(\mathit{x})={\partial }_{\nu }{u}_n^{id}(\mathit{x})\hfill \end{array}\right.$$

$$n=1,2,\cdots ,N,\mathit{x}\in \Gamma ,$$

(22)

where $u_n^{id}(\mathit{x})$ and ${\partial }_{\nu }{u}_n^{id}(\mathit{x})$ are some approximations to $u^{id}(\mathit{x},t_n)$ and ${\partial }_{\nu (\mathit{x})}{u}^{id}(\mathit{x},t_n)$, respectively. For the above Equation (22), applying the discrete Fourier transform to both sides, we finally get the following decoupled boundary integral equations

$$\left\{\begin{array}{c}({\mathrm{V}}_2(s_l;c)\widehat{{\lambda }_l})(\mathit{x})-({\mathrm{V}}_1(s_l;1)\widehat{{\varphi }_l})(\mathit{x})={\widehat{u}}_l^{id}(\mathit{x})\hfill \\ {\displaystyle \frac{1}{2}}({\kappa }_0{\widehat{\lambda }}_l(\mathit{x})+{\widehat{\varphi }}_l(\mathit{x}))+{\kappa }_0({\partial }_{\nu (\mathit{x})}{\mathrm{K}}_2^t(s_l;c){\widehat{\lambda }}_l)(\mathit{x})-({\partial }_{\nu (\mathit{x})}{\mathrm{K}}_1^t(s_l;1){\widehat{\varphi }}_l)(\mathit{x})\hfill \\ ={\partial }_{\nu }{\widehat{u}}_l^{id}(\mathit{x})\hfill \end{array}\right.$$

$$l=0,1,,\cdots ,N,\mathit{x}\in \Gamma$$

(23)

where ${\widehat{u}}_l^{id}$ and ${\partial }_{\nu }{\widehat{u}}_l^{id}$ are the scaled discrete Fourier transform

$${\widehat{u}}_l^{id}:=\sum _{n=0}^N{\varsigma }^n{u}_n^{id}{\zeta }_{N+1}^{-ln};{\partial }_{\nu }{\widehat{u}}_l^{id}:=\sum _{n=0}^N{\varsigma }^n{\partial }_{\nu }{u}_n^{id}{\zeta }_{N+1}^{-ln}.$$

(24)

Similarly, ${\widehat{\lambda }}_l$ and ${\widehat{\varphi }}_l$ are the scaled discrete Fourier transform

$${\widehat{\lambda }}_l:=\sum _{n=0}^N{\varsigma }^n{\lambda }_n{\zeta }_{N+1}^{-ln};{\widehat{\varphi }}_l:=\sum _{n=0}^N{\varsigma }^n{\varphi }_n{\zeta }_{N+1}^{-ln}.$$

(25)

Ultimately, we obtain the densities ${\lambda }_j$ and ${\varphi }_j$ by applying the inverse transform

$${\lambda }_j:={\displaystyle \frac{{\varsigma }^{-j}}{N+1}}\sum _{l=0}^N{\widehat{\lambda }}_l{\zeta }_{N+1}^{jl};{\varphi }_j:={\displaystyle \frac{{\varsigma }^{-j}}{N+1}}\sum _{l=0}^N{\widehat{\varphi }}_l{\zeta }_{N+1}^{jl}$$

(26)

Because the scaled discrete Fourier transform of $u_n(·)=u(·,t_n)$ has the expression

$${\widehat{u}}_l(\mathit{x})={\int }_{\Gamma }K(|\mathit{x}-\mathit{y}|,s_l){\widehat{\varphi }}_l(\mathit{y})d{s}_{\mathit{y}}:=(W{L}_{\Gamma }(s_l;c){\widehat{\varphi }}_l)(\mathit{x})$$

(27)

for $l=0,\cdots ,N$, $\mathit{x}\in {\mathbb{R}}^2\setminus \overline{\Omega }$, the scattering wave $u_n$ is given by

$$u_n:={\displaystyle \frac{{\varsigma }^{-n}}{N+1}}\sum _{l=0}^N{\widehat{u}}_n{\zeta }_{N+1}^{nl}$$

(28)

6. The Inverse Scheme

6.1. Nonlinear Integral Equations

For brevity, we use the symbols as follows:

$$\left\{\begin{array}{c}{A}_l(p_D,{\widehat{\lambda }}_l)(\theta ):={\displaystyle \frac{\mathrm{i}}{4}}{\int }_0^{2\pi }{H}_0^{(1)}({\displaystyle \frac{\mathrm{i}{s}_l}{c}}|p_D(\theta )-p_D(\mathit{\eta })|)Gr(\mathit{\eta }){\widehat{\lambda }}_l(\mathit{\eta })\mathrm{d}\mathit{\eta }\hfill \\ A_l(p_D,{\widehat{\varphi }}_l)(\theta ):={\displaystyle \frac{\mathrm{i}}{4}}{\int }_0^{2\pi }{H}_0^{(1)}(\mathrm{i}{s}_l|p_D(\theta )-p_D(\mathit{\eta })|)Gr(\mathit{\eta }){\widehat{\varphi }}_l(\mathit{\eta })\mathrm{d}\mathit{\eta }\hfill \end{array}\right.$$

and

$$\left\{\begin{array}{c}{B}_l(p_D,{\widehat{\lambda }}_l)(\theta ):={\displaystyle \frac{\mathrm{i}}{4}}{\int }_0^{2\pi }{\partial }_{\nu (\theta )}{H}_0^{(1)}({\displaystyle \frac{\mathrm{i}{s}_l}{c}}|p_D(\theta )-p_D(\mathit{\eta })|)Gr(\mathit{\eta }){\widehat{\lambda }}_l(\mathit{\eta })\mathrm{d}\mathit{\eta }\hfill \\ B_l(p_D,{\widehat{\varphi }}_l)(\theta ):={\displaystyle \frac{\mathrm{i}}{4}}{\int }_0^{2\pi }{\partial }_{\nu (\theta )}{H}_0^{(1)}(\mathrm{i}{s}_l|p_D(\theta )-p_D(\mathit{\eta })|)Gr(\mathit{\eta }){\widehat{\varphi }}_l(\mathit{\eta })\mathrm{d}\mathit{\eta }\hfill \end{array}\right.$$

where ${\widehat{\lambda }}_l(\mathit{\eta })={\widehat{\lambda }}_l(p(\mathit{\eta })),{\widehat{\varphi }}_l(\mathit{\eta })={\widehat{\varphi }}_l(p(\mathit{\eta }))$ and $Gr(\mathit{\eta })=(r^2(\mathit{\eta })+({\displaystyle \frac{\mathrm{d}r}{\mathrm{d}\mathit{\eta }}}(\mathit{\eta }){)}^2{)}^{1/2}.$ Then, we can get the parameterized expressions of Equation (23) as

$$\left\{\begin{array}{c}{A}_l(p_D,{\widehat{\lambda }}_l)(\theta )-A_l(p_D,{\widehat{\varphi }}_l)(\theta )={\widehat{\omega }}_l(\theta )\hfill \\ {\displaystyle \frac{1}{2}}({\kappa }_0{\widehat{\lambda }}_l(\theta )+{\widehat{\varphi }}_l(\theta ))+{\kappa }_0{B}_l(p_D,{\widehat{\lambda }}_l)(\theta )-B_l(p_D,{\widehat{\varphi }}_l)(\theta )={\widehat{\omega }}_l^{\prime }(\theta )\hfill \end{array}\right.$$

(29)

• where ${\widehat{\omega }}_l(\theta )={\widehat{u}}_l^{id}(p_{\Gamma }(\theta )),{\widehat{\omega }}_l^{\prime }(\theta )={\partial }_{\nu (\theta )}{\widehat{u}}_l^{id}(p_{\Gamma }(\theta )).$
• For simplicity, we use symbols to represent the retarded single layer potential (3)

$$(S{L}_{\Gamma }g)(\mathit{x},t;c):=(S\ast g)(\mathit{x},t;c).$$

(30)

Let $x\in \partial B_R$, assuming the observation curve is denoted by $\partial B_R$; then, we obtain the data equations

$$(S{L}_{\Gamma }\varphi )(\mathit{x},t_n)=u(\mathit{x},t_n),n=0,\cdots ,N,\mathit{x}\in \partial B_R.$$

(31)

Using the convolution quadrature method, a system of s-domain data equations are obtained (using the notation similiar as (27))

$$(W{L}_{\Gamma }(s_l;1){\widehat{\varphi }}_l)(\mathit{x})={\widehat{u}}_l(\mathit{x}),l=0,\cdots ,N,\mathit{x}\in \partial B_R.$$

(32)

Consequently, the parameterized operator $W{L}_{\Gamma }(s_l)$ is

$$A{L}_l(p_D,p_B,{\widehat{\varphi }}_l)(\mathit{\xi })={\displaystyle \frac{\mathrm{i}}{4}}{\int }_0^{2\pi }{H}_0^{(1)}(\mathrm{i}{s}_l|p_B(\mathit{\xi })-p_D(\mathit{\eta })|)Gr(\mathit{\eta }){\widehat{\varphi }}_l(\mathit{\eta })\mathrm{d}\mathit{\eta }.$$

(33)

And, we obtain the nolinear integral equation as

$$A{L}_l(p_D,p_B,{\widehat{\varphi }}_l)(\mathit{\xi })={\widehat{\omega }}_l^{\partial B_R}(\mathit{\xi }),l=0,\cdots ,N,$$

(34)

where ${\widehat{\omega }}_l^{\partial B_R}(\mathit{\xi })={\widehat{u}}_l^{id}(p_B(\mathit{\xi }))$. The linearization of Equation (34) with respect to $p_D$ requires the Frechet derivative of the operator $A{L}_l$, that is [31]

$$(A{L}_l^{\prime }(p_D,p_B,{\widehat{\varphi }}_l)q)(\mathit{\xi })$$

$$\begin{array}{cc}\hfill & =-{\displaystyle \frac{s_l}{4}}{\int }_0^{2\pi }{H}_1^{(1)}(\mathrm{i}{s}_l|p_B(\mathit{\xi })-p_D(\mathit{\eta })|){\displaystyle \frac{(p_B(\mathit{\xi })-p_D(\mathit{\eta }))·q(\mathit{\eta })}{|p_B(\mathit{\xi })-p_D(\mathit{\eta })|}}{\widehat{\varphi }}_l(\mathit{\eta })\mathrm{d}\mathit{\eta }\hfill \\ \hfill & =-{\displaystyle \frac{s_l}{4}}{\int }_0^{2\pi }{\displaystyle \frac{H_1^{(1)}(\mathrm{i}{s}_l|p_B(\mathit{\xi })-p_D(\mathit{\eta })|)}{|p_B(\mathit{\xi })-p_D(\mathit{\eta })|}}\hfill \end{array}$$

$$\begin{array}{cc}& \hfill \left[\right(b_1+R\cos \mathit{\xi }-c_1-r(\mathit{\eta })\cos \mathit{\eta }\left)\right(\Delta c_1+\Delta r(\mathit{\eta })\cos \mathit{\eta })\\ & \hfill +(b_2+R\sin \mathit{\xi }-c_2-r(\mathit{\eta })\sin \mathit{\eta })(\Delta c_2+\Delta r(\mathit{\eta })\sin \mathit{\eta })]{\widehat{\varphi }}_l(\mathit{\eta })\mathrm{d}\mathit{\eta }\end{array}$$

(35)

where

$$q(\mathit{\eta })=(\Delta c_1,\Delta c_2)+\Delta r(\mathit{\eta })(cos\mathit{\eta },sin\mathit{\eta }).$$

Then, the linearization of (34) leads to

$$A{L}_l^{\prime }(p_D,p_B,{\widehat{\varphi }}_l)q=f_l,$$

(36)

where

$$f_l={\widehat{\omega }}_l^{\partial B_R}-A{L}_l(p_D,p_B,{\widehat{\varphi }}_l).$$

Analogously to [38], we use the same relative error estimator for a stopping criterion:

$$E_{ll}:={\displaystyle \frac{\Vert {\widehat{\omega }}_l^{\partial B_R-A{L}_l(p_D^{ll},p_B,{\widehat{\varphi }}_l)}{\Vert }_{L^2}}{\Vert {\widehat{\omega }}_l^{\partial B_R}{\Vert }_{L^2}}}\le ϵ,$$

where $ϵ$ is a user-specified small positive constant on the noise level, and $P_D^{ll}$ is the $ll$th approximation of the boundary $\Gamma$. Iterative procedure for this scattering problem is similar to [31,38].

6.2. Fully Discretization

Next, we describe the full discretization of (29) and (36). Let ${\mathit{\eta }}_j^{\tilde{n}}:=\pi j/\tilde{n},j=0,\cdots ,2\tilde{n}-1$, ${\mathit{\xi }}_i^{\overline{n}}:=\pi i/\overline{n},i=0,\cdots ,2\overline{n}-1$ be equidistant sets of quadrature knots. Seting ${\widehat{\omega }}_{l,i}^{\tilde{n}}={\widehat{\omega }}_l({\mathit{\eta }}_i^{\tilde{n}})$, ${\widehat{\omega }}_{l,i}^{\prime \tilde{n}}={\widehat{\omega }}_l^{\prime }({\mathit{\eta }}_i^{\tilde{n}})$ and ${\widehat{\lambda }}_{l,j}^{(\tilde{n})}={\widehat{\lambda }}_l({\mathit{\eta }}_j^{(\tilde{n})})$, ${\widehat{\varphi }}_{l,j}^{(\tilde{n})}={\widehat{\varphi }}_l({\mathit{\eta }}_j^{(\tilde{n})})$ for $i,j=0,\cdots ,2\tilde{n}-1$ and $l=0,\cdots ,N$. Using the Nystrom method [36], we can get the full discretization of (29)

$$\begin{array}{cc}\hfill \sum _{j=0}^{2\tilde{n}-1}(R_{|i-j|}^{\tilde{n}}{K}_1^{(l)}& ({\mathit{\eta }}_i^{(\tilde{n})},{\mathit{\eta }}_j^{(\tilde{n})};c)+{\displaystyle \frac{\pi }{\tilde{n}}}{K}_2^{(l)}({\mathit{\eta }}_i^{(\tilde{n})},{\mathit{\eta }}_j^{(\tilde{n})};c)){\widehat{\lambda }}_{l,j}^{\tilde{n}}-\hfill \\ \hfill & \sum _{j=0}^{2\tilde{n}-1}(R_{|i-j|}^{\tilde{n}}{K}_1^{(l)}({\mathit{\eta }}_i^{(\tilde{n})},{\mathit{\eta }}_j^{(\tilde{n})};1)+{\displaystyle \frac{\pi }{\tilde{n}}}{K}_2^{(l)}({\mathit{\eta }}_i^{(\tilde{n})},{\mathit{\eta }}_j^{(\tilde{n})};1)){\widehat{\varphi }}_{l,j}^{\tilde{n}}={\widehat{\omega }}_{l,i}^{\tilde{n}}\hfill \end{array}$$

$$\begin{array}{cc}\hfill \sum _{j=0}^{2\tilde{n}-1}& (R_{|i-j|}^{\tilde{n}}{L}_1^{(l)}({\mathit{\eta }}_i^{(\tilde{n})},{\mathit{\eta }}_j^{(\tilde{n})};c)+{\displaystyle \frac{\pi }{\tilde{n}}}{L}_2^{(l)}({\mathit{\eta }}_i^{(\tilde{n})},{\mathit{\eta }}_j^{(\tilde{n})};c)){\widehat{\lambda }}_{l,j}^{\tilde{n}}-\hfill \\ \hfill & \sum _{j=0}^{2\tilde{n}-1}(R_{|i-j|}^{\tilde{n}}{L}_1^{(l)}({\mathit{\eta }}_i^{(\tilde{n})},{\mathit{\eta }}_j^{(\tilde{n})};1)+{\displaystyle \frac{\pi }{\tilde{n}}}{L}_2^{(l)}({\mathit{\eta }}_i^{(\tilde{n})},{\mathit{\eta }}_j^{(\tilde{n})};1)){\widehat{\varphi }}_{l,j}^{\tilde{n}}+{\displaystyle \frac{1}{2}}({\kappa }_0{\widehat{\lambda }}_{l,j}^{(\tilde{n})}+{\widehat{\varphi }}_{l,j}^{(\tilde{n})})={\widehat{\omega }}_{l,i}^{\prime \tilde{n}}\hfill \end{array}$$

for $i,j=0,\cdots ,2\tilde{n}-1$, $l=0,\cdots ,N$, where

$$R_j^{(\tilde{n})}=-{\displaystyle \frac{2\pi }{\tilde{n}}}\sum _{m=1}^{\tilde{n}-1}{\displaystyle \frac{1}{m}}cos{\displaystyle \frac{mj\pi }{\tilde{n}}}-{\displaystyle \frac{{(-1)}^j\pi }{{\tilde{n}}^2}},$$

$$K_1^{(l)}(\theta ,\mathit{\eta };c)=-{\displaystyle \frac{1}{4\pi }}{J}_0({\displaystyle \frac{\mathrm{i}{s}_l}{c}}|p_D(\theta )-p_D(\mathit{\eta })|),$$

$$K_2^{(l)}(\theta ,\mathit{\eta };c)={\displaystyle \frac{\mathrm{i}}{4}}{H}_0^{(1)}({\displaystyle \frac{\mathrm{i}{s}_l}{c}}|p_D(\theta )-p_D(\mathit{\eta })|)-K_1^{(l)}(\theta ,\mathit{\eta };c)ln(4{sin}^2{\displaystyle \frac{\theta -\mathit{\eta }}{2}}).$$

Meanwhile, we can decuce the diagonal term as

$$K_2^{(l)}(\mathit{\eta },\mathit{\eta })\{{\displaystyle \frac{\mathrm{i}}{4}}-{\displaystyle \frac{E_C}{2\pi }}-{\displaystyle \frac{1}{2\pi }}ln({\displaystyle \frac{\mathrm{i}{s}_l{G}_r(\mathit{\eta })}{2}})\},$$

with the Euler constant $E_C=0.57721\cdots$. Similarly,

$$L_1^{(l)}(\theta ,\mathit{\eta };c)={\displaystyle \frac{\mathrm{i}{s}_l}{4\pi c}}{\displaystyle \frac{x_2^{\prime }(\theta )(x_1(\theta )-x_1(\mathit{\eta }))-x_1^{\prime }(\theta )(x_2(\theta )-x_2(\mathit{\eta }))}{|x^{\prime }(\theta )|}}{\displaystyle \frac{J_1(\frac{\mathrm{i}{s}_l}{c}|x(\theta )-x(\mathit{\eta })|)}{|x(\theta )-x(\mathit{\eta })|}}$$

$$L_2^{(l)}(\theta ,\mathit{\eta };c)={\displaystyle \frac{\mathrm{i}}{4}}{\partial }_{\nu }(\theta )H_0^{(1)}({\displaystyle \frac{\mathrm{i}{s}_l}{c}}|x(\theta )-x(\mathit{\eta })|)-L_1^{(l)}(\theta ,\mathit{\eta };c)ln4{sin}^2{\displaystyle \frac{\theta -\mathit{\eta }}{2}},$$

and the diagonal term is

$$L_2^{(l)}(\mathit{\eta },\mathit{\eta };c)=-{\displaystyle \frac{1}{4\pi }}{\displaystyle \frac{1}{|x^{\prime }{(\mathit{\eta })|}^3}}\{x_2^{″}(\mathit{\eta })x_1^{\prime }(\mathit{\eta })-x_1^{″}(\mathit{\eta })x_2^{\prime }(\mathit{\eta })\}.$$

Next, we discretize the linearized Equation (36) to obtain the update by using least squares with Tikhonov regularization [37,38]. We choose the space of trigonometric polynomials of the form [31] to approximate the radial function r and its update $\Delta r$

$$\Delta r(\mathit{\eta })=\sum _{m=0}^M{\alpha }_{m}cosm\mathit{\eta }+\sum _{m=1}^M{\beta }_{m}sinm\mathit{\eta },$$

where the integer $M>1$ is the truncation number. We can reformulate Equation (36) as follows; for details of the fully discretization, we refer to [31].

$$T_1^{(l)}(\mathit{\xi },\mathit{\eta })=-{\displaystyle \frac{s_l}{4}}{\displaystyle \frac{H_1^{(1)}(\mathrm{i}{s}_l|p_B(\mathit{\xi })-p_D(\mathit{\eta })|)}{|p_B(\mathit{\xi })-p_D(\mathit{\eta })|}}(b_1+Rcos\mathit{\xi }-c_1-r(\mathit{\eta })cos\mathit{\eta }){\widehat{\varphi }}_l(\mathit{\eta }),$$

$$T_2^{(l)}(\mathit{\xi },\mathit{\eta })=-{\displaystyle \frac{s_l}{4}}{\displaystyle \frac{H_1^{(1)}(\mathrm{i}{s}_l|p_B(\mathit{\xi })-p_D(\mathit{\eta })|)}{|p_B(\mathit{\xi })-p_D(\mathit{\eta })|}}(b_2+Rsin\mathit{\xi }-c_2-r(\mathit{\eta })sin\mathit{\eta }){\widehat{\varphi }}_l(\mathit{\eta }),$$

$$\begin{array}{c}\hfill T_{3,m}^{(l)}(\mathit{\xi },\mathit{\eta })=-{\displaystyle \frac{s_l}{4}}{\displaystyle \frac{H_1^{(1)}(\mathrm{i}{s}_l|p_B(\mathit{\xi })-p_D(\mathit{\eta })|)}{|p_B(\mathit{\xi })-p_D(\mathit{\eta })|}}[(b_1-c_1)cos\mathit{\eta }+(b_2-c_2)sin\mathit{\eta }\\ \hfill +Rcos(\mathit{\xi }-\mathit{\eta })-r(\mathit{\eta })]cosm\mathit{\eta }{\widehat{\varphi }}_l(\mathit{\eta }),\end{array}$$

$$\begin{array}{c}\hfill T_{4,m}^{(l)}(\mathit{\xi },\mathit{\eta })=-{\displaystyle \frac{s_l}{4}}{\displaystyle \frac{H_1^{(1)}(\mathrm{i}{s}_l|p_B(\mathit{\xi })-p_D(\mathit{\eta })|)}{|p_B(\mathit{\xi })-p_D(\mathit{\eta })|}}[(b_1-c_1)cos\mathit{\eta }+(b_2-c_2)sin\mathit{\eta }\\ \hfill +Rcos(\mathit{\xi }-\mathit{\eta })-r(\mathit{\eta })]sinm\mathit{\eta }{\widehat{\varphi }}_l(\mathit{\eta }),\end{array}$$

Utilizing the trapezoidal rule, we obtain the discretized linear system from (33)–(36)

$$\begin{array}{c}\hfill G_1^{(l),\overrightarrow{c}}({\mathit{\xi }}_i^{(\overline{n})})\Delta c_1+G_2^{(l),\overrightarrow{c}}({\mathit{\xi }}_i^{(\overline{n})})\Delta c_2+\sum _{m=0}^M{\alpha }_m{G}_{3,m}^{(l),r}({\mathit{\xi }}_i^{(\overline{n})})+\\ \hfill \sum _{m=1}^M{\beta }_m{G}_{4,m}^{(l),r}({\mathit{\xi }}_i^{(\overline{n})})=f_l({\mathit{\xi }}_i^{(\overline{n})})\end{array}$$

(37)

to determine the real coefficients $\Delta c_1,\Delta c_2,{\alpha }_m$, and ${\beta }_m$, where

$$G_{kk}^{(l),\overrightarrow{c}}({\mathit{\xi }}_i^{(\overline{n})})={\displaystyle \frac{\pi }{\tilde{n}}}\sum _{j=0}^{2\tilde{n}-1}{T}_{kk}^{(l)}({\mathit{\xi }}_i^{(\tilde{n})},{\mathit{\eta }}_j),kk=1,2,$$

and

$$G_{kk,m}^{(l),r}({\mathit{\xi }}_i^{(\overline{n})})={\displaystyle \frac{\pi }{\tilde{n}}}\sum _{j=0}^{2\tilde{n}-1}{T}_{kk,m}^{(l)}({\mathit{\xi }}_i^{(\overline{n})},{\mathit{\eta }}_j),kk=3,4.$$

Since $2M+1\ll 2\overline{n}$, due to the ill-posedness, the Tikhonov regularization can be used to tackle the overdetermined system (37). Thus, the linear system (37) is turned into the following function [31,38]:

$$\begin{array}{c}\hfill \sum _{i=0}^{2\overline{n}-1}|G_1^{(l),\overrightarrow{c}}({\mathit{\xi }}_i^{(\overline{n})})\Delta c_1+G_2^{(l),\overrightarrow{c}}({\mathit{\xi }}_i^{(\overline{n})})\Delta c_2+\\ \hfill \sum _{m=0}^M{\alpha }_m{G}_{3,m}^{(l),r}({\mathit{\xi }}_i^{(\overline{n})})+\sum _{m=1}^M{\beta }_m{G}_{4,m}^{(l),r}({\mathit{\xi }}_i^{(\overline{n})})-f_l({\mathit{\xi }}_i^{\overline{n}}){|}^2\\ \hfill +\tilde{\lambda }(|\Delta c_1{|}^2+|\Delta c_2{|}^2+2\pi [{\alpha }_0^2+{\displaystyle \frac{1}{2}}\sum _{m=1}^M(1+m^2)({\alpha }_m^2+{\beta }_m^2)])\end{array}$$

(38)

with a positive regularization parameter $\tilde{\lambda }$ and $H^1$ penalty term. It is clear that the minimizer of above equation is the solution of the system

$$(\tilde{\lambda }\tilde{I}+\mathcal{R}({\tilde{G}}_l^{\ast }{\tilde{G}}_l))\tilde{\mathit{\xi }}=\mathcal{R}({\tilde{G}}_l^{\ast }{\tilde{f}}_l)$$

(39)

where

$${\tilde{G}}_l={(G_1^{(l),\overrightarrow{c}},G_2^{(l),\overrightarrow{c}},G_{3,0}^{(l),r},\cdots ,G_{3,M}^{(l),r},G_{4,1}^{(l),r},\cdots ,G_{4,M}^{(l),r})}_{2\overline{n}\times (2M+3)}$$

$$\tilde{\mathit{\xi }}={(\Delta c_1,\Delta c_2,{\alpha }_0,\cdots ,{\alpha }_M,{\beta }_1,\cdots ,{\beta }_M)}^T,$$

$$\tilde{I}=diag\{1,1,2\pi ,\pi (1+1^2),\cdots ,\pi (1+M^2),\cdots ,\pi (1+M^2)\}$$

$${\tilde{f}}_l={(f_l({\mathit{\xi }}_0^{(\overline{n})}),\cdots ,f_l({\mathit{\xi }}_{2\overline{n}-1}^{(\overline{n})}))}^T.$$

Consequently, the new approximation is obtained

$$p_D^{new}=(\overrightarrow{c}+\Delta \overrightarrow{c})+(r(\overrightarrow{\mathit{x}})+\Delta r(\overrightarrow{\mathit{x}}))\overrightarrow{\mathit{x}}$$

7. Numerical Experiments

For the numerical experiments, we considered the following income wave, defined by

$$u^{id}(x,t)=\left\{\begin{array}{cc}{\displaystyle \frac{sin[4(t-|\mathit{x}-{\mathit{x}}_{\mathbf{0}}{|)}^4]exp(-1.6(t-|\mathit{x}-{\mathit{x}}_{\mathbf{0}}{|-3)}^2)}{2\pi \sqrt{t^2-{|\mathit{x}-{\mathit{x}}_{\mathbf{0}}|}^2}}},\hfill & t>|\mathit{x}-{\mathit{x}}_{\mathbf{0}}|\hfill \\ 0,\hfill & t\le |\mathit{x}-{\mathit{x}}_{\mathbf{0}}|\hfill \end{array}\right.,$$

(40)

which is emitted from a single launch position ${\mathit{x}}_{\mathbf{0}}$ such that the incident wave clearly satisfies the two-dimensional time-domain wave equation. The scattered field data are numerically computed at 60 points, $\mathrm{i}$.e., $\overline{n}=30$. And 100 quadrature nodes are chosen for the direct problem and 64 quadrature nodes for the inverse problem. We construct the noisy data $v_{\sigma }(\mathrm{y},t)$ in the following way [38]: $v_{\sigma }=v(1+\delta \Theta )$, where $\Theta$ are random numbers ranging in $[-1,1]$, $\delta >0$ is the relative noise level. By using the Tikhonov regularization and $H^1$ penalty term, we will get the update $\mathit{\xi }$ from a scaled Newton step, i.e.,

$$\tilde{\mathit{\xi }}=\rho {(\tilde{\lambda }\tilde{I}+\mathcal{R}({\tilde{G}}_l^{\ast }{\tilde{G}}_l))}^{-1}\mathcal{R}({\tilde{G}}_l^{\ast }{\tilde{f}}_l),$$

where the scaling factor $\rho \ge 0$ is a fixed number throughout the iteration. Analogously to [31], the regularization parameters $\tilde{\lambda }$ in (39) are chosen as

$${\tilde{\lambda }}_{ll}={\parallel {\widehat{\omega }}_{l,\delta }^{\partial B_R}-A{L}_l(p^{ll},p_B,{\widehat{\varphi }}_l)\parallel }_{L^2},l=[ll/loop],ll=0,1,\cdots$$

Analogously to [31,38], we choose the update $\mathit{\xi }=0$ when the norm of the right hand ${\widehat{\omega }}_l(\theta )$ is less than the tolerance: $\tilde{\epsilon }={10}^{-6}$.

Numerical Examples. Inverse acoustic scattering problem for a homogeneous penetrable obstacle with a single launch position. We investigate the inverse scattering problem of reconstructing a homogeneous penetrable obstacle from the scattered-field data in the time domain, including the shape and the position. The reconstructions with different noise levels for the acorn-shaped, bean-shaped, peanut-shaped, and kite-shaped obstacles are shown in Figure 1, Figure 2, Figure 3, and Figure 4, respectively. The numerical results show that the reconstruction of a homogeneous penetrable obstacle deals with the angle of the incident wave, the position and radius of the initial guess, the shape of the obstacle, and the size of the noise level. These results show that the reconstruction is not sensitive to the initial guess or the launch position of the incident wave, although there will be a large error in reconstructing the shape of the obstacle at some cusp positions, and the error is within our control. Note that all the programs are written using Matlab.

In all the following figures, the exact boundary curves are expressed by solid lines while the reconstructed boundary curves are displayed by dashed lines $--$, and all the initial guesses are taken to be a circle with radius $r_0$, which is indicated by the dash-dotted lines $·-$. The position of incident waves is marked by an asterisk *. We choose $T=15$, the radius of the observation curve $R=2$, the center of the observation curve $(0,0)$, the scaling factor $\rho =0.9$, the cycle index for per $l,loop=6$, and the truncation $M=4$.

8. Conclusions

In this paper, we have studied the two-dimensional inverse acoustic scattering from a homogeneous penetrable obstacle. The challenge of the time domain inverse problem is due to the time dependence. The fact that we obtain a convolution equation suggests the possibility of using CQ methods for the time variable. Although we obtain expressions for the time-domain potentials, we will write all the equations in the frequency domain. However, it has to be understood that this method works in the time domain, and the solutions are computed in a time-stepping fashion. The fact that we use the operators in the frequency domain (which is a requisite of CQ) does not mean that we are going to solve at that domain and then invert the Laplace transform.

That scattered waves outside the obstacle and the total wave inside the obstacle are both represented by a single-layer potential, resulting in the boundary integral equation reformulating to an integral equation of the first kind. Thus, we only prove the existence of the solution of the boundary integral equation under regularization. But this is not conducive to accurate numerical analysis. This is one of the problems that we will solve in our follow-up work. A nonlinear integral equation method is developed for the inverse problem, and numerical examples are presented to demonstrate the effectiveness and stability of the method. In summary, the angle of the incident wave, the position and radius of the initial guess, the shape of the obstacle, and the size of the noise level all affect the reconstruction results. All the numerical experiment results show the method presented in this paper is very effective for retrieving the position of the obstacle, and there will be a large error in reconstructing the shape of the obstacle at some cusp positions, which is also related to the angle of the incident wave. For example, the results of reconstruction for kite-shaped obstacles do not perform as well as for bean-shaped obstacles. From the results of numerical experiments, we can also see that for a bean-shaped obstacle, even if we enhance the noise level to $20\%$, the method still works, while for a kite-shaped obstacle, the results are barely acceptable for a $1\%$ noise level. Similar results can be found in the literature [31], which uses the same method to reconstruct the position and shape of sound soft obstacles. In general, these results show that the reconstruction is not sensitive to the initial guess or the launch position of the incident wave. Future work includes the application to other inverse acoustic scattering problems for non-homogeneous penetrable obstacles and error analysis.