A Nyström Method for Scattering by a Two-layered Medium with a Rough Boundary

This paper is concerned with the well-posedness and the numerical method for the problems of scattering of time-harmonic acoustic waves in a two-layered medium in two dimensions. The two-layered medium is composed of two unbounded media with different physical properties and the interface between the two media is considered to be a planar surface. The boundary of the two-layered medium is assumed to be a rough surface, which is a non-local perturbation with a finite height from a planar surface. Such scattering problems occur in various scientific and engineering applications, such as ground-penetrating radar, seismic exploration, ocean exploration, photonic crystal, and diffraction by gratings. For an introduction and historical remarks, we refer to [14, 34, 16, 32, 35, 33].

There are many works concerning the well-posedness of the rough surface scattering problems for acoustic waves. The rough surface scattering problems with Dirichlet or impedance boundary conditions have been studied in [10, 37, 6, 7] by using the integral equation methods. In each of these works, the layer potential technique was applied to transform the scattering problem into an equivalent boundary integral equation. [36, 9, 11] considered the rough surface scattering problems by penetrable interfaces and inhomogeneous layers, using the integral equation methods. In [4, 3], the authors studied the rough surface scattering problem with a sound-soft boundary by employing the variational approach in the classical Sobolev space or the weighted Sobolev space. Moreover, the method in [4] was extended in [24] to study the scattering problem by an inhomogeneous layer of a finite height, where the Neumann or generalized impedance boundary condition was imposed on the lower boundary of the inhomogeneous layer. For more works on the well-posedness of the rough surface scattering problems for electromagnetic or elastic waves, we refer to [17, 18, 21, 28, 23].

Some numerical methods have also been developed for the rough surface scattering problems. In [29], the authors introduced the Nyström method for the second-kind integral equation defined on the real line. Based on this, numerical algorithms were proposed for the rough surface scattering problems; see [29] for the sound-soft case and [26] for the penetrable case. An adaptive finite element method with a perfectly matched layer (PML) was proposed in [13] for the wave scattering by periodic structures. In [39], the authors proposed the Nyström method for the scattering problem by penetrable diffraction gratings. In this method, a fast FFT-based algorithm developed in [38] was utilized for efficient computation of the quasi-periodic Green’s functions. In [5], the authors investigated the use of the PML to truncate the rough surface scattering problem and proved the linear rate of convergence for the proposed PML-based method.

In this paper, we consider the scattering problems in a two-layered medium, where the Dirichlet or impedance boundary condition is imposed on the rough boundary. First, we formulate the considered scattering problems as the boundary value problems and prove that each boundary value problem has a unique solution by utilizing the integral equation method associated with the two-layered Green function. Our proofs follow the ideas in [7, 10, 37], which are based on an integral equation theory on unbounded domains given in [8]. We note that different from [7, 10, 37], in this paper we use the two-layered Green function rather than the free-space fundamental solution to the Helmholtz equation in the proposed integral equation formulations, which is due to the presence of the two-layered medium with the planar interface. It is also worth noting that in the proofs of the uniqueness and existence results of this paper, an essential role is played by the investigation of the asymptotic properties of the two-layered Green function for small and large arguments. Second, based on the proposed integral equation formulations, we develop the Nyström method for numerically solving the considered boundary value problems, where the relevant integral equations are discretized by using the method given in [29]. With the aid of the convergence theory of the Nyström method given in [29], we establish the convergence results of our method with the convergence rates depending on the smoothness of the rough boundary. It should be noted that the asymptotic properties of the two-layered Green function obtained in this paper provide a theoretical foundation for our convergence results. Finally, numerical experiments are carried out to show the effectiveness of our Nyström method.

The rest of the paper is organized as follows. In Section 2, we introduce the considered scattering problems and formulate them as the boundary value problems. In Section 3, we study the properties of the two-layered Green function. Based on these properties, we establish the well-posedness of the considered boundary value problems in Section 4. Section 5 is devoted to the Nyström method for the considered boundary value problems. The convergence results and the numerical experiments of the Nyström method are also given in Section 5. Some concluding remarks are given in Section 6. In Appendixes A and B, we present the potential theory and the solvability of integral operators on the real line, respectively, associated with the two-layered Green function.

In this section, we introduce the mathematical models of the scattering problems. To this end, we give some notations, which will be used throughout the paper. Let $V\subset\mathbb{R}^{m}$ ($m=1,2$). We denote by $BC(V)$ the set of functions bounded and continuous on $V$, a Banach space under the norm $\|\phi\|_{\infty,V}:=\sup_{x\in V}|\phi(x)|$, and by $BUC(V)$ the closed subspace of $BC(V)$ containing functions that are bounded and uniformly continuous on $V$. We abbreviate $\|\cdot\|_{\infty,\mathbb{R}}$ by $\|\cdot\|_{\infty}$. For $0<\alpha\leq 1$, we denote by $C^{0,\alpha}(V)$ the Banach space of functions $\phi\in BC(V)$, which are uniformly Hölder continuous with exponent $\alpha$ and with norm $\|\cdot\|_{C^{0,\alpha}(V)}$ defined by $\|\phi\|_{C^{0,\alpha}(V)}:=\|\phi\|_{\infty,V}+\sup_{x,y\in V,x\neq y}[|\phi(x)-\phi(y)|/|x-y|^{\alpha}]$. We let $C^{1,\alpha}(\mathbb{R}):=\{\phi\in BC(\mathbb{R})\cap C^{1}(\mathbb{R}):\phi^{\prime}\in C^{0,\alpha}(\mathbb{R})\}$ be a Banach space under the norm $\|\phi\|_{C^{1,\alpha}(\mathbb{R})}:=\|\phi\|_{\infty}+\|\phi^{\prime}\|_{C^{0,\alpha}(\mathbb{R})}$. For any $a\in\mathbb{R}$, define $\Gamma_{a}:=\{(x_{1},a)\,:\,x_{1}\in\mathbb{R}\}$ and $U_{a}:=\left\{(x_{1},x_{2})\,:\,x_{1}\in\mathbb{R},x_{2}>a\right\}$. In particular, the notation $\Gamma_{0}$ denotes the plane $x_{2}=0$. Let $\mathbb{R}_{\pm}^{2}:=\{(x_{1},x_{2})\in\mathbb{R}^{2}\,:\,x_{2}\gtrless 0\}$ be the upper and lower half-spaces, respectively. For any $x,y\in\mathbb{R}^{2}$, let $x=(x_{1},x_{2})$ and $y=(y_{1},y_{2})$. For any $x\in\mathbb{R}^{2}$ with $x\neq 0$, let $\hat{x}:=x/|x|$ denote the direction of $x$. Define $\mathbb{S}^{1}_{\pm}:=\{x=(x_{1},x_{2})\in\mathbb{R}^{2}:|x|=1,x_{2}\gtrless 0\}$. Let $C(\Omega)$ represent the space of continuous functions on $\Omega$ and let $C^{i}(\Omega)$ represent the space of $C^{i}$-continuous functions on $\Omega$ for $i=1,2$.

The geometry of the scattering problems we consider is shown in Figure 2.1. Let $\mathbb{R}_{+}^{2}$ and $\mathbb{R}_{-}^{2}$ denote the homogeneous media above and below $\Gamma_{0}$, respectively. The wave numbers of the media in the upper and lower half-spaces are $k_{+}$ and $k_{-}$, respectively, with $k_{+},k_{-}>0$ and $k_{+}\neq k_{-}$. Define $n:=k_{-}/k_{+}$. Assume that a rough surface $\Gamma:=\left\{(x_{1},x_{2})\,:\,x_{2}=f(x_{1}),x_{1}\in\mathbb{R}\right\}$ is fully embeded in the lower half-space $\mathbb{R}^{2}_{-}$, where $f\in C^{1,1}(\mathbb{R})$ with $f_{+}:=\sup_{x\in\mathbb{R}}f(x)<0$. Let $f_{-}:=\inf_{x\in\mathbb{R}}f(x)$ and the Lipschitz constant $L:=\|f^{\prime}\|_{\infty}$. Define the domain $D:=\{(x_{1},x_{2})\,:\,x_{2}>f(x_{1})\}$.

Consider the scattering problems with time-harmonic incident waves in the domain $D$. In this paper, we assume that the incident wave $u^{i}$ is either a plane wave or a point-source wave. The reference wave $u^{0}$ is generated by the incident wave $u^{i}$ and the two-layered medium. The explicit expressions of the incident wave and its corresponding reference wave will be described later. Then the total field $u^{tot}=u^{0}+u^{s}$ is the sum of the reference wave $u^{0}$ and the scattered wave $u^{s}$, where $u^{s}$ satisfies the following Helmholtz equations

Moreover, we assume the total field $u^{tot}$ satisfies the following boundary conditions on the interface $\Gamma_{0}$, i.e.,

where ’+/-’ denote the limits from $\mathbb{R}^{2}_{+}$ and $\mathbb{R}^{2}_{-}$, respectively. Furthermore, the boundary condition imposed on $\Gamma$ is given by $\mathscr{B}(u^{tot})=0$ on $\Gamma$. Here, $\mathscr{B}$ denotes one of the following two boundary conditions:

where $\beta\in BC(\Gamma)$, $\nu(x)$ denotes the unit normal at $x\in\Gamma$ pointing out of $D$ and $\partial u^{tot}/\partial\nu$ denotes the normal derivative of $u^{tot}$ with respect to $\nu$.

To guarantee the uniqueness of the considered scattering problems, the scattered wave $u^{s}$ is required to satisfy a radiation condition. In contrast to the bounded obstacle scattering problems, which utilize the Sommerfeld radiation condition, $u^{s}$ needs to satisfy the so-called upward propagating radiation condition in $U_{0}$ with respect to $k_{+}$, that is,

for some $h>0$ and $\phi\in L^{\infty}(\Gamma_{h})$, where $\Phi_{k_{+}}(x,y):=\frac{i}{4}H_{0}^{(1)}(k_{+}|x-y|)$ with $x,y\in\mathbb{R}^{2}$ and $x\neq y$ is the free-space Green function for the Helmholtz equation $\Delta w+k^{2}_{+}w=0$ with the wave number $k_{+}$ and $H_{0}^{(1)}(t)$ with $t\in\mathbb{R}$ denotes the Hankel function of the first kind of order zero. We also need $u^{s}$ to satisfy the following boundedness condition

for some $\alpha\in\mathbb{R}$.

Furthermore, if $\Gamma$ is an impedance boundary, the scattered wave $u^{s}$ needs to satisfy that for some $\theta\in(0,1)$ and some constant $C_{\theta}>0$,

for $x\in D\backslash\overline{U_{b}}$, where $b=f_{+}/2$.

Now we describe the reference wave $u^{0}$ more specifically. The reference wave is the total field of the scattering problem in the two-layered medium without the rough surface $\Gamma$ and is generated by the incident wave $u^{i}$. In this paper, we consider two types of incident waves, that is, the plane wave and the point-source wave.

First, we describe the reference wave in the case when the incident wave $u^{i}$ is the plane wave $u_{pl}^{i}(x):=e^{ik_{+}x\cdot d}$, where $d:=(\cos(\theta_{d}),\sin(\theta_{d}))$, $\theta_{d}\in(\pi,2\pi)$. In this case, the reference wave $u^{0}=u^{0}_{pl}$ is given by (see, e.g., (2.13a) and (2.13b) in [31] or Section 4 in [27])

with

where $d^{r}=(\cos(\theta_{d}),-\sin(\theta_{d}))$ is the reflected direction, $d^{t}=n^{-1}(\cos(\theta_{d}),-i\mathcal{S}(\cos\theta_{d},n))$ and where $\mathcal{R}\left(\pi+\theta_{d}\right)$ and $\mathcal{T}\left(\pi+\theta_{d}\right)$ are called the reflection and transmission coefficients, respectively, with $\mathcal{R}$ and $\mathcal{T}$ defined by

Here, $S(z,a)$ with $z\in\mathbb{R}$ and $a>0$ is defined by

The definition of $\mathcal{S}$ gives that

In particular, if $|n^{-1}\cos(\theta_{d})|\leq 1$, then $d^{t}=(\cos(\theta_{d}^{t}),\sin(\theta_{d}^{t}))$ is the transmitted direction with $\theta_{d}^{t}\in[\pi,2\pi]$ satisfying $\cos(\theta_{d}^{t})=n^{-1}\cos(\theta_{d})$. It is easy to see that such reference wave $u^{0}$ satisfies the following conditions

Second, we describe the reference wave in the case when the incident wave $u^{i}$ is the point source wave $u_{ps}^{i}(x)$, where $u_{ps}^{i}(x)=\frac{i}{4}H_{0}^{(1)}(k_{+}|x-y|)$ if the source point $y\in\mathbb{R}^{2}_{+}$ and $u_{ps}^{i}(x)=\frac{i}{4}H_{0}^{(1)}(k_{-}|x-y|)$ if the source point $y\in D\cap\mathbb{R}^{2}_{-}$. In this case, the reference wave $u^{0}(x)=G(x,y)$, where $G(x,y)$ denotes the so-called two-layered Green function. Precisely, for any $y\in\mathbb{R}^{2}_{+}\cup\mathbb{R}^{2}_{-}$, the two-layered Green function $G(x,y)$ is the solution of the following scattering problem (see page 17 in [31])

where $k(x)=k_{\pm}$ for $x\in\mathbb{R}^{2}_{\pm}$, $\delta$ denotes the Dirac delta distribution, $\nu$ denotes the unit normal on $\Gamma_{0}$ pointing into $\mathbb{R}^{2}_{+}$ and $[\cdot]$ denotes the jump across the interface $\Gamma_{0}$. Here, (2.10) is called the Sommerfeld radiation condition. The explicit expression of $G(x,y)$ is given by (see, e.g., [31, formula (2.27)])

where $G_{\mathcal{R}}(x,y)$ and $G_{\mathcal{Q}}(x,y)$ are given by

Now the above scattering problems can be formulated as the following two boundary value problems (DBVP) and (IBVP) for the scattered wave $u^{s}$.

Dirichlet Boundary Value Problem (DBVP). Given $g\in BC(\Gamma)$, determine $u^{s}\in\mathscr{R}_{d}(D)$ such that:

(i) $u^{s}$ is a solution of the Helmholtz equations in (2.1);

(ii) $u^{s}|_{+}=u^{s}|_{-},\partial u^{s}/\partial x_{2}\left|{}_{+}=\partial u^{s}/\partial x_{2}\right|_{-}$ on $\Gamma_{0}$;

(iii) $u^{s}=g$ on $\Gamma$;

(iv) $u^{s}$ satisfies (2.4) for some $\alpha\in\mathbb{R}$;

(v) $u^{s}$ satisfies the upward propagating radiation condition (2.3) in $U_{0}$ with the wave number $k_{+}$.

Impedance Boundary Value Problem (IBVP). Given $g\in BC(\Gamma),\beta\in BC(\Gamma)$, determine $u^{s}\in\mathscr{R}_{i}(D)$ such that:

(i) $u^{s}$ is a solution of the Helmholtz equations in (2.1);

(ii) $u^{s}|_{+}=u^{s}|_{-},\partial u^{s}/\partial x_{2}\left|{}_{+}=\partial u^{s}/\partial x_{2}\right|_{-}$ on $\Gamma_{0}$;

(iii) $\partial u^{s}/\partial\nu-ik_{-}\beta u^{s}=g$ on $\Gamma$;

(iv) $u^{s}$ satisfies (2.4) for some $\alpha\in\mathbb{R}$;

(v) For some $\theta\in(0,1)$ and some constant $C_{\theta}>0$, $u^{s}$ satisfies (2.5) for $x\in D\backslash\overline{U_{b}}$, where $b=f_{+}/2$;

(vi) $u^{s}$ satisfies the upward propagating radiation condition (2.3) in $U_{0}$ with the wave number $k_{+}$.

In this section, we derive some properties of the two-layered Green function $G(x,y)$, which are useful for the investigation of the well-posedness of the considered boundary value problems and the convergence of the Nyström method in the following two sections. To this end, we give some notations. Define the angle $\theta_{c}:=\mathrm{arccos}(n)$ if $0<n<1$, where $n=k_{-}/k_{+}$ is given as in Section 2. For any $R_{0}>0$, define $B_{R_{0}}:=\{y\in\mathbb{R}^{2}\,:\,|y|<R_{0}\}$ and $B_{R_{0}}^{\pm}:=\{y\in\mathbb{R}_{\pm}^{2}:|y|<R_{0}\}$. Throughout the paper, the constants may be different at different places.

Let the two-layered Green function $G(x,y)$ with $y\in\mathbb{R}^{2}_{+}\cup\mathbb{R}^{2}_{-}$ be given as in Section 2. For any source point $y$ lying on the interface $\Gamma_{0}$, due to the well-posedness of the scattering problem in a two-layered medium (see [2]), we can define the two-layered Green function $G(x,y)$ as the unique solution that satisfies $G(\cdot,y)-G_{0}(\cdot,y)\in H^{1}_{loc}(\mathbb{R}^{2})$, $\Delta_{x}G(x,y)+k^{2}(x)G(x,y)=-\delta(x,y)$ in $\mathbb{R}^{2}$ (in the distributional sense) and the Sommerfeld radiation condition (2.10), where $G_{0}(\cdot,y):=-1/(2\pi)\ln|\cdot-y|$ denotes the fundamental solution of the Laplace equation $\Delta w=0$ in $\mathbb{R}^{2}$. Here, $H^{1}_{loc}(\mathbb{R}^{2})$ denotes the space of all functions $\phi:\mathbb{R}^{2}\to\mathbb{C}$ such that $\phi\in H^{1}(B)$ for all open balls $B\subset\mathbb{R}^{2}$. Moreover, by the expression of the Hankel function $H^{1}_{0}(t)$ given in [15, Section 3.5] and the expression of $G(x,y)$ given in (2.11), it is clear that for any $y\in\mathbb{R}^{2}_{+}\cup\mathbb{R}^{2}_{-}$, $G(x,y)$ also satisfies $G(\cdot,y)-G_{0}(\cdot,y)\in H^{1}_{loc}(\mathbb{R}^{2})$.

Let $x,y\in\mathbb{R}^{2}$ with $x=(x_{1},x_{2}),y=(y_{1},y_{2})$. For any $y=(y_{1},y_{2})$, let $y^{\prime}:=(y_{1},-y_{2})$. Using the following integral representation of Hankel function (see [14, formula (2.2.11)])

for $\kappa>0$, $x,y\in\mathbb{R}^{2}$ with $x\neq y$, the formula (2.11) for $G(x,y)$ can be written as

where $G_{\mathcal{D},\kappa}(x,y)$ is the half-space Dirichlet Green function for $\kappa>0$ (see [6]) and is defined as

for $x,y\in\mathbb{R}^{2}$ with $x\neq y$ and where $G_{\mathcal{P}}$ is given by

Further, with the help of (3.1), we write $G_{\mathcal{Q}}(x,y)$ as

where $G_{\mathcal{S}}(x,y)$ is defined by

for $x\in\mathbb{R}^{2}_{-},y\in\mathbb{R}^{2}_{+}\text{ or }x\in\mathbb{R}^{2}_{+},y\in\mathbb{R}^{2}_{-}.$

The following lemma presents the continuity properties of $G(x,y)$.