Consider the scattering of a time-harmonic acoustic plane wave by a bounded penetrable obstacle, which is immersed in an open space occupied by a homogeneous acoustic medium such as some compressible inviscid air or fluid. The obstacle is assumed to be a homogeneous and isotropic elastic medium. When the incident wave impinges the obstacle, a scattered acoustic wave will be generated in the open space and an elastic wave is induced simultaneously inside the obstacle. This scattering phenomenon leads to an acoustic-elastic interaction problem (AEIP). Given the incident wave and the obstacle, the direct acoustic-elastic interaction problem (DAEIP) is to determine the pressure of the acoustic wave field and the displacement of the elastic wave field in the open space and in the obstacle, respectively; the inverse acoustic-elastic interaction problem (IAEIP) is to determine the elastic obstacle from the far-field pattern of the acoustic wave field. The AEIPs have received ever-increasing attention due to their significant applications in seismology and geophysics 
. Despite many work done so far for both of the DAEIP and IAEIP, they still present many challenging mathematical and computational problems due to the complex of the model equations and the associated Green tensor, as well as the nonlinearity and ill-posedness.
The phased IAEIP referres to the IAEIP that determines the location and shape of the elastic obstacle from the phased far-field data, which contains both the phase and amplitude information. It has been extensively studied in the recent decades. In [10, 11], an optimization based variational method and a decomposition method were proposed to the IAEIP. The direct imaging methods, such as the linear sampling method [34, 35] and the factorization method [23, 40, 13], were also developed to the corresponding inverse problems with far-field and near-field data. For the theoretical analysis, the uniqueness results may be found in [34, 36] for the phased IAEIP.
The phaseless IAEIP is to determine the location and shape of the elastic obstacle from the modulus of the far-field acoustic scattering data, which contains only the amplitude information. Due to the translation invariance property of the phaseless far-field field, it is impossible to uniquely determine the location of the unknown object by plane incident wave, which makes the phaseless inverse problem much more challenging than the phased counterpart. Various numerical methods have been proposed to solve the phaseless inverse obstacle scattering problems, especially for the acoustic waves which are governed by the scalar Helmholtz equation. For the shape reconstruction with one incident plane wave, we refer to the Newton iterative method , the nonlinear integral equation method [14, 15], the fundamental solution method , and the hybrid method . In particular, the nonlinear integral equation method, which was proposed by Johansson and Sleeman , was extended to reconstruct the shape of a sound-soft crack by using phaseless far-field data from a single incident plane wave . To reconstruct the location and shape simultaneously, Zhang et al. [45, 46] proposed an iterative method by using the superposition of two plane waves with different incident directions to reconstruct the unknown object. In , a phase retrieval technique combined with the direct sampling method was proposed to reconstruct the location and shape of an obstacle from phaseless far-field data. The method was extended to the phaseless inverse elastic scattering problem and phaseless IAEIP . We refer to [38, 41, 37, 44, 43] for the uniqueness results on the inverse scattering problems by using phaseless data. Related phaseless inverse scattering problems as well as numerical methods can be found in [1, 31, 18, 5, 3, 4, 42, 24]. Recently, a reference ball technique based nonlinear integral equations method was proposed in  to break the translation invariance from phaseless far-field data by one incident plane wave. In our recent work , we extended this method to the inverse elastic scattering problem with phaseless far-field data by using a single incident plane wave to recover both the location and shape of a rigid elastic obstacle.
In this paper, we consider both the DAEIP and IAEIP. In particular, we study the IAEIP of determining the location and shape of an elastic obstacle from the phased or phaseless far-field data with a single incident plane wave. The goal of this work is fivefold:
deduce the jump relations for the second derivatives of the single-layer potential and the coupled system of boundary integral equations;
prove the well-posedness of the solution for the coupled system and develop a Nyström-type discretization for the boundary integral equations;
show the translation invariance of the phaseless far-field pattern and present a uniqueness result for the phaseless IAEIP;
propose a numerical method of nonlinear integral equations to reconstruct the obstacle’s location and shape by using the phased far-field data from a single plane incident wave;
develop a reference ball based method to reconstruct both the obstacle’s location and shape by using phaseless far-field data from a single plane incident wave.
For the direct problem, instead of considering directly the coupled acoustic and elastic wave equations, we make use of the Helmholtz decomposition and reduce the model problem into a coupled boundary value problem of the Helmholtz equations. The method of boundary integral equations is adopted to solve the coupled Helmholtz system. However, the boundary conditions are more complicated, since the second derivatives of surface potentials are involved due to the traction operator. Therefore, we investigate carefully the jump relations for the second derivatives of the single-layer potential and establish coupled boundary integral equations. Moreover, we prove the existence and uniqueness for the solution of the coupled boundary integral equations, and develop a Nyström-type discretization to efficiently and accurately solve the direct acoustic-elastic interaction problem. The proposed method is extremely efficient for the direct scattering problem since we only need to solve the scalar Helmholtz equations instead of solving the vector Navier equations. Related work on the direct acoustic-elastic interaction problems and time-domain acoustic-elastic interaction problem can be found in[2, 19, 33, 39].
For the inverse problem, motivated by the reference ball technique [32, 41] and the recent work [8, 9], we give a uniqueness result for the phaseless IAEIP by introducing an elastic reference ball, and also propose a nonlinear integral equations based iterative numerical scheme to solve the phased and phaseless IAEIP. Since the location of reference ball is known, the method breaks the translation invariance and is able to recover the location information of the obstacle with negligible additional computational costs. Numerical results show that the method is effective and robust to reconstruct the obstacle with either the phased or phaseless far-field data.
The paper is organized as follows. In Section 2, we introduce the coupled acoustic-elastic interaction problem and show the uniqueness for the coupled boundary value problem by using the Helmholtz decomposition. In Section 3, we study the jump properties for the second derivatives of the single-layer potential and establish the coupled boundary integral equations. The existence and uniqueness of the solution for the coupled boundary integral equations are given. Section 4 is devoted to the translation invariance and the uniqueness for the phaseless IAEIP. Section 5 presents a high order Nyström-type discretization to solve the coupled boundary value problem. In Section 6, a method of nonlinear integral equations and a reference ball based method are developed to solve the phased and phaseless inverse problems, respectively. Numerical experiments are provided to demonstrate the effectiveness of the proposed methods in Section 7. The paper is concluded with some general remarks and directions for future work in Section 8.
2. Problem formulation
Consider the scattering problem of a time-harmonic acoustic plane wave by a two-dimensional elastic obstacle with boundary . The elastic obstacle is assumed to be homogeneous and isotropic with a mass density ; the exterior domain is assumed to be filled with a homogeneous and compressible inviscid air or fluid with a mass density . Denote by and the unit normal vector and the tangential vector on , respectively. Let . Given a vector function and a scalar function , we introduce the scalar and vector curl operators
Specifically, the time-harmonic acoustic plane wave is given by , where is the propagation direction vector, is the incident angle. Given the incident field , the direct problem is to find the elastic wave displacement and the acoustic wave pressure , which satisfy the Navier equation and the Helmholtz equation, respectively:
Moreover, and are required to satisfy the transmission conditions
The scattered acoustic wave pressure is required to satisfy the Sommerfeld radiation condition
Here is the angular frequency, is the wavenumber in the air/fluid with the sound speed , and are the Lamé parameters satisfying . The traction operator is defined by
It can be shown (cf. [33, 39]) that the scattering problem (2.1)–(2.4) admits a unique solution for all but some particular frequencies , which are called the Jones frequencies . At the Jones frequency, the acoustic wave field is unique, but the elastic field is not unique. Since the Jones frequency happens only for some special geometries , for simplicity, we assume that does not admit any Jones mode in this work.
For any solution of the elastic wave equation (2.1), we introduce the Helmholtz decomposition
which is fulfilled if and satisfy the Helmholtz equation with a different wavenumber, respectively:
are the compressional wavenumber and the shear wavenumber, respectively.
Substituting the Helmholtz decomposition into (2.3) and taking the dot product with and , respectively, we obtain
In summary, the scalar potential functions and the scattered acoustic wave satisfy the following coupled boundary value problem
The following result concerns the uniqueness of the boundary value problem (2.6).
The coupled boundary value problem (2.6) has at most one solution for .
It suffices to show that when . It follows from straightforward calculations that
where is the Frobenius inner product of square matrices and . The last two identities follow from Green’s formula and the Navier equation (2.1). Taking the imaginary part of the above equation yields
which gives that in by Rellich’s lemma. Using the continuity conditions (2.3), we conclude that is identically zero in provided that there is no Jones mode in . Hence,
which implies and . The proof is completed by noting that and in . ∎
It is known that a radiating solution of the Helmholtz equation (2.2) has the asymptotic behaviour of the form
uniformly in all directions . The function , defined on the unit circle , is known as the far-field pattern of . Let be an artificially added elastic ball centered at such that . The problem geometry is shown in Figure 1. For brevity, we denote the boundary of and by and , respectively. The phased and phaseless IAEIP can be stated as follows:
Problem 1 (Phased IAEIP).
Given an incident plane wave with a single incident direction and the corresponding far-field pattern due to the unknown obstacle , the inverse problem is to determine the location and shape of the boundary .
Problem 2 (Phaseless IAEIP).
Given an incident plane wave with a single incident direction and the corresponding phaseless far-field pattern due to the scatterer , the inverse problem is to determine the location and shape of the boundary .
3. Boundary integral equations
In this section, we derive the boundary integral equations for the coupled boundary value problem (2.6) and discuss their well-posedness.
3.1. Jump relations
We begin with investigating the jump relations for the surface potentials at the boundary .
For given vectors and , denote
For a given scalar function , define
Denote the fundamental solution of the two-dimensional Helmholtz equation by
where is the Hankel function of the first kind of order zero. The single- and double-layer potentials with density are defined by
In addition, we define the tangential-layer potential by
The jump relations can be found in  for the single- and double-layer potentials as . It is necessary to study the jump properties for the derivatives of those layer potentials in order to derive the boundary integral equations for the coupled boundary value problem (2.6).
The first derivatives of the single-layer potential with density , , can be uniformly extended in a Hölder continuous fashion from into and from into with the limiting values
Moreover, for the single-layer potential with density , , we have
The first derivatives of the double-layer potential with density , , can be uniformly extended in a Hölder continuous fashion from into and from into with the limiting values
For the tangential-layer potential with density , , we have
The second derivatives of the single-layer potential with density , , can be uniformly extended in a Hölder continuous fashion from into and from into with the limiting values
In view of , and Theorem 3.4, we have the following result.
For the single-layer potential with density , , we have on that
3.2. Boundary integral equations
We introduce the single-layer integral operator and the corresponding far-field integral operator
the normal derivative integral operator
and the tangential derivative integral operator
Let the solution of (2.6) be given in the form of single-layer potentials, i.e.,
where the densities , , and .
Letting tend to boundary in (3.9)–(3.10) and tend to boundary in (3.11), using the jump relations of the single-layer potentials, Lemmas 3.1–3.2, Corollary 3.5, and the boundary conditions of (2.6), we obtain on that
We point out that and inside of or are with respect to the variable ; otherwise and are taken with respect to the variable . For brevity, we shall adopt the same notations in the rest of the paper but they should be clear from the context. The far-field pattern is
Now we discuss the uniqueness and existence of the solution for the integral equations (3.12).
It suffices to show that if for equations in (3.12). For , we define single-layer potentials