Abstract
Consider the two-dimensional inverse elastic scattering problem of recovering a piecewise linear rigid rough or periodic surface of rectangular type for which the neighboring line segments are always perpendicular. We prove the global uniqueness with at most two incident elastic plane waves by using near-field data. If the Lame constants satisfy a certain condition, then the data of a single plane wave is sufficient to imply the uniqueness. Our proof is based on a transcendental equation for the Navier equation, which is derived from the expansion of analytic solutions to the Helmholtz equation. The uniqueness results apply also to an inverse scattering problem for non-convex bounded rigid bodies of rectangular type.
Highlights
This paper is concerned with the inverse scattering of timeharmonic elastic waves from rigid unbounded periodic and rough surfaces of rectangular type, which has a wide field of applications, in geophysics, seismology and nondestructive testing
Using an elastic plane wave as an incoming source, we obtain a two-dimensional inverse problem of recovering a rectangular interface from the knowledge of near-field data measured above the periodic structure; see [17]
The associated direct scattering problem is formulated as a Dirichlet boundary value problem for the time-harmonic Navier equation in the unbounded domain above the surface, which can be considered as a simple model problem in linear elasticity
Summary
This paper is concerned with the inverse scattering of timeharmonic elastic waves from rigid unbounded periodic and rough surfaces of rectangular type (see Sections 2.1 and 3 for a precise description), which has a wide field of applications, in geophysics, seismology and nondestructive testing. Uniqueness, Navier equation, linear elasticity, Dirichlet boundary condition, rough surface, diffraction grating. Sending a single incident point source wave always leads to the uniqueness of the inverse problem within polygonal periodic or rough surfaces; see [12] for the Helmholtz equation. Such an argument applies so far only to the third or fourth kind boundary value problems of the Navier equation, and it still remains a challenging problem to prove the uniqueness under the more practical Dirichlet or Neumann-type boundary conditions, due to the lack of corresponding reflection principles. It is supposed that Λ is of rectangular type, i.e., the neighboring line segments are always perpendicular
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