Direct and inverse acoustic scattering by global rough surfaces
Chengyu Wu, Jiaqinq Yang
TL;DR
The paper analyzes direct and inverse acoustic scattering by an unbounded penetrable rough surface in a lossless medium, addressing both $\\mu eq 1$ and $\\mu = 1$ scenarios. It develops a boundary-integral equation framework to prove well-posedness of the direct problem, provides a novel singularity analysis for point- and hypersingular- incidence, and establishes a global uniqueness result showing that the unbounded surface, the transmission coefficient $\\mu$, and the lower-wave number $k_-$ can be uniquely recovered from near-field measurements on a line at a fixed frequency. The approach combines boundary-integral representations, energy estimates, and interior transmission problem techniques to handle unbounded geometries and the transmission condition. These results enhance theoretical understanding of rough-surface scattering and provide a rigorous foundation for inverse imaging of unbounded penetrable interfaces in acoustics and related wave problems.
Abstract
In this paper, we investigate on the direct and inverse scattering problem by an unbounded penetrable rough surface in a lossless medium. The cases that the transmission coefficient $μ\neq1$ and $μ=1$, which creates certain difficulties in the direct and inverse problem, respectively, are both considered. We first estalish the well-posedness of the direct problem using the integral equation method through an elaborate analysis. Then we carefully consider the singularity of the solutions to the problem with incident point source or hypersingular point source, where a simple and novel perspective is given for the derivation of the singularity. Finally, a global uniqueness result is proven for the inverse problem on the unique determination of the unbounded rough surface, the transmission coefficient and the wave number in the lower half plane from the measurements of the near field only on a line segment above the interface at a fixed frequency.