Inverse problem for the Rayleigh system with spectral data

TL;DR

This paper investigates an inverse problem for the Rayleigh system in elastic media, using spectral data and a Gel'fand-Levitan approach to uniquely recover material parameters with limited frequency data.

Contribution

It introduces a novel method employing the Markushevich substitution and spectral data to solve the inverse problem for the Rayleigh system, establishing uniqueness with minimal frequency information.

Findings

Derived a Gel'fand-Levitan type equation for the problem

Proved uniqueness of the solution with two distinct frequencies

Linked the spectral data to the physical parameters of the system

Abstract

We analyze an inverse problem associated with the time-harmonic Rayleigh system on a flat elastic half-space concerning the recovery of Lam\'{e} parameters in a slab beneath a traction-free surface. We employ the Markushevich substitution, while the data are captured in a Jost function, and point out parallels with a corresponding problem for the Schr\"{o}dinger equation. The Jost function can be identified with spectral data. We derive a Gel'fand-Levitan type equation and obtain uniqueness with two distinct frequencies.