Well-posedness for the biharmonic scattering problem for a penetrable obstacle

TL;DR

This paper investigates the mathematical well-posedness of the biharmonic scattering problem for a penetrable obstacle in a thin elastic plate, combining operator theory and boundary value analysis, with numerical validation.

Contribution

It establishes well-posedness and reciprocity relations for the biharmonic scattering problem in elastic plates, using operator factorization and boundary value techniques.

Findings

Proves well-posedness of the scattering problem

Derives reciprocity relations for the model

Provides numerical examples validating the theory

Abstract

We address the direct scattering problem for a penetrable obstacle in an infinite elastic two--dimensional Kirchhoff--Love plate. Under the assumption that the plate's thickness is small relative to the wavelength of the incident wave, the propagation of perturbations on the plate is governed by the two-dimensional biharmonic wave equation, which we study in the frequency domain. With the help of an operator factorization, the scattering problem is analyzed from the perspective of a coupled boundary value problem involving the Helmholtz and modified Helmholtz equations. Well-posedness and reciprocity relations for the problem are established. Numerical examples for some special cases are provided to validate the theoretical findings.