The solid-fluid transmission problem

Abstract

We study microlocally the transmission problem at the interface between an isotropic linear elastic solid and a linear inviscid fluid. We set up a system of evolution equations describing the particle displacement and velocity in the solid, and pressure and velocity in the fluid, coupled by suitable transmission conditions at the interface. We show well posedness for the coupled system and study the problem microlocally, constructing a parametrix for it using geometric optics. This construction describes the reflected and transmitted waves, including mode converted ones, related to incoming waves from either side. We also study formation of surface Scholte waves. Finally, we prove that under suitable assumptions, we can recover the s-and the p-speeds, as well as the speed of the liquid, from boundary measurements.

Recovery of piecewise smooth density and Lamé parameters from high-frequency exterior Cauchy data

“…For the piecewise smooth case, a novel scattering control method was developed in [6] in order to show in [4] that uniqueness holds as well for piecewise smooth wave speeds with conormal singularities, under mild geometric conditions. Less is known in the elastic setting as will be described, but several works such as [11,26,23] show how to construct an FIO representation of the solution to the elastic wave equation near an interface, which is useful for inverse problems in the hyperbolic elastic setting where coefficients have conormal singularities. An additional challenge of recovering a coefficient that is not in the principal symbol of the operator is that one needs to solve a tensor tomography problem at some stage of the argument, which has a gauge freedom that obstructs uniqueness [14].…”