Selective focusing of elastic cavities based on the time reversal far field model

TL;DR

This paper extends the DORT method to inverse elastic scattering, enabling selective focusing on small cavities using far field measurements, supported by rigorous mathematical analysis and numerical verification.

Contribution

It introduces a mathematically justified extension of the DORT method for elastic scattering to achieve cavity-specific focusing based on eigenfunctions.

Findings

Each cavity produces five significant eigenvalues.

Eigenfunctions can focus incident waves on individual cavities.

Numerical experiments confirm theoretical predictions.

Abstract

This paper is concerned with the inverse time harmonic elastic scattering of multiple small and well-resolved cavities in two dimensions. We extend the so-called DORT method to the inverse elastic scattering so that selective focusing can be achieved on each cavity with far field measurements. A rigorous mathematical justification that relates the corresponding eigenfunctions of the time reversal operator to the locations of cavities is presented based on the asymptotic analysis of the far field operator and decaying property of oscillatory integrals. We show that in the regime $a\ll k^{-1}\ll L$, where $a$ denotes the size of cavity, $k$ is the compressional wavenumber $\kp$ or shear wavenumber $\ks$, and $L$ is the minimal distance between the cavities, each cavity gives rise to five significant eigenvalues and the corresponding eigenfunction generates an incident wave focusing selectively on that cavity. Numerical experiments are given to verify the theoretical result.