Construction of artificial point sources for a linear wave equation in unknown medium

TL;DR

This paper develops a method to construct artificial point sources for the wave equation in unknown media using boundary measurements, enabling wave focusing and aiding inverse problem solutions, with numerical demonstration in 1D.

Contribution

It introduces a novel boundary control technique to generate artificial point sources in unknown media, advancing inverse problem methodologies.

Findings

Successfully constructs waves converging to a delta distribution at time T

Demonstrates wave focusing numerically in 1D case

Provides a new approach for inverse problems in wave equations

Abstract

We study the wave equation on a bounded domain of $\mathbb R^m$ and on a compact Riemannian manifold $M$ with boundary. We assume that the coefficients of the wave equation are unknown but that we are given the hyperbolic Neumann-to-Dirichlet map $\Lambda$ that corresponds to the physical measurements on the boundary. Using the knowledge of $\Lambda$ we construct a sequence of Neumann boundary values so that at a time $T$ the corresponding waves converge to zero while the time derivative of the waves converge to a delta distribution. Such waves are called an artificial point source. The convergence of the wave takes place in the function spaces naturally related to the energy of the wave. We apply the results for inverse problems and demonstrate the focusing of the waves numerically in the 1-dimensional case.