Detecting a hidden obstacle via the time domain enclosure method. A scalar wave case

TL;DR

This paper introduces a method to detect hidden obstacles behind known objects using a single wave observation over time, providing criteria for existence and distance estimates based on the time domain enclosure method.

Contribution

It develops a novel approach for obstacle detection in wave equations, enabling existence determination and distance bounds from limited wave data.

Findings

Can determine the existence of an unknown obstacle behind a known one.

Provides a lower bound on the distance to the obstacle.

Uses the time domain enclosure method with heat kernel estimates.

Abstract

The characterization problem of the existence of an unknown obstacle behind a known obstacle is considered by using a singe observed wave at a place where the wave is generated. The unknown obstacle is invisible from the place by using visible ray. A mathematical formulation of the problem using the classical wave equation is given. The main result consists of two parts: (i) one can make a decision whether the unknown obstacle exists or not behind a known impenetrable obstacle by using a single wave over a finite time interval under some a-priori information on the position of the unknown obstacle; (ii) one can obtain a lower bound of the Euclidean distance of the unknown obstacle to the center point of the support of the initial data of the wave. The proof is based on the idea of the time domain enclosure method and employs some previous results on the Gaussian lower/upper estimates for the heat kernels and domination of semigroups.