On inverse problem of waves identification by measurements at one point vicinity

TL;DR

This paper formulates and analyzes an inverse wave identification problem using measurements at a single point, employing projection techniques and regularization to reconstruct wave origins in various wave equations.

Contribution

It introduces a novel approach using projecting operator techniques and regularization for wave source reconstruction from single-point measurements.

Findings

Effective wave source reconstruction algorithm developed.

Application to acoustic and electromagnetic wave problems demonstrated.

Analysis of dissipation and entropy modes in wave extraction included.

Abstract

A problem of a wave identification is formulated. An example is considered in conditions of one-dimensional Cauchy problem for conventional string equation in matrix form and its inhomogeneous two-component version. The acoustic and electromagnetic problems are discussed within the restrictions outlined. The projecting operator technique is used to split the solution space and analyze input of a wave monitoring in vicinity of an observation point. The solution space is supplied by $L_2$ norm via the problem conservation law; its finite-dimensional analog is used as a measure of a given mode presence and information about form. The algorithm of the problem solution is presented in terms of appropriate regularization to reconstruct an incoming pulses origin. The dissipation and entropy mode account in the problem of acoustic waves extraction is also discussed in terms of correspondent projecting technique.