Explicit inversion for variable-speed wave equations on bounded domains

TL;DR

This paper develops explicit formulas to reconstruct the initial pressure in variable-speed wave equations from boundary measurements, unifying different boundary conditions and measurement types.

Contribution

It introduces a unified framework for explicit spectral coefficient recovery of initial pressure using boundary data in variable sound speed wave models.

Findings

Explicit formulas for spectral coefficient recovery are derived.

The framework handles both Dirichlet and Robin boundary conditions.

It enables direct reconstruction from boundary measurements in variable media.

Abstract

We study the reconstruction of the initial pressure $f(x)=p(x,0)$ for the wave model \[ \partial_t^2 p(x,t)=c(x)\Delta_{x}p(x,t)\qquad (x,t)\in\Omega\times[0,\infty), \] posed on a bounded domain $\Omega$ with variable sound speed $c(\cdot)$. From time-resolved boundary measurements, we consider two settings: (i) measurement of $p|_{\partial\Omega\times[0,\infty)}$ under a Robin boundary condition $p+\alpha\,\partial_\nu p=0$ on $\partial\Omega\times[0,\infty)$ with $\alpha\gneq 0$, and (ii) measurement of $\partial_\nu p|_{\partial\Omega\times[0,\infty)}$ under a Dirichlet boundary condition $p=0$ on $\partial\Omega\times[0,\infty)$. Within a unified framework, we present explicit formulas that recover the spectral coefficients $\langle f,\phi_k^B\rangle$ of $f$ with respect to the eigenfunction bases of the operator $-c(\cdot)\Delta_{x}$ for boundary types $B\in\{D,R\}$. The framework integrates variable sound speed with Dirichlet/Robin boundary conditions in a single setting, enabling direct coefficient-level recovery from boundary data.