Explicit inversion for variable-speed wave equations on bounded domains

Abstract

We study the reconstruction of the initial pressure $f(x)=p(x,0)$ for the wave model \[ \partial_t^2 p(x,t)=c(x)Δ_{x}p(x,t)\qquad (x,t)\inΩ\times[0,\infty), \] posed on a bounded domain $\hat{I}\text{c◯}$ with variable sound speed $c(\cdot )$. From time-resolved boundary measurements, we consider two settings: (i) measurement of $p{|}_{\partial \hat{I}\text{c◯}\times [0,\infty )}$ under a Robin boundary condition $p+\hat{I}\pm {\partial }_{\hat{I}}½p=0$ on $\partial \hat{I}\text{c◯}\times [0,\infty )$ with $\hat{I}\pm ⪈0$, and (ii) measurement of ${\partial }_{\hat{I}}½p{|}_{\partial \hat{I}\text{c◯}\times [0,\infty )}$ under a Dirichlet boundary condition $p=0$ on $\partial \hat{I}\text{c◯}\times [0,\infty )$. Within a unified framework, we present explicit formulas that recover the spectral coefficients $⟨f,\ddot{I}{•}_k^B⟩$ of $f$ with respect to the eigenfunction bases of the operator $-c(\cdot )\hat{I}{”}_x$ 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.

To appear in Bull. Korean Math. Soc., 11 pages