Explicit inversion for variable-speed wave equations on bounded domains

We study the reconstruction of the initial pressure $f(x)=p(x,0)$ for the wave model $${\mathrm{\partial }}_t^{2}p(x,t)=c(x){\mathrm{\Delta }}_{x}p(x,t)(x,t)\in \mathrm{\Omega }\times [0,\mathrm{\infty }),$$ posed on a bounded domain $\mathrm{\Omega }$ with variable sound speed $c(\cdot )$. From time-resolved boundary measurements, we consider two settings: (i) measurement of $p{\mathrm{\mid }}_{\mathrm{\partial }\mathrm{\Omega }\times [0,\mathrm{\infty })}$ under a Robin boundary condition $p+\alpha {\mathrm{\partial }}_{\nu }p=0$ on $\mathrm{\partial }\mathrm{\Omega }\times [0,\mathrm{\infty })$ with $\alpha ⪈0$, and (ii) measurement of ${\mathrm{\partial }}_{\nu }p{\mathrm{\mid }}_{\mathrm{\partial }\mathrm{\Omega }\times [0,\mathrm{\infty })}$ under a Dirichlet boundary condition $p=0$ on $\mathrm{\partial }\mathrm{\Omega }\times [0,\mathrm{\infty })$. Within a unified framework, we present explicit formulas that recover the spectral coefficients $⟨f,{\phi }_k^B⟩$ of $f$ with respect to the eigenfunction bases of the operator $-c(\cdot ){\mathrm{\Delta }}_x$ 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.