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| Main Authors: | , |
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| Format: | Preprint |
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2023
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| Online Access: | https://arxiv.org/abs/2308.06968 |
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| _version_ | 1866913066220781568 |
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| author | Moon, Sunghwan Seo, Ihyeok |
| author_facet | Moon, Sunghwan Seo, Ihyeok |
| contents | 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 $Ω$ with variable sound speed $c(\cdot)$. From time-resolved boundary measurements, we consider two settings: (i) measurement of $p|_{\partialΩ\times[0,\infty)}$ under a Robin boundary condition $p+α\,\partial_νp=0$ on $\partialΩ\times[0,\infty)$ with $α\gneq 0$, and (ii) measurement of $\partial_νp|_{\partialΩ\times[0,\infty)}$ under a Dirichlet boundary condition $p=0$ on $\partialΩ\times[0,\infty)$. Within a unified framework, we present explicit formulas that recover the spectral coefficients $\langle f,ϕ_k^B\rangle$ of $f$ with respect to the eigenfunction bases of the operator $-c(\cdot)Δ_{x}$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2308_06968 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Explicit inversion for variable-speed wave equations on bounded domains Moon, Sunghwan Seo, Ihyeok Analysis of PDEs 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 $Ω$ with variable sound speed $c(\cdot)$. From time-resolved boundary measurements, we consider two settings: (i) measurement of $p|_{\partialΩ\times[0,\infty)}$ under a Robin boundary condition $p+α\,\partial_νp=0$ on $\partialΩ\times[0,\infty)$ with $α\gneq 0$, and (ii) measurement of $\partial_νp|_{\partialΩ\times[0,\infty)}$ under a Dirichlet boundary condition $p=0$ on $\partialΩ\times[0,\infty)$. Within a unified framework, we present explicit formulas that recover the spectral coefficients $\langle f,ϕ_k^B\rangle$ of $f$ with respect to the eigenfunction bases of the operator $-c(\cdot)Δ_{x}$ 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. |
| title | Explicit inversion for variable-speed wave equations on bounded domains |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2308.06968 |