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arXiv · 2503.18515

Recovering a (1+1)-dimensional wave equation from a single white noise boundary measurement

Emilia L. K. Blåsten·Tapio Helin·Antti Kujanpää·Lauri Oksanen·Jesse Railo
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Abstract

We consider the following inverse problem: Suppose a $(1+1)$-dimensional wave equation on $\mathbb{R}_+$ with zero initial conditions is excited with a Neumann boundary data modelled as a white noise process. Given also the Dirichlet data at the same point, determine the unknown first order coefficient function of the system. We first establish that direct problem is well-posed. The inverse problem is then solved by showing that correlations of the boundary data determine the Neumann-to-Dirichlet operator in the sense of distributions, which is known to uniquely identify the coefficient. This approach has applications in acoustic measurements of internal cross-sections of fluid pipes such as pressurised water supply pipes and vocal tract shape determination.

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Emilia L. K. Blåsten, Tapio Helin, Antti Kujanpää, Lauri Oksanen, Jesse Railo. 2025-03-24. Recovering a (1+1)-dimensional wave equation from a single white noise boundary measurement. https://arxiv.org/abs/2503.18515

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