On the Cauchy Problem for the Wave Equation with Data on the Boundary

Abstract

We consider the Cauchy problem for the wave equation in Ω × R with data given on some part of the boundary ∂Ω × R. We provide a reconstruction algorithm for this problem based on analytic expressions. Our result is applicable to the problem of determining nonstationary wave field arising in geophysics, photoacoustic tomography, tsunami wave source recovery.

Determination of a wave field in a laterally inhomogeneous medium from boundary data

“…One of the pioneering results is that of R. Courant concerning the ultrahyperbolic equations in the half-space (see [4]). The Cauchy problem for the wave equation in a domain was considered in [5,6]. The inversion formula for the problem in the three-dimensional half-space obtained in [7] allows determining a solution to the wave equation from the Cauchy data given on a certain unbounded subset of the space-time boundary.…”