Translator Disclaimer
2012 An inverse problem for the wave equation with one measurement and the pseudorandom source
Tapio Helin, Matti Lassas, Lauri Oksanen
Anal. PDE 5(5): 887-912 (2012). DOI: 10.2140/apde.2012.5.887

Abstract

We consider the wave equation (t2Δg)u(t,x)=f(t,x), in n, u|×n=0, where the metric g=(gjk(x))j,k=1n is known outside an open and bounded set Mn with smooth boundary M. We define a source as a sum of point sources, f(t,x)=j=1ajδxj(x)δ(t), where the points xj,j+, form a dense set on M. We show that when the weights aj are chosen appropriately, u|×M determines the scattering relation on M, that is, it determines for all geodesics which pass through M the travel times together with the entering and exit points and directions. The wave u(t,x) contains the singularities produced by all point sources, but when aj=λλj for some λ>1, we can trace back the point source that produced a given singularity in the data. This gives us the distance in (n,g) between a source point xj and an arbitrary point yM. In particular, if (M¯,g) is a simple Riemannian manifold and g is conformally Euclidian in M¯, these distances are known to determine the metric g in M. In the case when (M¯,g) is nonsimple, we present a more detailed analysis of the wave fronts yielding the scattering relation on M.

Citation

Download Citation

Tapio Helin. Matti Lassas. Lauri Oksanen. "An inverse problem for the wave equation with one measurement and the pseudorandom source." Anal. PDE 5 (5) 887 - 912, 2012. https://doi.org/10.2140/apde.2012.5.887

Information

Received: 10 November 2010; Accepted: 26 May 2011; Published: 2012
First available in Project Euclid: 20 December 2017

zbMATH: 1329.35350
MathSciNet: MR3022845
Digital Object Identifier: 10.2140/apde.2012.5.887

Subjects:
Primary: 35R30, 58J32
Secondary: 35A18

Rights: Copyright © 2012 Mathematical Sciences Publishers

JOURNAL ARTICLE
26 PAGES


SHARE
Vol.5 • No. 5 • 2012
MSP
Back to Top