On the determination of the sound speed and a damping coefficient by two measurements

Abstract

We discuss an inverse problem of finding the sound speed c(x) and a damping coefficient σ( x ) in a second-order hyperbolic equation from two boundary observations. The coefficients are assumed to be unknown inside a ball in

with n ≥ 2. On a suitable bounded part of the cylindrical surface, Cauchy data for solutions to a hyperbolic equation with zero initial data and a source located on the plane

, are supposed to be given for two different unit vectors

, k = 1, 2. We obtain a conditional stability estimate under a priori assumptions on the smallness of c(x) − 1 and σ(x).

Keywords:

• Inverse hyperbolic problem
• Stability
• Impulsive input
• Mathmatical Subject Classifications: 35R25
• 35R30
• 35R12

Acknowledgements

This article was completed when as a Foreign Special Visiting Professor, the first author stayed at the Graduate School of Mathematical Sciences of the University of Tokyo from December 2002 till March 2003. He expresses his thanks to the University of Tokyo and the Japanese Ministry of Education, Culture, Sports, Science and Technology. The second author was supported partly by Grants 15340027 and 15654015 from the Japan Society for Promotion of Science.