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

Abstract We consider the following inverse problem: suppose a ( 1 + 1 ) -dimensional wave equation on R + with zero initial conditions is excited with 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 the 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 pressurized water supply pipes and vocal tract shape determination.