  We consider the inverse problem of reconstructing the boundary curve of a
cavity embedded in a bounded domain. The problem is formulated in two
dimensions for the wave equation. We combine the Laguerre transform with the
integral equation method and we reduce the inverse problem to a system of
boundary integral equations. We propose an iterative scheme that linearizes the
equation using the Fr\'echet derivative of the forward operator. The
application of special quadrature rules results to an ill-conditioned linear
system which we solve using Tikhonov regularization. The numerical results show
that the proposed method produces accurate and stable reconstructions.
