The one-dimensional inverse scattering problem for the wave equation

A constructive method is given for solving the inverse scattering problem for the wave equation on the line and half-line. The slowness function is assumed to have a derivative everywhere except at a finite number of points, and both it and its derivative are assumed to be functions of bounded variation. In addition, the slowness $n(x)$ is required to tend to 1 sufficiently rapidly as $x\to\infty$. In this case the slowness function can be reconstructed from the reflection coefficient.