Geodesic
The one-dimensional inverse scattering problem for the wave equation
Sbornik. Mathematics, Tome 70 (1991) no. 2, pp. 557-572

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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. 
@article{SM_1991_70_2_a12,
     author = {N. I. Grinberg},
     title = {The one-dimensional inverse scattering problem for the wave equation},
     journal = {Sbornik. Mathematics},
     pages = {557--572},
     publisher = {mathdoc},
     volume = {70},
     number = {2},
     year = {1991},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_1991_70_2_a12/}
}
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N. I. Grinberg. The one-dimensional inverse scattering problem for the wave equation. Sbornik. Mathematics, Tome 70 (1991) no. 2, pp. 557-572. http://geodesic.mathdoc.fr/item/SM_1991_70_2_a12/