Determination of the wave velocity in an inhomogeneous medium from the reflection coefficient

In this paper, we study the problem of obtaining the velocity of the wave propagation in an inhomogeneous medium from the reflection coefficient (given for all frequencies) for a one-dimensional wave equation. Following the work of Kay and Moses, we reduce the problem to that of determining a ’’potential’’ for a Schrödinger-type equation. However, in our work this ’’potential’’ depends quadratically on the wavenumber, and this dependence changes the structure of the spectral function and integral equations of the inverse problem. In the first part of the paper, we assume that the wave velocity c (x) has the same value at x=+∞ and at x=−∞. Then we relax this condition on c (x) and consider the cases where c (+∞) is different from c (−∞), but both quantities have well-defined values. For certain types of velocities, we obtain the reflection coefficient analytically. We then use these reflection coefficients to get approximate solutions to the inverse problem. Finally, we present an example of a heterogeneous medium which is transparent for all frequencies of the incident wave. Subject Classification: 20.30, 20.35, 20.15.