Inverse problem of the scattering of plane waves for a class of layered media

M. I. Belishev

Zapiski Nauchnykh Seminarov POMI, Mathematical problems in the theory of wave propagation. Part 9, Tome 78 (1978), pp. 30-53

Résumé

The direct and inverse problems of the scattering of plane waves in a layered, inhomogeneous medium are considered in the paper. In the appropriate variables the wave equation of the problem has the form $u_{\xi\xi}(z,\xi)=Q(z)u_{zz}(z,\xi)$, $-\infty$, $\xi\infty$, $Q(z){z0}|_{z0}\equiv1$. A special feature of the case considered, in contrast to those studied earlier, is that $Q(z)|_{z\geqslant0}$ may change sign; because of this, the equation of the problem is, in general, an equation of mixed type. The correct formulation of the direct problem for such an equation and the study of the properties of its solution form a necessary step in the investigation. For a very broad class of media including cases of $Q(z)$ of variable sign ($Q(z)$ can change sign by a jump a finite number of times without vanishing anywhere) a procedure is developed for solving the corresponding inverse problem of determining $Q(z)$ on the basis of the scattering data $u(0,\xi)|_{\xi\in(-\infty,\infty)}$. This procedure makes it possible to recover $Q(z)$ for all $Z\in[0,\infty)$. The solution of the inverse problem is unique in this class.