Inverse wave propagation in an inhomogeneous waveguide

Abstract

A solution is given for the problem of inverse propagation in an inhomogeneous rectangular two-dimensional waveguide. The sound speed is assumed to vary in depth and inverse propagation means the calculation of the field at range xl in tenns of the field at range x2 where x2 > xl. The method is analogous to that used by Wolf, Shewell, and Lalor for the inverse diffraction problem in a homogeneous half space. It is found that the field at xl can be expressed in terms of two integrals over the field at x2. The kernel of the first integral is bounded and expresses physically the result at xl of the waves at x2 reversing their direction of propagation and decay, ie they now propagate and decay in the direction of xl. A reciprocity relation for this term is possible. The kernel of the second integral is singular and expresses the mathematical fact of the residual effect of the evanescent waves at xl as they reverse their direction at x2 and now grow exponentially. Consequences of the neglect of thissingular term are discussed.