Inverse wave propagation in an inhomogeneous waveguide

Abstract

A solution is given for the problem of inverse propagation of a scalar wave in 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 x1 in terms of the field at range x2 where x2≳x1. 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 x1 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 x1 of the waves at x2 reversing their direction of propagation and decay, i.e., they now propagate and decay in the direction of x1. 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 x1 as they reverse their direction at x2 and now grow exponentially. Consequences of the neglect of this singular term are discussed.