On the asymptotic solution of a wave damping problem arising in lnhomogeneous media

In studying a wave damping problem in inhomogeneous media, the following equation arises: (Formula presented.) This paper considers the asymptotic solution of the above equation for large complex values of u which is uniformly valid in x. Attention is focused on the solution which tends to zero as x tends to infinity and for values of u in the half plane Re(u) >0. The results obtained are simply expressed in terms of the Macdonald type Bessel function K_u(x). Our method utilizes a relation between two types of the modified Bessel functions and a uniform asymptotic formula for K_u(x), and depends for its success on a novel method to obtain the uniform bounds for the Green function that arises in the reformulated integral equation.