Method of propagating waves for a one-dimensional inhomogeneous medium

The aim of this work is to develop a method of propagating waves based on the idea of a wave as a changing state of a medium. This method allows us to represent a solution of the one-dimensional wave equation in an inhomogeneous medium as the sum of two constantly deformed waves, the "right wave" and the "left wave," transported from point to point with coefficients depending on the points and the transport time. By the propagating-wave method we obtain explicit (as far as possible) formulas for solutions of the mixed problem with homogeneous and inhomogeneous boundary conditions and solutions of the Goursat problem. The derivation of these formulas is based on special convolution formulas for the transport coefficients that are similar to the addition identities for trigonometric functions. © 2005 Springer Science+Business Media, Inc.