The method of successive integration of the linear inhomogeneous wave equations in the theory of transient wave propagation in non-linear hereditary elastic media

A moderate distortion of the initial pulse form which takes place when a one-dimensional longitudinal pulse propagates through a sufficiently small distance in a non-linear hereditary clastic medium is considered. The governing equation is a quasi-linear integro-differential equation. Its first- and second-order asymptotic solutions arc derived with the aid of a method of successive integration of the linear inhomogeneous wave equations. Besides the constants which define the wave speed and the non-linear properties of the medium, the asymptotic solutions suggested in this paper contain two arbitrary functions whose properties are restricted only by certain smoothness conditions. One of them is the kernel function which defines the hereditary properties of the medium. and the other is the function which defines the initial form (shape) of the pulse. An example of the use of the asymptotic solutions is presented in which these two functions are given explicitly.