On the evolution equation of longitudinal shock waves in elastic media with weak inhomogeneity

Several problems of shock deformation in a nonlinearly elastic compressible medium with inhomogeneous properties are considered. The method of matched asymptotic expansions is used to show that the weak inhomogeneity and a certain relation between its order and the model nonlinearity order lead to different types of evolution quasilinear wave equations in regions far from the loaded boundary. The most interesting version of the arising evolution equation was obtained by joint change of the spatial coordinate scale and the related type of the semicharacteristic variable. The solution ideas are illustrated by an example of plane longitudinal shock wave in a medium with inhomogeneity in the wave motion direction. The obtained evolution equations become the well-known Cole-Hopf equation in the limit when passing to the isotropic medium.