General theory of the solutions of the equations of motion of an elastic medium of different moduli
Consider the one dimensional partial differential equation \(u_{tt}=(u_ x-a| u_ x|)_ x\) with \(a=const.\), which describes the longitudinal motion of an elastic medium. Depending upon the value of a, the medium will be of different modulii, granular or elasto-plastic and will be applicable to geophysical and two-phases systems. The general solution is attempted through integral transforms with piecewise smooth solutions, leading to the Hugoniot conditions. The study of the local structure is then carried out. For the case \(0<a<1\), shocks, simple discontinuities, signotons, semi-signotons and other solutions are classified. For \(a=1\), an additional type, a discontinuity in the continuity of displacement, occurs. The evolution of the bifurcation of these features is next considered. Reflections of a compressional wave from a free edge or a rigid wall and the particle motion in an elastic granular medium with body forces are considered in detail in which substantial nonlinear effects occur. Comments are also made on the general Cauchy and associated boundary/initial value problems.
discontinuity in the continuity of displacement
scientific article; zbMATH DE number 3984687