Anharmonic Waves in a Mindlin–Herrmann Rod Immersed in a Nonlinearly Elastic Medium

Abstract

Propagation of longitudinal waves in a Mindlin–Herrmann rod immersed in a nonlinearly elastic medium is studied. Various options of the stiffness ratio of the rod and external medium in which the rod is placed are considered and three limiting cases are obtained. It is shown that, if the stiffness of the external medium significantly exceeds the rod stiffness, then the evolutionary equation becomes the Ostrovsky equation that is well-known in the nonlinear dynamics. The equation has no exact solutions, but allows a qualitative study when the senior derivative is equal to zero. In this case, the solution in the form of a nonlinear periodic stationary wave is found and analyzed. If the stiffness of external medium is significantly lower than the rod stiffness, then the evolutionary equation is the equation different from the Ostrovsky equation in the nonlinear part. In this case, it is demonstrated soliton propagation of a classical profile in the rod is possible. It is pointed out that, when the stiffnesses of external medium and rod are of the same order of magnitude, the nonlinear stationary waves cannot be generated.