Longitudinal–Torsional Waves in Nonlinear Elastic Rods

Previously, we have obtained a system of fourth-order hyperbolic equations describing long nonlinear small-amplitude longitudinal–torsional waves propagating along an elastic rod. Waves of two types, fast and slow, propagate in each direction along the rod. In the present paper, based on this system of equations, we derive a second-order hyperbolic system that describes longitudinal–torsional waves propagating in one direction along the rod at close velocities. The waves propagating in the opposite direction along the rod are assumed to have a negligible amplitude. We show that the variation of quantities in simple and shock waves described by the system of second-order equations obtained in this paper exactly coincides with the variation of the same quantities in the corresponding waves described by the original system of fourth-order equations, and the velocities of these waves are close. We also analyze the variation of quantities in simple (Riemann) waves and the overturning conditions for these waves.