Special Discontinuities in the Solution of a Self-similar Problem on Waves in a Rod

We consider a self-similar problem on small-amplitude long nonlinear longitudinal–torsional waves arising in a semi-infinite elastic rod under a change of stresses at its end. A specific feature of this problem is that among possible discontinuities in its solution, in addition to shocks, a special discontinuity may appear with an additional boundary condition that expresses the existence condition for a stationary viscous structure of this discontinuity.