Characteristic crossing in a boundary-generated elastic simple wave ($t_{\mathrm{sh}}={\inf }_{{\dot{c}}_b>0}(s+c_b/{\dot{c}}_b)$)

In a right-going simple wave, the state emitted at boundary time $s$ travels along $X=c_b(s)(t-s)$. The mapping from emission time to position first becomes singular when ${\dot{c}}_b(t-s)-c_b=0$, giving the displayed shock time and position $X=c_b^2/{\dot{c}}_b$. A smooth increase of traction produces steepening only if it increases the propagation speed. This describes the first gradient singularity; continuation as a shock wave needs admissible jump conditions.