= Characteristic crossing in a boundary-generated elastic simple wave
{title2=$t_{\rm sh}=\inf_{\dot c_b>0}(s+c_b/\dot c_b)$}
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.
Back to article page
