 Rectangular pulse solution of the inviscid Burgers equation

Start the Inviscid Burgers equation with height $a>0$ on $[x_L,x_L+L]$, $L>0$, and zero elsewhere. The upward left jump gives a rarefaction wave $u=(x-x_L)/t$, while the downward right jump gives an entropy shock at $x_L+L+at/2$. They meet at $t_{\ast }=2L/a$. Afterward the entire nonzero profile is the triangular fan on $x_L<x<s(t)$, with $s(t)=x_L+\sqrt{2aLt}$ and zero outside. To prove the continuation, the Rankine-Hugoniot condition gives $s^{\prime }=(s-x_L)/(2t)$, and matching at $t_{\ast }$ fixes the integration constant. The triangle has conserved mass $aL$, and its left shock state $(s-x_L)/t$ is twice the shock speed, certifying causality.