= Rectangular pulse solution of the inviscid Burgers equation
{title2=$t_*=2L/a,\quad s(t)=x_L+\sqrt{2aLt}\quad(t\geq t_*)$}
= Rectangular pulse solutions of the inviscid Burgers equation
{synonym}
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 $t_*=2L/a$. 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 $s'=(s-x_L)/(2t)$, and matching at $t_*$ 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.
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