 Rarefaction wave

A rarefaction wave is a continuous expanding fan of characteristic curves joining constant states of a hyperbolic system or scalar conservation law. For $u_t+f(u{)}_x=0$ with $f^{\prime \prime }>0$ between $u_L<u_R$, the Riemann problem has the self-similar solutionTo derive the fan, substitute $u(t,x)=V(\xi )$, $\xi =x/t$, into the smooth equation. It gives $(f^{\prime }(V)-\xi )V^{\prime }=0$, so the nonconstant branch has $f^{\prime }(V)=\xi$. Strict convexity makes $f^{\prime }$ invertible on the state interval, and the two fan edges match the exterior states continuously. Derivatives can jump at those edges.

For the Inviscid Burgers equation with $u_L=0$ and $u_R=1$, the solution is $0$ for $x\le 0$, $x/t$ for $0<x<t$, and $1$ for $x\ge t$. The characteristic speed equals $u$, so the gap between the outgoing exterior characteristic curves fills with all intermediate speeds. This is the selected entropy solution, whereas the corresponding expansion discontinuity is only a weak solution.^[ref]