= Rarefaction wave
{wiki}
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 $f''>0$ between $u_L<u_R$, the <Riemann problem> has the <self-similar solution>
$$
u(t,x)=\begin{cases}u_L,&x/t\leq f'(u_L),\\(f')^{-1}(x/t),&f'(u_L)<x/t<f'(u_R),\\u_R,&x/t\geq f'(u_R),\end{cases}\qquad t>0.
$$
To derive the fan, substitute $u(t,x)=V(\xi)$, $\xi=x/t$, into the smooth equation. It gives $(f'(V)-\xi)V'=0$, so the nonconstant branch has $f'(V)=\xi$. Strict <convexity> makes $f'$ 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 $x\leq0$, $x/t$ for $0<x<t$, and $1$ for $x\geq t$. 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>.<Riemann Problems and Jupyter Solutions by David I. Ketcheson, Randall J. LeVeque, and Mauricio J. del Razo/Burgers' equation>[Rarefaction wave; The entropy condition]{ref}
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