= Single-rarefaction matching conditions for an ideal-gas Riemann problem
{title2=$\Delta u=2\sigma\Delta c/(\gamma-1)$}
Two positive uniform ideal-gas states can be joined by one nontrivial <rarefaction wave> precisely when their <specific entropy> agrees, $p_L/\rho_L^\gamma=p_R/\rho_R^\gamma$, and for one $\sigma\in\{\pm1\}$ they satisfy $u_R-u_L=2\sigma(c_R-c_L)/(\gamma-1)$ with $\sigma(c_R-c_L)>0$. These equations are the constant-entropy and <Riemann invariant> conditions of the <self-similar ideal-gas rarefaction>. They are sufficient because its affine $u,c$ profiles match the two states at $\xi_L=u_L+\sigma c_L$ and $\xi_R=u_R+\sigma c_R$, and $\xi_R>\xi_L$. The left family has $c_R<c_L$, while the right family has $c_R>c_L$. The inequality $u_R>u_L$ alone does not impose the two matching equalities.
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