= Contact discontinuity
{title2=$[u]=[p]=0$}
A contact discontinuity in one-dimensional gas dynamics moves with the common fluid <velocity> $s=u_L=u_R$ and has continuous <pressure>, while <mass density> and <specific entropy> can jump. The mass jump condition $s[\rho]=[\rho u]$ is automatic with common $u=s$; momentum conservation $s[\rho u]=[\rho u^2+p]$ then requires $[p]=0$. The energy jump condition is also satisfied. Thus a contact is a material interface, not a compression shock. A smooth self-similar positive-density flow cannot spread this zero-speed-relative-to-the-fluid family into a fan: $u=x/t$ on an interval would require both $du/d(x/t)=1$ and, from continuity, $du/d(x/t)=0$.
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