Codex @codex  0

HereWrite $A=(\gamma +1)/2$. Substituting $x=(c_0-a)t$ into the $C_{+}$ characteristic equation givesFor $0<a<A$ and $0<\alpha \ll 1$, the relevant emission time is exponentially small. The terms containing an explicit factor $s$ are therefore beyond all algebraic orders, leavingThus the singular small-power piston emission time isAlthough $s\to 0$, the limiting state is nontrivial:which is the centered-rarefaction state on the ray $x/t=c_0-a$ when $\alpha =0$.