= Vacuum formation at an accelerating withdrawing piston
{title2=$t_v=2c_0/[f(\gamma-1)]$}
An initially resting perfect gas with adiabatic index $\gamma>1$ next to a piston $X(t)=-ft^2/2$ develops a smooth rarefaction. The incoming invariant gives $c_p=c_0-(\gamma-1)ft/2$. Pressure falls as $p_p=p_0(1-t/t_v)^{2\gamma/(\gamma-1)}$ until $t_v$, when the piston speed is $2c_0/(\gamma-1)$. Thereafter the gas edge moves at that limiting constant speed while the piston accelerates away, opening a vacuum gap.
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