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For isentropic flow of a perfect gas, $p=K{\rho }^{\gamma }$ with $\gamma >1$ and $c^2=dp/d\rho$. The mass and momentum equations implyAdding and subtracting the appropriate multiples therefore provesThese are the Riemann invariants for one-dimensional isentropic flow.

In the piston-driven rarefaction wave, the incoming minus invariant has its initial value zero. Thus $c=c_0+(\gamma -1)u/2$. At the piston $u_p=-ft$, giving $c_p=c_0-(\gamma -1)ft/2$. A plus characteristic emitted at time $\tau$ has constant $u=-f\tau$ and $c=c_0-(\gamma -1)f\tau /2$, so its straight trajectory isThe first characteristic is $x=c_{0}t$, ahead of which the gas is undisturbed. Later characteristics have decreasing propagation speed and spread into a rarefaction. More explicitly ${\partial }_{\tau }x=-c_0-(\gamma +1)ft/2+\gamma f\tau <0$ for $\tau \le t<t_v$, with equality only at the vacuum endpoint. They never intersect, so no compressive shock forms.

Since $p/p_0=(c/c_0{)}^{2\gamma /(\gamma -1)}$, the vacuum formation at an accelerating withdrawing piston givesAt $t_v$, density, sound speed and pressure at the gas edge vanish, and the piston velocity is $\boxed{\dot{X}(t_v)=-2c_0/(\gamma -1)}$. Its speed is the positive magnitude $2c_0/(\gamma -1)$. Afterwards the gas edge continues at this limiting velocity, while the piston keeps accelerating away; a vacuum gap of width $f(t-t_v{)}^2/2$ opens and the gas pressure on the piston stays zero.