Rankine-Hugoniot conditions for a perfect gas

In the rest frame of a one-dimensional shock, states with velocities $w_0,w_1$ satisfyand

For a stationary normal shock in a polytropic perfect gas, the upstream and downstream normal speeds obey $u_1{u}_2=c_{\mathrm{cr}}^2$, where $c_{\mathrm{cr}}$ is the sonic speed on their common Bernoulli streamline.

When the upstream pressure is negligible, a perfect-gas shock has density ratio $(\gamma +1)/(\gamma -1)$. For $\gamma =5/3$, the downstream density is four times the upstream density.

The shock compression ratio is the downstream-to-upstream density ratio. For a strong perfect-gas shock,In the upstream rest frame, the downstream velocity is $2V_s/(\gamma +1)$ and its pressure is $2{\rho }_0{V}_s^2/(\gamma +1)$.

If a piston moves at speed $V$ into stationary perfect gas and drives a shock at speed $U$, then the shock-frame upstream and downstream speeds are $U$ and $U-V$. If $p_2-p_1=\beta p_1$, the jump conditions give

If a shock travels into stationary gas at speed $V$ while the downstream piston and gas travel at $V/3$, then for $\gamma =3$,

Let $P=p_2/p_1$ and $D={\rho }_2/{\rho }_1$ be the pressure and mass density ratios across a normal shock wave in a perfect gas with specific-heat ratio $\gamma$. Eliminating the velocities from the Rankine-Hugoniot conditions for a perfect gas givesAs $P\to \infty$, the ratio tends to the shock compression ratio $(\gamma +1)/(\gamma -1)$.

For the pressure-density Hugoniot relation for a perfect gas, the change in specific entropy is $[s]/c_v=\log P-\gamma \log D(P)$, with $c_v$ the specific heat capacity at constant volume. Its derivative isIt is positive for a compressive normal shock wave with $P>1$. For a weak shock with $P=1+\delta$, integrating the leading term givesThus the entropy production is cubic in the small pressure jump, even though the mass density and temperature changes already appear at first order.

For $N$ equal weak shocks with individual pressure ratio $1+\delta$ and total ratio $P_s=(1+\delta {)}^N$, entropy production in a perfect-gas shock givesAt fixed $P_s>1$, this tends to zero as the compression is divided into increasingly many smaller weak shocks. A single finite normal shock wave instead gives the positive value $\log P_s-\gamma \log D(P_s)$. Gradual compression can consequently approach reversible isentropic flow while an abrupt compression generates finite entropy.