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 is
$$
\frac{d([s]/c_v)}{dP}
=\frac{(\gamma^2-1)(P-1)^2}{P[(\gamma+1)P+\gamma-1][(\gamma-1)P+\gamma+1]}.
$$
It is positive for a compressive <normal shock wave> with $P>1$. For a <weak shock> with $P=1+\delta$, integrating the leading term gives
$$
\frac{[s]}{c_v}=\frac{\gamma^2-1}{12\gamma^2}\delta^3+O(\delta^4).
$$
Thus the <entropy production> is cubic in the small <pressure> jump, even though the <mass density> and <temperature> changes already appear at first order.

Entropy production in a perfect-gas shock

Let $m=\rho_1u_1=\rho_2u_2$ be the signed mass flux. Momentum conservation gives
$$
m^2\left(\frac1{\rho_1}-\frac1{\rho_2}\right)=p_2-p_1.
$$
Dividing the energy-flux equation by $m$ and eliminating $m^2$ yields the <pressure-density Hugoniot relation for a perfect gas>:
$$
\frac{\gamma}{\gamma-1}\left(\frac{p_2}{\rho_2}-\frac{p_1}{\rho_1}\right)
=\frac{p_2-p_1}{2}\left(\frac1{\rho_1}+\frac1{\rho_2}\right).
$$
With $P=p_2/p_1$ and $D=\rho_2/\rho_1$, this becomes
$$
\frac{2\gamma}{\gamma-1}(P-D)=(P-1)(D+1).
$$
Solving for the <mass density> ratio gives
$$
\boxed{D=\frac{(\gamma+1)P+\gamma-1}{(\gamma-1)P+\gamma+1}.}
$$
For a compressive <normal shock wave>, $P>1$ and $D>1$; as $P\to\infty$, the finite <shock compression ratio> is $(\gamma+1)/(\gamma-1)$.

Solution

Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 314 / 2 / b / Solution

For a <perfect gas> with constant <specific heat capacity> $c_v$, the <first law of thermodynamics> gives
$$
ds=c_v\,d\log\left(\frac p{\rho^\gamma}\right).
$$
Using the <pressure-density Hugoniot relation for a perfect gas>, the <entropy production in a perfect-gas shock> is therefore
$$
\boxed{\frac{[s]}{c_v}=\log P-\gamma\log\left[\frac{(\gamma+1)P+\gamma-1}{(\gamma-1)P+\gamma+1}\right].}
$$
This is positive for $P>1$, as required by the <Second law of thermodynamics> for the physical compressive <normal shock wave>.

Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 314 / 2 / c / Solution

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> gives
$$
\frac{\gamma}{\gamma-1}(p_2/\rho_2-p_1/\rho_1)
=\frac12(p_2-p_1)(1/\rho_1+1/\rho_2),
\qquad
D=\frac{(\gamma+1)P+\gamma-1}{(\gamma-1)P+\gamma+1}.
$$
As $P\to\infty$, the ratio tends to the <shock compression ratio> $(\gamma+1)/(\gamma-1)$.

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 Incoming links: Pressure-density Hugoniot relation for a perfect gas