= 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>.
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