= Entropy production in successive weak shocks
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> gives
$$
\frac{\Delta s}{c_v}
=\frac{\gamma^2-1}{12\gamma^2}\delta^2\log P_s\,[1+O(\delta)].
$$
At 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>.
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