Past exam of the theoretical mechanics course of the University of Oxford / 2009 / B / 6 / c / Solution

The second law requires the specific entropy to increase along the fluid's passage through a nontrivial physical shock: $s_1>s_0$. For a perfect gas this selects a compressive shock, whose upstream shock-frame Mach number is greater than one and whose downstream shock-frame flow is subsonic.

Here the upstream Mach number is $M_0=V/c_0$. The quadratic gives $V^2-c_0^2=(\gamma +1)UV/2>0$, so $\boxed{M_0>1}$. Density and pressure both rise. With $c_1^2=\gamma p_1/{\rho }_1$, one also has $V-U<c_1$. For example, put $M=V/c_0$ and use the normal-shock identityThe characteristic geometry is therefore $c_0<V<U+c_1$. The plus-family characteristics from the gas ahead are overtaken by the shock, while those behind catch up with it; they converge into the discontinuity. The minus-family speeds are $-c_0$ ahead and $U-c_1$ behind. The piston is the line $x=Ut$ and the shock is the steeper-in-$x/t$ line $x=Vt$.