Past exam of the mathematics course of the University of Cambridge / 2020 / ii / Paper 4 / 39B / b / i / Solution

Forthe adiabatic sound speed isSince $\int (c/\rho )d\rho =-B/\rho =-c$, choose invariants that vanish in the equilibrium state $(u,c)=(0,c_0)$:ConsequentlyIn a right-moving simple wave with $R_{-}=0$, the $R_{+}$ characteristic speed is therefore $u+c=c_0$, independent of amplitude. The transport equation becomes $({\partial }_t+c_0{\partial }_x)R_{+}=0$, so the whole velocity profile translates rigidly at the equilibrium sound speed $c_0$ without nonlinear distortion.