Past exam of the mathematics course of the University of Cambridge / 2020 / ii / Paper 4 / 39B (Waves)

The one-dimensional Isentropic Euler equations areDefine the Riemann invariants for one-dimensional isentropic flowCombining the two equations directly givesso $R_{\pm }$ is constant on the corresponding characteristic curve $dx/dt=u\pm c$.

Initially both invariant disturbances are supported in $0\le x\le L$. Once the two characteristic families have separated, the left-moving packet is a simple wave with $R_{+}$ at its equilibrium value, while the right-moving packet is a simple wave with $R_{-}$ at equilibrium. Between and outside them, both invariants retain their equilibrium values. The schematic spacetime diagram below uses the equilibrium slopes $\pm c_0$ and shows separation after $t=L/(2c_0)$.

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.

Let $\tau$ be the time at which a right-running characteristic curve leaves the piston. The boundary velocity and the characteristic it emits areWiththese relations becomeBecause $d\varphi /d\theta =1-ϵ\cos \theta >0$ and $\varphi (\theta +2\pi )=\varphi (\theta )+2\pi$, $U$ is single-valued and $2\pi$-periodic behind the wavefront. Ahead of the first characteristic, where $\varphi <0$, the fluid remains undisturbed and $U=0$. The implicit relation is Kepler equation; unless $ϵ=0$, its inverse is not a pure sinusoid.

Series reversion givesExpanding the cosine then givesand hence