# Waves

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Paper 1, Section II, 40A

commentCompressible fluid of equilibrium density $\rho_{0}$, pressure $p_{0}$ and sound speed $c_{0}$ is contained in the region between an inner rigid sphere of radius $R$ and an outer elastic sphere of equilibrium radius $2 R$. The elastic sphere is made to oscillate radially in such a way that it exerts a spherically symmetric, perturbation pressure $\tilde{p}=\epsilon p_{0} \cos \omega t$ on the fluid at $r=2 R$, where $\epsilon \ll 1$ and the frequency $\omega$ is sufficiently small that

$\alpha \equiv \frac{\omega R}{c_{0}} \leqslant \frac{\pi}{2}$

You may assume that the acoustic velocity potential satisfies the wave equation

$\frac{\partial^{2} \phi}{\partial t^{2}}=c_{0}^{2} \nabla^{2} \phi$

(a) Derive an expression for $\phi(r, t)$.

(b) Hence show that the net radial component of the acoustic intensity (wave-energy flux) $\mathbf{I}=\tilde{p} \mathbf{u}$ is zero when averaged appropriately in a way you should define. Interpret this result physically.

(c) Briefly discuss the possible behaviour of the system if the forcing frequency $\omega$ is allowed to increase to larger values.

$\left[\right.$ For a spherically symmetric variable $\left.\psi(r, t), \nabla^{2} \psi=\frac{1}{r} \frac{\partial^{2}}{\partial r^{2}}(r \psi) .\right]$

Paper 2, Section II, 40A

commentA semi-infinite elastic medium with shear modulus $\mu$ and shear-wave speed $c_{s}$ lies in $z \leqslant 0$. Above it, there is a layer $0 \leqslant z \leqslant h$ of a second elastic medium with shear modulus $\bar{\mu}$ and shear-wave speed $\bar{c}_{s}<c_{s}$. The top boundary is stress-free. Consider a monochromatic SH-wave propagating in the $x$-direction at speed $c$ with wavenumber $k>0$.

(a) Derive the dispersion relation

$\tan \left[k h \sqrt{c^{2} / \bar{c}_{s}^{2}-1}\right]=\frac{\mu}{\bar{\mu}} \frac{\sqrt{1-c^{2} / c_{s}^{2}}}{\sqrt{c^{2} / \bar{c}_{s}^{2}-1}}$

for trapped modes with no disturbance as $z \rightarrow-\infty$.

(b) Show graphically that there is always a zeroth mode, and show that the other modes have cut-off frequencies

$\omega_{c}^{(n)}=\frac{n \pi \bar{c}_{s} c_{s}}{h \sqrt{c_{s}^{2}-\bar{c}_{s}^{2}}}$

where $n$ is a positive integer. Sketch a graph of frequency $\omega$ against $k$ for the $n=1$ mode showing the behaviour near cut-off and for large $k$.

Paper 3, Section II, 39A

commentConsider a two-dimensional stratified fluid of sufficiently slowly varying background density $\rho_{b}(z)$ that small-amplitude vertical-velocity perturbations $w(x, z, t)$ can be assumed to satisfy the linear equation

$\nabla^{2}\left(\frac{\partial^{2} w}{\partial t^{2}}\right)+N^{2}(z) \frac{\partial^{2} w}{\partial x^{2}}=0, \quad \text { where } N^{2}=\frac{-g}{\rho_{0}} \frac{d \rho_{b}}{d z}$

and $\rho_{0}$ is a constant. The background density profile is such that $N^{2}$ is piecewise constant with $N^{2}=N_{0}^{2}>0$ for $|z|>L$ and with $N^{2}=0$ in a layer $|z|<L$ of uniform density $\rho_{0}$.

A monochromatic internal wave of amplitude $A_{I}$ is incident on the intermediate layer from $z=-\infty$, and produces velocity perturbations of the form

$w(x, z, t)=\widehat{w}(z) e^{i(k x-\omega t)}$

where $k>0$ and $0<\omega<N_{0}$.

(a) Show that the vertical variations have the form

$\widehat{w}(z)= \begin{cases}A_{I} \exp [-i m(z+L)]+A_{R} \exp [i m(z+L)] \quad \text { for } z<-L \\ B_{C} \cosh k z+B_{S} \sinh k z & \text { for }|z|<L \\ A_{T} \exp [-i m(z-L)] & \text { for } z>L\end{cases}$

where $A_{R}, B_{C}, B_{S}$ and $A_{T}$ are (in general) complex amplitudes and

$m=k \sqrt{\frac{N_{0}^{2}}{\omega^{2}}-1}$

In particular, you should justify the choice of signs for the coefficients involving $m$.

(b) What are the appropriate boundary conditions to impose on $\widehat{w}$at $z=\pm L$ to determine the unknown amplitudes?

(c) Apply these boundary conditions to show that

$\frac{A_{T}}{A_{I}}=\frac{2 i m k}{2 i m k \cosh 2 \alpha+\left(k^{2}-m^{2}\right) \sinh 2 \alpha},$

where $\alpha=k L$.

(d) Hence show that

$\left|\frac{A_{T}}{A_{I}}\right|^{2}=\left[1+\left(\frac{\sinh 2 \alpha}{\sin 2 \psi}\right)^{2}\right]^{-1}$

where $\psi$ is the angle between the incident wavevector and the downward vertical.

Paper 4, Section II, 39A

commentA plane shock is moving with speed $U$ into a perfect gas. Ahead of the shock the gas is at rest with pressure $p_{1}$ and density $\rho_{1}$, while behind the shock the velocity, pressure and density of the gas are $u_{2}, p_{2}$ and $\rho_{2}$ respectively.

(a) Write down the Rankine-Hugoniot relations across the shock, briefly explaining how they arise.

(b) Show that

$\frac{\rho_{1}}{\rho_{2}}=\frac{2 c_{1}^{2}+(\gamma-1) U^{2}}{(\gamma+1) U^{2}}$

where $c_{1}^{2}=\gamma p_{1} / \rho_{1}$ and $\gamma$ is the ratio of the specific heats of the gas.

(c) Now consider a change of frame such that the shock is stationary and the gas has a component of velocity $U$ parallel to the shock on both sides. Deduce that a stationary shock inclined at a 45 degree angle to an incoming stream of Mach number $M=\sqrt{2} U / c_{1}$ deflects the flow by an angle $\delta$ given by

$\tan \delta=\frac{M^{2}-2}{\gamma M^{2}+2}$

$\left[\right.$ Note that $\left.\tan (\alpha-\beta)=\frac{\tan \alpha-\tan \beta}{1+\tan \alpha \tan \beta} .\right]$

Paper 1, Section II, B

comment(a) Write down the linearised equations governing motion of an inviscid compressible fluid at uniform entropy. Assuming that the velocity is irrotational, show that the velocity potential $\phi(\mathbf{x}, t)$ satisfies the wave equation and identify the wave speed $c_{0}$. Obtain from these linearised equations the energy-conservation equation

$\frac{\partial E}{\partial t}+\nabla \cdot \mathbf{I}=0$

and give expressions for the acoustic-energy density $E$ and the acoustic-energy flux, or intensity, I.

(b) Inviscid compressible fluid with density $\rho_{0}$ and sound speed $c_{0}$ occupies the regions $y<0$ and $y>0$, which are separated by a thin elastic membrane at an undisturbed position $y=0$. The membrane has mass per unit area $m$ and is under a constant tension $T$. Small displacements of the membrane to $y=\eta(x, t)$ are coupled to small acoustic disturbances in the fluid with velocity potential $\phi(x, y, t)$.

(i) Write down the (linearised) kinematic and dynamic boundary conditions at the membrane. [Hint: The elastic restoring force on the membrane is like that on a stretched string.]

(ii) Show that the dispersion relation for waves proportional to $\cos (k x-\omega t)$ propagating along the membrane with $|\phi| \rightarrow 0$ as $y \rightarrow \pm \infty$ is given by

$\left\{m+\frac{2 \rho_{0}}{\left(k^{2}-\omega^{2} / c_{0}^{2}\right)^{1 / 2}}\right\} \omega^{2}=T k^{2}$

Interpret this equation by explaining physically why all disturbances propagate with phase speed $c$ less than $(T / m)^{1 / 2}$ and why $c(k) \rightarrow 0$ as $k \rightarrow 0$.

(iii) Show that in such a wave the component $\left\langle I_{y}\right\rangle$ of mean acoustic intensity perpendicular to the membrane is zero.

Paper 2, Section II, 39B

commentSmall displacements $\mathbf{u}(\mathbf{x}, t)$ in a homogeneous elastic medium are governed by the equation

$\rho \frac{\partial^{2} \mathbf{u}}{\partial t^{2}}=(\lambda+2 \mu) \boldsymbol{\nabla}(\boldsymbol{\nabla} \cdot \mathbf{u})-\mu \boldsymbol{\nabla} \wedge(\boldsymbol{\nabla} \wedge \mathbf{u})$

where $\rho$ is the density, and $\lambda$ and $\mu$ are the Lamé constants.

(a) Show that the equation supports two types of harmonic plane-wave solutions, $\mathbf{u}=\mathbf{A} \exp [i(\mathbf{k} \cdot \mathbf{x}-\omega t)]$, with distinct wavespeeds, $c_{P}$ and $c_{S}$, and distinct polarizations. Write down the direction of the displacement vector A for a $P$-wave, an $S V$-wave and an $S H$-wave, in each case for the wavevector $(k, 0, m)$.

(b) Given $k$ and $c$, with $c>c_{P}\left(>c_{S}\right)$, explain how to construct a superposition of $P$-waves with wavenumbers $\left(k, 0, m_{P}\right)$ and $\left(k, 0,-m_{P}\right)$, such that

$\mathbf{u}(x, z, t)=e^{i k(x-c t)}\left(f_{1}(z), 0, i f_{3}(z)\right)$

where $f_{1}(z)$ is an even function, and $f_{1}$ and $f_{3}$ are both real functions, to be determined. Similarly, find a superposition of $S V$-waves with $\mathbf{u}$ again in the form $(*)$.

(c) An elastic waveguide consists of an elastic medium in $-H<z<H$ with rigid boundaries at $z=\pm H$. Using your answers to part (b), show that the waveguide supports propagating eigenmodes that are a mixture of $P$ - and $S V$-waves, and have dispersion relation $c(k)$ given by

$a \tan (a k H)=-\frac{\tan (b k H)}{b}, \quad \text { where } \quad a=\left(\frac{c^{2}}{c_{P}^{2}}-1\right)^{1 / 2} \quad \text { and } \quad b=\left(\frac{c^{2}}{c_{S}^{2}}-1\right)^{1 / 2}$

Sketch the two sides of the dispersion relationship as functions of $c$. Explain briefly why there are infinitely many solutions.

Paper 3, Section II, B

commentThe dispersion relation for capillary waves on the surface of deep water is

$\omega^{2}=S^{2}|k|^{3}$

where $S=(T / \rho)^{1 / 2}, \rho$ is the density and $T$ is the coefficient of surface tension. The free surface $z=\eta(x, t)$ is undisturbed for $t<0$, when it is suddenly impacted by an object, giving the initial conditions at time $t=0$ :

$\eta=0 \quad \text { and } \quad \frac{\partial \eta}{\partial t}= \begin{cases}-W, & |x|<\epsilon \\ 0, & |x|>\epsilon\end{cases}$

where $W$ is a constant.

(i) Use Fourier analysis to find an integral expression for $\eta(x, t)$ when $t>0$.

(ii) Use the method of stationary phase to find the asymptotic behaviour of $\eta(V t, t)$ for fixed $V>0$ as $t \rightarrow \infty$, for the case $V \ll \epsilon^{-1 / 2} S$. Show that the result can be written in the form

$\eta(x, t) \sim \frac{W \epsilon S t^{2}}{x^{5 / 2}} F\left(\frac{x^{3}}{S^{2} t^{2}}\right)$

and determine the function $F$.

(iii) Give a brief physical interpretation of the link between the condition $\epsilon V^{2} / S^{2} \ll$ 1 and the simple dependence on the product $W \epsilon$.

[You are given that $\int_{-\infty}^{\infty} e^{\pm i a u^{2}} d u=(\pi / a)^{1 / 2} e^{\pm i \pi / 4} \quad$ for $a>0 .$ ]

Paper 4, Section II, B

comment(a) Show that the equations for one-dimensional unsteady flow of an inviscid compressible fluid at constant entropy can be put in the form

$\left(\frac{\partial}{\partial t}+(u \pm c) \frac{\partial}{\partial x}\right) R_{\pm}=0$

where $u$ and $c$ are the fluid velocity and the local sound speed, respectively, and the Riemann invariants $R_{\pm}$are to be defined.

Such a fluid occupies a long narrow tube along the $x$-axis. For times $t<0$ it is at rest with uniform pressure $p_{0}$, density $\rho_{0}$ and sound speed $c_{0}$. At $t=0$ a finite segment, $0 \leqslant x \leqslant L$, is disturbed so that $u=U(x)$ and $c=c_{0}+C(x)$, with $U=C=0$ for $x \leqslant 0$ and $x \geqslant L$. Explain, with the aid of a carefully labelled sketch, how two independent simple waves emerge after some time. You may assume that no shock waves form.

(b) A fluid has the adiabatic equation of state

$p(\rho)=A-\frac{B^{2}}{\rho}$

where $A$ and $B$ are positive constants and $\rho>B^{2} / A$.

(i) Calculate the Riemann invariants for this fluid, and express $u \pm c$ in terms of $R_{\pm}$ and $c_{0}$. Deduce that in a simple wave with $R_{-}=0$ the velocity field translates, without any nonlinear distortion, at the equilibrium sound speed $c_{0}$.

(ii) At $t=0$ this fluid occupies $x>0$ and is at rest with uniform pressure, density and sound speed. For $t>0$ a piston initially at $x=0$ executes simple harmonic motion with position $x(t)=a \sin \omega t$, where $a \omega<c_{0}$. Show that $u(x, t)=U(\phi)$, where $\phi=\omega\left(t-x / c_{0}\right)$, for some function $U$ that is zero for $\phi<0$ and is $2 \pi$-periodic, but not simple harmonic, for $\phi>0$. By approximately inverting the relationship between $\phi$ and the time $\tau$ that a characteristic leaves the piston for the case $\epsilon=a \omega / c_{0} \ll 1$, show that

$U(\phi)=a \omega\left(\cos \phi-\epsilon \sin ^{2} \phi-\frac{3}{2} \epsilon^{2} \sin ^{2} \phi \cos \phi+O\left(\epsilon^{3}\right)\right) \quad \text { for } \quad \phi>0$

Paper 1, Section II, A

commentThe equation of state relating pressure $p$ to density $\rho$ for a perfect gas is given by

$\frac{p}{p_{0}}=\left(\frac{\rho}{\rho_{0}}\right)^{\gamma}$

where $p_{0}$ and $\rho_{0}$ are constants, and $\gamma>1$ is the specific heat ratio.

(a) Starting from the equations for one-dimensional unsteady flow of a perfect gas of uniform entropy, show that the Riemann invariants,

$R_{\pm}=u \pm \frac{2}{\gamma-1}\left(c-c_{0}\right)$

are constant on characteristics $C_{\pm}$given by

$\frac{d x}{d t}=u \pm c$

where $u(x, t)$ is the velocity of the gas, $c(x, t)$ is the local speed of sound, and $c_{0}$ is a constant.

(b) Such an ideal gas initially occupies the region $x>0$ to the right of a piston in an infinitely long tube. The gas and the piston are initially at rest. At time $t=0$ the piston starts moving to the left with path given by

$x=X_{p}(t), \quad \text { with } X_{p}(0)=0 .$

(i) Solve for $u(x, t)$ and $\rho(x, t)$ in the region $x>X_{p}(t)$ under the assumptions that $-\frac{2 c_{0}}{\gamma-1}<\dot{X}_{p}<0$ and that $\left|\dot{X}_{p}\right|$ is monotonically increasing, where dot indicates a time derivative.

[It is sufficient to leave the solution in implicit form, i.e. for given $x, t$ you should not attempt to solve the $C_{+}$characteristic equation explicitly.]

(ii) Briefly outline the behaviour of $u$ and $\rho$ for times $t>t_{c}$, where $t_{c}$ is the solution to $\dot{X}_{p}\left(t_{c}\right)=-\frac{2 c_{0}}{\gamma-1}$.

(iii) Now suppose,

$X_{p}(t)=-\frac{t^{1+\alpha}}{1+\alpha}$

where $\alpha \geqslant 0$. For $0<\alpha \ll 1$, find a leading-order approximation to the solution of the $C_{+}$characteristic equation when $x=c_{0} t-a t, 0<a<\frac{1}{2}(\gamma+1)$ and $t=O(1)$.

[Hint: You may find it useful to consider the structure of the characteristics in the limiting case when $\alpha=0$.]

Paper 2, Section II, A

commentThe linearised equation of motion governing small disturbances in a homogeneous elastic medium of density $\rho$ is

$\rho \frac{\partial^{2} \mathbf{u}}{\partial t^{2}}=(\lambda+\mu) \nabla(\boldsymbol{\nabla} \cdot \mathbf{u})+\mu \nabla^{2} \mathbf{u}$

where $\mathbf{u}(\mathbf{x}, t)$ is the displacement, and $\lambda$ and $\mu$ are the Lamé moduli.

(a) The medium occupies the region between a rigid plane boundary at $y=0$ and a free surface at $y=h$. Show that $S H$ waves can propagate in the $x$-direction within this region, and find the dispersion relation for such waves.

(b) For each mode, deduce the cutoff frequency, the phase velocity and the group velocity. Plot the latter two velocities as a function of wavenumber.

(c) Verify that in an average sense (to be made precise), the wave energy flux is equal to the wave energy density multiplied by the group velocity.

[You may assume that the elastic energy per unit volume is given by

$\left.E_{p}=\frac{1}{2} \lambda e_{i i} e_{j j}+\mu e_{i j} e_{i j} \cdot\right]$

Paper 3, Section II, A

comment(a) Derive the wave equation for perturbation pressure for linearised sound waves in a compressible gas.

(b) For a single plane wave show that the perturbation pressure and the velocity are linearly proportional and find the constant of proportionality, i.e. the acoustic impedance.

(c) Gas occupies a tube lying parallel to the $x$-axis. In the regions $x<0$ and $x>L$ the gas has uniform density $\rho_{0}$ and sound speed $c_{0}$. For $0<x<L$ the temperature of the gas has been adjusted so that it has uniform density $\rho_{1}$ and sound speed $c_{1}$. A harmonic plane wave with frequency $\omega$ and unit amplitude is incident from $x=-\infty$. If $T$ is the (in general complex) amplitude of the wave transmitted into $x>L$, show that

$|T|=\left(\cos ^{2} k_{1} L+\frac{1}{4}\left(\lambda+\lambda^{-1}\right)^{2} \sin ^{2} k_{1} L\right)^{-\frac{1}{2}}$

where $\lambda=\rho_{1} c_{1} / \rho_{0} c_{0}$ and $k_{1}=\omega / c_{1}$. Discuss both of the limits $\lambda \ll 1$ and $\lambda \gg 1$.

Paper 4, Section II, A

comment(a) Assuming a slowly-varying two-dimensional wave pattern of the form

$\varphi(\mathbf{x}, t)=A(\mathbf{x}, t ; \varepsilon) \exp \left[\frac{i}{\varepsilon} \theta(\mathbf{x}, t)\right]$

where $0<\varepsilon \ll 1$, and a local dispersion relation $\omega=\Omega(\mathbf{k} ; \mathbf{x}, t)$, derive the ray tracing equations,

$\frac{d x_{i}}{d t}=\frac{\partial \Omega}{\partial k_{i}}, \quad \frac{d \omega}{d t}=\frac{\partial \Omega}{\partial t}, \quad \frac{d k_{i}}{d t}=-\frac{\partial \Omega}{\partial x_{i}}, \quad \frac{1}{\varepsilon} \frac{d \theta}{d t}=-\omega+k_{j} \frac{\partial \Omega}{\partial k_{j}}$

for $i, j=1,2$, explaining carefully the meaning of the notation used.

(b) For a homogeneous, time-independent (but not necessarily isotropic) medium, show that all rays are straight lines. When the waves have zero frequency, deduce that if the point $\mathbf{x}$ lies on a ray emanating from the origin in the direction given by a unit vector $\widehat{\mathbf{c}}_{\mathrm{g}}$, then

$\theta(\mathbf{x})=\theta(\mathbf{0})+\widehat{\mathbf{c}}_{\mathbf{g}} \cdot \mathbf{k}|\mathbf{x}|$

(c) Consider a stationary obstacle in a steadily moving homogeneous medium which has the dispersion relation

$\Omega=\alpha\left(k_{1}^{2}+k_{2}^{2}\right)^{1 / 4}-V k_{1}$

where $(V, 0)$ is the velocity of the medium and $\alpha>0$ is a constant. The obstacle generates a steady wave system. Writing $\left(k_{1}, k_{2}\right)=\kappa(\cos \phi, \sin \phi)$, with $\kappa>0$, show that the wave satisfies

$\kappa=\frac{\alpha^{2}}{V^{2} \cos ^{2} \phi}, \quad \widehat{\mathbf{c}}_{\mathrm{g}}=(\cos \psi, \sin \psi)$

where $\psi$ is defined by

$\tan \psi=-\frac{\tan \phi}{1+2 \tan ^{2} \phi}$

with $\frac{1}{2} \pi<\psi<\frac{3}{2} \pi$ and $-\frac{1}{2} \pi<\phi<\frac{1}{2} \pi$. Deduce that the wave pattern occupies a wedge of semi-angle $\tan ^{-1}\left(2^{-3 / 2}\right)$, extending in the negative $x_{1}$-direction.

Paper 1, Section II, 39C

commentDerive the wave equation governing the velocity potential for linearised sound waves in a perfect gas. How is the pressure disturbance related to the velocity potential?

A high pressure gas with unperturbed density $\rho_{0}$ is contained within a thin metal spherical shell which makes small amplitude spherically symmetric vibrations. Let the metal shell have radius $a$, mass $m$ per unit surface area, and an elastic stiffness which tries to restore the radius to its equilibrium value $a_{0}$ with a force $\kappa\left(a-a_{0}\right)$ per unit surface area. Assume that there is a vacuum outside the spherical shell. Show that the frequencies $\omega$ of vibration satisfy

$\theta^{2}\left(1+\frac{\alpha}{\theta \cot \theta-1}\right)=\frac{\kappa a_{0}^{2}}{m c_{0}^{2}}$

where $\theta=\omega a_{0} / c_{0}, \alpha=\rho_{0} a_{0} / m$, and $c_{0}$ is the speed of sound in the undisturbed gas. Briefly comment on the existence of solutions.

[Hint: In terms of spherical polar coordinates you may assume that for a function $\psi \equiv \psi(r)$,

$\nabla^{2} \psi=\frac{1}{r} \frac{\partial^{2}}{\partial r^{2}}(r \psi)$

Paper 2, Section II, C

commentA perfect gas occupies the region $x>0$ of a tube that lies parallel to the $x$-axis. The gas is initially at rest, with density $\rho_{1}$, pressure $p_{1}$, speed of sound $c_{1}$ and specific heat ratio $\gamma$. For times $t>0$ a piston, initially at $x=0$, is pushed into the gas at a constant speed $V$. A shock wave propagates at constant speed $U$ into the undisturbed gas ahead of the piston. Show that the excess pressure in the gas next to the piston, $p_{2}-p_{1} \equiv \beta p_{1}$, is given implicitly by the expression

$V^{2}=\frac{2 \beta^{2}}{2 \gamma+(\gamma+1) \beta} \frac{p_{1}}{\rho_{1}}$

Show also that

$\frac{U^{2}}{c_{1}^{2}}=1+\frac{\gamma+1}{2 \gamma} \beta$

and interpret this result.

[Hint: You may assume for a perfect gas that the speed of sound is given by

$c^{2}=\frac{\gamma p}{\rho}$

and that the internal energy per unit mass is given by

$e=\frac{1}{\gamma-1} \frac{p}{\rho}$

Paper 3, Section II, 40C

commentDerive the ray-tracing equations

$\frac{d x_{i}}{d t}=\frac{\partial \Omega}{\partial k_{i}}, \quad \frac{d k_{i}}{d t}=-\frac{\partial \Omega}{\partial x_{i}}, \quad \frac{d \omega}{d t}=\frac{\partial \Omega}{\partial t}$

for wave propagation through a slowly-varying medium with local dispersion relation $\omega=\Omega(\mathbf{k} ; \mathbf{x}, t)$, where $\omega$ and $\mathbf{k}$ are the frequency and wavevector respectively, $t$ is time and $\mathbf{x}=(x, y, z)$ are spatial coordinates. The meaning of the notation $d / d t$ should be carefully explained.

A slowly-varying medium has a dispersion relation $\Omega(\mathbf{k} ; \mathbf{x}, t)=k c(z)$, where $k=|\mathbf{k}|$. State and prove Snell's law relating the angle $\psi$ between a ray and the $z$-axis to $c$.

Consider the case of a medium with wavespeed $c=c_{0}\left(1+\beta^{2} z^{2}\right)$, where $\beta$ and $c_{0}$ are positive constants. Show that a ray that passes through the origin with wavevector $k(\cos \phi, 0, \sin \phi)$, remains in the region

$|z| \leqslant z_{m} \equiv \frac{1}{\beta}\left[\frac{1}{|\cos \phi|}-1\right]^{1 / 2}$

By considering an approximation to the equation for a ray in the region $\left|z_{m}-z\right| \ll \beta^{-1}$, or otherwise, determine the path of a ray near $z_{m}$, and hence sketch rays passing through the origin for a few sample values of $\phi$ in the range $0<\phi<\pi / 2$.

Paper 4, Section II, C

commentA physical system permits one-dimensional wave propagation in the $x$-direction according to the equation

$\left(1-2 \frac{\partial^{2}}{\partial x^{2}}+\frac{\partial^{4}}{\partial x^{4}}\right) \frac{\partial^{2} \varphi}{\partial t^{2}}+\frac{\partial^{4} \varphi}{\partial x^{4}}=0$

Derive the corresponding dispersion relation and sketch graphs of frequency, phase velocity and group velocity as functions of the wavenumber. Waves of what wavenumber are at the front of a dispersing wave train arising from a localised initial disturbance? For waves of what wavenumbers do wave crests move faster or slower than a packet of waves?

Find the solution of the above equation for the initial disturbance given by

$\varphi(x, 0)=\int_{-\infty}^{\infty} 2 A(k) e^{i k x} d k, \quad \frac{\partial \varphi}{\partial t}(x, 0)=0$

where $A^{*}(-k)=A(k)$, and $A^{*}$ is the complex conjugate of $A$. Let $V=x / t$ be held fixed. Use the method of stationary phase to obtain a leading-order approximation to this solution for large $t$ when $0<V<V_{m}=(3 \sqrt{3}) / 8$, where the solutions for the stationary points should be left in implicit form.

Very briefly discuss the nature of the solutions for $-V_{m}<V<0$ and $|V|>V_{m}$.

[Hint: You may quote the result that the large time behaviour of

$\Phi(x, t)=\int_{-\infty}^{\infty} A(k) e^{i k x-i \omega(k) t} d k$

due to a stationary point $k=\alpha$, is given by

$\Phi(x, t) \sim\left(\frac{2 \pi}{\left|\omega^{\prime \prime}(\alpha)\right| t}\right)^{\frac{1}{2}} A(\alpha) e^{i \alpha x-i \omega(\alpha) t+i \sigma \pi / 4}$

where $\left.\sigma=-\operatorname{sgn}\left(\omega^{\prime \prime}(\alpha)\right) .\right]$

Paper 1, Section II, B

commentDerive the wave equation governing the pressure disturbance $\tilde{p}$, for linearised, constant entropy sound waves in a compressible inviscid fluid of density $\rho_{0}$ and sound speed $c_{0}$, which is otherwise at rest.

Consider a harmonic acoustic plane wave with wavevector $\mathbf{k}_{I}=k_{I}(\sin \theta, \cos \theta, 0)$ and unit-amplitude pressure disturbance. Determine the resulting velocity field $\mathbf{u}$.

Consider such an acoustic wave incident from $y<0$ on a thin elastic plate at $y=0$. The regions $y<0$ and $y>0$ are occupied by gases with densities $\rho_{1}$ and $\rho_{2}$, respectively, and sound speeds $c_{1}$ and $c_{2}$, respectively. The kinematic boundary conditions at the plate are those appropriate for an inviscid fluid, and the (linearised) dynamic boundary condition is

$m \frac{\partial^{2} \eta}{\partial t^{2}}+B \frac{\partial^{4} \eta}{\partial x^{4}}+[\tilde{p}(x, 0, t)]_{-}^{+}=0$

where $m$ and $B$ are the mass and bending moment per unit area of the plate, and $y=\eta(x, t)$ (with $\left|\mathbf{k}_{I} \eta\right| \ll 1$ ) is its perturbed position. Find the amplitudes of the reflected and transmitted pressure perturbations, expressing your answers in terms of the dimensionless parameter

$\beta=\frac{k_{I} \cos \theta\left(m c_{1}^{2}-B k_{I}^{2} \sin ^{4} \theta\right)}{\rho_{1} c_{1}^{2}}$

(i) If $\rho_{1}=\rho_{2}=\rho_{0}$ and $c_{1}=c_{2}=c_{0}$, under what condition is the incident wave perfectly transmitted?

(ii) If $\rho_{1} c_{1} \gg \rho_{2} c_{2}$, comment on the reflection coefficient, and show that waves incident at a sufficiently large angle are reflected as if from a pressure-release surface (i.e. an interface where $\tilde{p}=0$ ), no matter how large the plate mass and bending moment may be.

Paper 2, Section II, B

commentShow that, for a one-dimensional flow of a perfect gas (with $\gamma>1$ ) at constant entropy, the Riemann invariants $R_{\pm}=u \pm 2\left(c-c_{0}\right) /(\gamma-1)$ are constant along characteristics $d x / d t=u \pm c .$

Define a simple wave. Show that in a right-propagating simple wave

$\frac{\partial u}{\partial t}+\left(c_{0}+\frac{1}{2}(\gamma+1) u\right) \frac{\partial u}{\partial x}=0$

In some circumstances, dissipative effects may be modelled by

$\frac{\partial u}{\partial t}+\left(c_{0}+\frac{1}{2}(\gamma+1) u\right) \frac{\partial u}{\partial x}=-\alpha u$

where $\alpha$ is a positive constant. Suppose also that $u$ is prescribed at $t=0$ for all $x$, say $u(x, 0)=u_{0}(x)$. Demonstrate that, unless a shock develops, a solution of the form

$u(x, t)=u_{0}(\xi) e^{-\alpha t}$

can be found, where, for each $x$ and $t, \xi$ is determined implicitly as the solution of the equation

$x-c_{0} t=\xi+\frac{\gamma+1}{2 \alpha}\left(1-e^{-\alpha t}\right) u_{0}(\xi)$

Deduce that, despite the presence of dissipative effects, a shock will still form at some $(x, t)$ unless $\alpha>\alpha_{c}$, where

$\alpha_{c}=\frac{1}{2}(\gamma+1) \max _{u_{0}^{\prime}<0}\left|u_{0}^{\prime}(\xi)\right|$

Paper 3, Section II, B

commentWaves propagating in a slowly-varying medium satisfy the local dispersion relation $\omega=\Omega(\mathbf{k} ; \mathbf{x}, t)$ in the standard notation. Derive the ray-tracing equations

$\frac{d x_{i}}{d t}=\frac{\partial \Omega}{\partial k_{i}}, \quad \frac{d k_{i}}{d t}=-\frac{\partial \Omega}{\partial x_{i}}, \quad \frac{d \omega}{d t}=\frac{\partial \Omega}{\partial t}$

governing the evolution of a wave packet specified by $\varphi(\mathbf{x}, t)=A(\mathbf{x}, t ; \varepsilon) e^{i \theta(\mathbf{x}, t) / \varepsilon}$, where $0<\varepsilon \ll 1$. A formal justification is not required, but the meaning of the $d / d t$ notation should be carefully explained.

The dispersion relation for two-dimensional, small amplitude, internal waves of wavenumber $\mathbf{k}=(k, 0, m)$, relative to Cartesian coordinates $(x, y, z)$ with $z$ vertical, propagating in an inviscid, incompressible, stratified fluid that would otherwise be at rest, is given by

$\omega^{2}=\frac{N^{2} k^{2}}{k^{2}+m^{2}},$

where $N$ is the Brunt-Väisälä frequency and where you may assume that $k>0$ and $\omega>0$. Derive the modified dispersion relation if the fluid is not at rest, and instead has a slowly-varying mean flow $(U(z), 0,0)$.

In the case that $U^{\prime}(z)>0, U(0)=0$ and $N$ is constant, show that a disturbance with wavenumber $\mathbf{k}=(k, 0,0)$ generated at $z=0$ will propagate upwards but cannot go higher than a critical level $z=z_{c}$, where $U\left(z_{c}\right)$ is equal to the apparent wave speed in the $x$-direction. Find expressions for the vertical wave number $m$ as $z \rightarrow z_{c}$ from below, and show that it takes an infinite time for the wave to reach the critical level.

Paper 4, Section II, 38B

commentConsider the Rossby-wave equation

$\frac{\partial}{\partial t}\left(\frac{\partial^{2}}{\partial x^{2}}-\ell^{2}\right) \varphi+\beta \frac{\partial \varphi}{\partial x}=0,$

where $\ell>0$ and $\beta>0$ are real constants. Find and sketch the dispersion relation for waves with wavenumber $k$ and frequency $\omega(k)$. Find and sketch the phase velocity $c(k)$ and the group velocity $c_{g}(k)$, and identify in which direction(s) the wave crests travel, and the corresponding direction(s) of the group velocity.

Write down the solution with initial value

$\varphi(x, 0)=\int_{-\infty}^{\infty} A(k) e^{i k x} d k$

where $A(k)$ is real and $A(-k)=A(k)$. Use the method of stationary phase to obtain leading-order approximations to $\varphi(x, t)$ for large $t$, with $x / t$ having the constant value $V$, for

(i) $0<V<\beta / 8 \ell^{2}$,

(ii) $-\beta / \ell^{2}<V \leqslant 0$,

where the solutions for the stationary points should be left in implicit form. [It is helpful to note that $\omega(-k)=-\omega(k)$.]

Briefly discuss the nature of the solution for $V>\beta / 8 \ell^{2}$ and $V<-\beta / \ell^{2}$. [Detailed calculations are not required.]

[Hint: You may assume that

$\int_{-\infty}^{\infty} e^{\pm i \gamma u^{2}} d u=\left(\frac{\pi}{\gamma}\right)^{\frac{1}{2}} e^{\pm i \pi / 4}$

for $\gamma>0 .]$

Paper 1, Section II, D

commentWrite down the linearised equations governing motion of an inviscid compressible fluid at uniform entropy. Assuming that the velocity is irrotational, show that it may be derived from a velocity potential $\phi(\mathbf{x}, t)$ satisfying the wave equation

$\frac{\partial^{2} \phi}{\partial t^{2}}=c_{0}^{2} \nabla^{2} \phi$

and identify the wave speed $c_{0}$. Obtain from these linearised equations the energyconservation equation

$\frac{\partial E}{\partial t}+\nabla \cdot \mathbf{I}=0$

and give expressions for the acoustic-energy density $E$ and the acoustic-energy flux $\mathbf{I}$ in terms of $\phi$.

Such a fluid occupies a semi-infinite waveguide $x>0$ of square cross-section $0<y<a$, $0<z<a$ bounded by rigid walls. An impenetrable membrane closing the end $x=0$ makes prescribed small displacements to

$x=X(y, z, t) \equiv \operatorname{Re}\left[e^{-i \omega t} A(y, z)\right]$

where $\omega>0$ and $|A| \ll a, c_{0} / \omega$. Show that the velocity potential is given by

$\phi=\operatorname{Re}\left[e^{-i \omega t} \sum_{m=0}^{\infty} \sum_{n=0}^{\infty} \cos \left(\frac{m \pi y}{a}\right) \cos \left(\frac{n \pi z}{a}\right) f_{m n}(x)\right]$

where the functions $f_{m n}(x)$, including their amplitudes, are to be determined, with the sign of any square roots specified clearly.

If $0<\omega<\pi c_{0} / a$, what is the asymptotic behaviour of $\phi$ as $x \rightarrow+\infty$ ? Using this behaviour and the energy-conservation equation averaged over both time and the crosssection, or otherwise, determine the double-averaged energy flux along the waveguide,

$\left\langle\overline{I_{x}}\right\rangle(x) \equiv \frac{\omega}{2 \pi a^{2}} \int_{0}^{2 \pi / \omega} \int_{0}^{a} \int_{0}^{a} I_{x}(x, y, z, t) \mathrm{d} y \mathrm{~d} z \mathrm{~d} t$

explaining why this is independent of $x$.

Paper 2, Section II, 37D

commentStarting from the equations for one-dimensional unsteady flow of a perfect gas at constant entropy, show that the Riemann invariants

$R_{\pm}=u \pm \frac{2\left(c-c_{0}\right)}{\gamma-1}$

are constant on characteristics $C_{\pm}$given by $d x / d t=u \pm c$, where $u(x, t)$ is the speed of the gas, $c(x, t)$ is the local speed of sound, $c_{0}$ is a constant and $\gamma>1$ is the exponent in the adiabatic equation of state for $p(\rho)$.

At time $t=0$ the gas occupies $x>0$ and is at rest at uniform density $\rho_{0}$, pressure $p_{0}$ and sound speed $c_{0}$. For $t>0$, a piston initially at $x=0$ has position $x=X(t)$, where

$X(t)=-U_{0} t\left(1-\frac{t}{2 t_{0}}\right)$

and $U_{0}$ and $t_{0}$ are positive constants. For the case $0<U_{0}<2 c_{0} /(\gamma-1)$, sketch the piston path $x=X(t)$ and the $C_{+}$characteristics in $x \geqslant X(t)$ in the $(x, t)$-plane, and find the time and place at which a shock first forms in the gas.

Do likewise for the case $U_{0}>2 c_{0} /(\gamma-1)$.

Paper 3, Section II, $37 \mathrm{D}$

commentSmall disturbances in a homogeneous elastic solid with density $\rho$ and Lamé moduli $\lambda$ and $\mu$ are governed by the equation

$\rho \frac{\partial^{2} \mathbf{u}}{\partial t^{2}}=(\lambda+2 \mu) \boldsymbol{\nabla}(\boldsymbol{\nabla} \cdot \mathbf{u})-\mu \boldsymbol{\nabla} \times(\boldsymbol{\nabla} \times \mathbf{u})$

where $\mathbf{u}(\mathbf{x}, t)$ is the displacement. Show that a harmonic plane-wave solution

$\mathbf{u}=\operatorname{Re}\left[\mathbf{A} e^{i(\mathbf{k} \cdot \mathbf{x}-\omega t)}\right]$

must satisfy

$\omega^{2} \mathbf{A}=c_{P}^{2} \mathbf{k}(\mathbf{k} \cdot \mathbf{A})-c_{S}^{2} \mathbf{k} \times(\mathbf{k} \times \mathbf{A}),$

where the wavespeeds $c_{P}$ and $c_{S}$ are to be identified. Describe mathematically how such plane-wave solutions can be classified into longitudinal $P$-waves and transverse $S V$ - and $S H$-waves (taking the $y$-direction as the vertical direction).

The half-space $y<0$ is filled with the elastic solid described above, while the slab $0<y<h$ is filled with a homogeneous elastic solid with Lamé moduli $\bar{\lambda}$ and $\bar{\mu}$, and wavespeeds $\bar{c}_{P}$ and $\bar{c}_{S}$. There is a rigid boundary at $y=h$. A harmonic plane $S H$-wave propagates from $y<0$ towards the interface $y=0$, with displacement

$\operatorname{Re}\left[A e^{i(\ell x+m y-\omega t)}\right](0,0,1)$

How are $\ell, m$ and $\omega$ related? The total displacement in $y<0$ is the sum of $(*)$ and that of the reflected $S H$-wave,

$\operatorname{Re}\left[R A e^{i(\ell x-m y-\omega t)}\right](0,0,1)$

Write down the form of the displacement in $0<y<h$, and determine the (complex) reflection coefficient $R$. Verify that $|R|=1$ regardless of the parameter values, and explain this physically.

Paper 3, Section II, $37 \mathrm{D}$

commentSmall disturbances in a homogeneous elastic solid with density $\rho$ and Lamé moduli $\lambda$ and $\mu$ are governed by the equation

$\rho \frac{\partial^{2} \mathbf{u}}{\partial t^{2}}=(\lambda+2 \mu) \boldsymbol{\nabla}(\boldsymbol{\nabla} \cdot \mathbf{u})-\mu \boldsymbol{\nabla} \times(\boldsymbol{\nabla} \times \mathbf{u})$

where $\mathbf{u}(\mathbf{x}, t)$ is the displacement. Show that a harmonic plane-wave solution

$\mathbf{u}=\operatorname{Re}\left[\mathbf{A} e^{i(\mathbf{k} \cdot \mathbf{x}-\omega t)}\right]$

must satisfy

$\omega^{2} \mathbf{A}=c_{P}^{2} \mathbf{k}(\mathbf{k} \cdot \mathbf{A})-c_{S}^{2} \mathbf{k} \times(\mathbf{k} \times \mathbf{A}),$

where the wavespeeds $c_{P}$ and $c_{S}$ are to be identified. Describe mathematically how such plane-wave solutions can be classified into longitudinal $P$-waves and transverse $S V$ - and $S H$-waves (taking the $y$-direction as the vertical direction).

The half-space $y<0$ is filled with the elastic solid described above, while the slab $0<y<h$ is filled with a homogeneous elastic solid with Lamé moduli $\bar{\lambda}$ and $\bar{\mu}$, and wavespeeds $\bar{c}_{P}$ and $\bar{c}_{S}$. There is a rigid boundary at $y=h$. A harmonic plane $S H$-wave propagates from $y<0$ towards the interface $y=0$, with displacement

$\operatorname{Re}\left[A e^{i(\ell x+m y-\omega t)}\right](0,0,1)$

How are $\ell, m$ and $\omega$ related? The total displacement in $y<0$ is the sum of $(*)$ and that of the reflected $S H$-wave,

$\operatorname{Re}\left[R A e^{i(\ell x-m y-\omega t)}\right](0,0,1)$

Write down the form of the displacement in $0<y<h$, and determine the (complex) reflection coefficient $R$. Verify that $|R|=1$ regardless of the parameter values, and explain this physically.

Paper 4, Section II, D

commentA duck swims at a constant velocity $(-V, 0)$, where $V>0$, on the surface of infinitely deep water. Surface tension can be neglected, and the dispersion relation for the linear surface water waves (relative to fluid at rest) is $\omega^{2}=g|\mathbf{k}|$. Show that the wavevector $\mathbf{k}$ of a plane harmonic wave that is steady in the duck's frame, i.e. of the form

$\operatorname{Re}\left[A e^{i\left(k_{1} x^{\prime}+k_{2} y\right)}\right]$

where $x^{\prime}=x+V t$ and $y$ are horizontal coordinates relative to the duck, satisfies

$\left(k_{1}, k_{2}\right)=\frac{g}{V^{2}} \sqrt{p^{2}+1}(1, p)$

where $\hat{\mathbf{k}}=(\cos \phi, \sin \phi)$ and $p=\tan \phi$. [You may assume that $|\phi|<\pi / 2 .$ ]

Assume that the wave pattern behind the duck can be regarded as a Fourier superposition of such steady waves, i.e., the surface elevation $\eta$ at $\left(x^{\prime}, y\right)=R(\cos \theta, \sin \theta)$ has the form

$\eta=\operatorname{Re} \int_{-\infty}^{\infty} A(p) e^{i \lambda h(p ; \theta)} \mathrm{d} p \quad \text { for }|\theta|<\frac{1}{2} \pi$

where

$\lambda=\frac{g R}{V^{2}}, \quad h(p ; \theta)=\sqrt{p^{2}+1}(\cos \theta+p \sin \theta)$

Show that, in the limit $\lambda \rightarrow \infty$ at fixed $\theta$ with $0<\theta<\cot ^{-1}(2 \sqrt{2})$,

$\eta \sim \sqrt{\frac{2 \pi}{\lambda}} \operatorname{Re}\left\{\frac{A\left(p_{+}\right)}{\sqrt{h_{p p}\left(p_{+} ; \theta\right)}} e^{i\left(\lambda h\left(p_{+} ; \theta\right)+\frac{1}{4} \pi\right)}+\frac{A\left(p_{-}\right)}{\sqrt{-h_{p p}\left(p_{-} ; \theta\right)}} e^{i\left(\lambda h\left(p_{-} ; \theta\right)-\frac{1}{4} \pi\right)}\right\},$

where

$p_{\pm}=-\frac{1}{4} \cot \theta \pm \frac{1}{4} \sqrt{\cot ^{2} \theta-8}$

and $h_{p p}$ denotes $\partial^{2} h / \partial p^{2}$. Briefly interpret this result in terms of what is seen.

Without doing detailed calculations, briefly explain what is seen as $\lambda \rightarrow \infty$ at fixed $\theta$ with $\cot ^{-1}(2 \sqrt{2})<\theta<\pi / 2$. Very briefly comment on the case $\theta=\cot ^{-1}(2 \sqrt{2})$.

[Hint: You may find the following results useful.

$\begin{gathered} h_{p}=\left\{p \cos \theta+\left(2 p^{2}+1\right) \sin \theta\right\}\left(p^{2}+1\right)^{-1 / 2} \\ \left.h_{p p}=(\cos \theta+4 p \sin \theta)\left(p^{2}+1\right)^{-1 / 2}-\left\{p \cos \theta+\left(2 p^{2}+1\right) \sin \theta\right\} p\left(p^{2}+1\right)^{-3 / 2} \cdot\right] \end{gathered}$

Paper 1, Section II, 37B

commentAn acoustic plane wave (not necessarily harmonic) travels at speed $c_{0}$ in the direction $\hat{\mathbf{k}}$, where $|\hat{\mathbf{k}}|=1$, through an inviscid, compressible fluid of unperturbed density $\rho_{0}$. Show that the velocity $\tilde{\mathbf{u}}$ is proportional to the perturbation pressure $\tilde{p}$, and find $\tilde{\mathbf{u}} / \tilde{p}$. Define the acoustic intensity $\mathbf{I}$.

A harmonic acoustic plane wave with wavevector $\mathbf{k}=k(\cos \theta, \sin \theta, 0)$ and unitamplitude perturbation pressure is incident from $x<0$ on a thin elastic membrane at unperturbed position $x=0$. The regions $x<0$ and $x>0$ are both occupied by gas with density $\rho_{0}$ and sound speed $c_{0}$. The kinematic boundary conditions at the membrane are those appropriate for an inviscid fluid, and the (linearized) dynamic boundary condition

$m \frac{\partial^{2} X}{\partial t^{2}}-T \frac{\partial^{2} X}{\partial y^{2}}+[\tilde{p}(0, y, t)]_{-}^{+}=0$

where $T$ and $m$ are the tension and mass per unit area of the membrane, and $x=X(y, t)$ (with $|k X| \ll 1$ ) is its perturbed position. Find the amplitudes of the reflected and transmitted pressure perturbations, expressing your answers in terms of the dimensionless parameter

$\alpha=\frac{\rho_{0} c_{0}^{2}}{k \cos \theta\left(m c_{0}^{2}-T \sin ^{2} \theta\right)} .$

Hence show that the time-averaged energy flux in the $x$-direction is conserved across the membrane.

Paper 2, Section II, 36B

A uniform elastic solid with density $\rho$