• # B1.26

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

$\frac{\partial^{2} \psi}{\partial t^{2}}-\alpha^{2} \frac{\partial^{6} \psi}{\partial x^{6}}=0$

where $\alpha$ is a real positive constant. Derive the corresponding dispersion relation and sketch graphs of frequency, phase velocity and group velocity as functions of the wave number. Is it the shortest or the longest waves that are at the front of a dispersing wave train arising from a localised initial disturbance? Do the wave crests move faster or slower than a packet of waves?

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

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

where $A(k)$ is real and $A(-k)=A(k)$.

Use the method of stationary phase to obtain a leading-order approximation to this solution for large $t$ when $V=x / t$ is held fixed.

[Note that

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

comment
• # B2.26

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

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

where $\mathbf{u}(\mathbf{x}, t)$ is the displacement, and $\lambda$ and $\mu$ are the Lamé constants. Derive solutions for plane longitudinal waves $P$ with wavespeed $c_{P}$, and plane shear waves $S$ with wavespeed $c_{S}$.

The half-space $y<0$ is filled with the elastic solid described above, while the slab $0 is filled with an elastic solid with shear modulus $\bar{\mu}$, and wavespeeds $\bar{c}_{P}$ and $\bar{c}_{S}$. There is a vacuum in $y>h$. A harmonic plane $S H$ wave of frequency $\omega$ and unit amplitude propagates from $y<0$ towards the interface $y=0$. The wavevector is in the $x y$-plane, and makes an angle $\theta$ with the $y$-axis. Derive the complex amplitude, $R$, of the reflected $S H$ wave in $y<0$. Evaluate $|R|$ for all possible values of $\bar{c}_{S} / c_{S}$, and explain your answer.

comment
• # B3.25

The dispersion relation for sound waves of frequency $\omega$ in a stationary, homogeneous gas is $\omega=c|\mathbf{k}|$, where $c$ is the speed of sound and $\mathbf{k}$ is the wavevector. Derive the dispersion relation for sound waves of frequency $\omega$ in a uniform flow with velocity U.

For a slowly-varying medium with a local dispersion relation $\omega=\Omega(\mathbf{k} ; \mathbf{x}, t)$, derive the ray-tracing equations

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

The meaning of the notation $d / d t$ should be carefully explained.

Suppose that two-dimensional sound waves with initial wavevector $\left(k_{0}, l_{0}\right)$ are generated at the origin in a gas occupying the half-space $y>0$. The gas has a mean velocity $(\gamma y, 0)$, where $0<\gamma \ll\left(k_{0}^{2}+l_{0}^{2}\right)^{\frac{1}{2}}$. Show that

(a) if $k_{0}>0$ and $l_{0}>0$ then an initially upward propagating wavepacket returns to the level $y=0$ within a finite time, after having reached a maximum height that should be identified;

(b) if $k_{0}<0$ and $l_{0}>0$ then an initially upward propagating wavepacket continues to propagate upwards for all time.

For the case of a fixed frequency disturbance comment briefly on whether or not there is a quiet zone.

comment
• # B4.27

A 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. Derive the Rankine-Hugoniot relations across the shock. 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. Now consider a change of frame such that the shock is stationary and the gas has a component of velocity $V$ parallel to the shock. Deduce that the angle of deflection $\delta$ of the flow which is produced by a stationary shock inclined at an angle $\alpha=\tan ^{-1}(U / V)$ to an oncoming stream of Mach number $M=\left(U^{2}+V^{2}\right)^{\frac{1}{2}} / c_{1}$ is given by

$\tan \delta=\frac{2 \cot \alpha\left(M^{2} \sin \alpha^{2}-1\right)}{2+M^{2}(\gamma+\cos 2 \alpha)}$

[Note that

$\left.\tan (\theta+\phi)=\frac{\tan \theta+\tan \phi}{1-\tan \theta \tan \phi} . \quad\right]$

comment

• # B1.26

Consider the equation

$\frac{\partial^{2} \phi}{\partial t^{2}}+\alpha^{2} \frac{\partial^{4} \phi}{\partial x^{4}}+\beta^{2} \phi=0$

with $\alpha$ and $\beta$ real constants. Find the dispersion relation for waves of frequency $\omega$ and wavenumber $k$. Find the phase velocity $c(k)$ and the group velocity $c_{g}(k)$, and sketch the graphs of these functions.

By multiplying $(*)$ by $\partial \phi / \partial t$, obtain an energy equation in the form

$\frac{\partial E}{\partial t}+\frac{\partial F}{\partial x}=0$

where $E$ represents the energy density and $F$ the energy flux.

Now let $\phi(x, t)=A \cos (k x-\omega t)$, where $A$ is a real constant. Evaluate the average values of $E$ and $F$ over a period of the wave to show that

$\langle F\rangle=c_{g}\langle E\rangle$

Comment on the physical meaning of this result.

comment
• # B2.25

Starting from the equations for the one-dimensional unsteady flow of a perfect gas of uniform entropy, derive the Riemann invariants

$u \pm \frac{2}{\gamma-1} c=\text { constant }$

on characteristics

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

A piston moves smoothly down a long tube, with position $x=X(t)$. Gas occupies the tube ahead of the piston, $x>X(t)$. Initially the gas and the piston are at rest, and the speed of sound in the gas is $c_{0}$. For $t>0$, show that the $C_{+}$characteristics are straight lines, provided that a shock-wave has not formed. Hence find a parametric representation of the solution for the velocity $u(x, t)$ of the gas.

comment
• # B3.25

Derive the wave equation governing the velocity potential for linearised sound in a perfect gas. How is the pressure disturbance related to the velocity potential? Write down the spherically symmetric solution to the wave equation with time dependence $e^{i \omega t}$, which is regular at the origin.

A high pressure gas is contained, at density $\rho_{0}$, within a thin metal spherical shell which makes small amplitude spherically symmetric vibrations. Ignore the low pressure gas outside. Let the metal shell have radius $a$, mass $m$ per unit surface area, and 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. Show that the frequency of these vibrations is given by

$\omega^{2}\left(m+\frac{\rho_{0} a_{0}}{\theta \cot \theta-1}\right)=\kappa \quad \text { where } \theta=\omega a_{0} / c_{0}$

comment
• # B4.27

Show that the equations governing isotropic linear elasticity have plane-wave solutions, identifying them as $\mathrm{P}, \mathrm{SV}$ or $\mathrm{SH}$ waves.

A semi-infinite elastic medium in $y<0$ (where $y$ is the vertical coordinate) with density $\rho$ and Lamé moduli $\lambda$ and $\mu$ is overlaid by a layer of thickness $h$ (in $0 ) of a second elastic medium with density $\rho^{\prime}$ and Lamé moduli $\lambda^{\prime}$ and $\mu^{\prime}$. The top surface at $y=h$ is free, i.e. the surface tractions vanish there. The speed of S-waves is lower in the layer, i.e. $c_{S}^{\prime}{ }^{2}=\mu^{\prime} / \rho^{\prime}<\mu / \rho=c_{S}^{2}$. For a time-harmonic SH-wave with horizontal wavenumber $k$ and frequency $\omega$, which oscillates in the slow top layer and decays exponentially into the fast semi-infinite medium, derive the dispersion relation for the apparent wave speed $c(k)=\omega / k$,

$\tan \left(k h \sqrt{\frac{c^{2}}{c_{S}^{2}}-1}\right)=\frac{\mu \sqrt{1-\frac{c^{2}}{c_{S}^{2}}}}{\mu^{\prime} \sqrt{\frac{c^{2}}{c_{S}^{\prime}}-1}} .$

Show graphically that there is always one root, and at least one higher mode if $\sqrt{c_{S}^{2} / c_{S}^{\prime 2}-1}>\pi / k h$.

comment

• # B1.26

Starting from the equations governing sound waves linearized about a state with density $\rho_{0}$ and sound speed $c_{0}$, derive the acoustic energy equation, giving expressions for the local energy density $E$ and energy flux $\mathbf{I}$.

A sphere executes small-amplitude vibrations, with its radius varying according to

$r(t)=a+\operatorname{Re}\left(\epsilon e^{i \omega t}\right)$

with $0<\epsilon \ll a$. Find an expression for the velocity potential of the sound, $\tilde{\phi}(r, t)$. Show that the time-averaged rate of working by the surface of the sphere is

$2 \pi a^{2} \rho_{0} \omega^{2} \epsilon^{2} c_{0} \frac{\omega^{2} a^{2}}{c_{0}^{2}+\omega^{2} a^{2}}$

Calculate the value at $r=a$ of the dimensionless ratio $c_{0} \bar{E} /|\overline{\mathbf{I}}|$, where the overbars denote time-averaged values, and comment briefly on the limits $c_{0} \ll \omega a$ and $c_{0} \gg \omega a$.

comment
• # B2 25

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 $\gamma$ is the specific heat ratio.

Such a 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 at a constant speed $V$. Find $u(x, t)$ and $c(x, t)$ in the three regions

\begin{aligned} \text { (i) } \quad & c_{0} t \leq x \\ \text { (ii) } \quad a t & \leq x

where $a=c_{0}-\frac{1}{2}(\gamma+1) V$. What is the largest value of $V$ for which $c$ is positive throughout region (iii)? What happens if $V$ exceeds this value?

comment
• # B3.25

Consider the equation

$\frac{\partial \phi}{\partial t}+\frac{\partial \phi}{\partial x}-\frac{\partial^{3} \phi}{\partial x^{3}}=0$

Find the dispersion relation for waves of frequency $\omega$ and wavenumber $k$. Do the wave crests move faster or slower than a packet of waves?

Write down the solution with initial value

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

where $A(k)$ is real and $A(-k)=A(k)$.

Use the method of stationary phase to obtain an approximation to $\phi(x, t)$ for large $t$, with $x / t$ having the constant value $V$. Explain, using the notion of group velocity, the constraint that must be placed on $V$.

comment
• # B4.27

Write down the equation governing linearized displacements $\mathbf{u}(\mathbf{x}, t)$ in a uniform elastic medium of density $\rho$ and Lamé constants $\lambda$ and $\mu$. Derive solutions for monochromatic plane $P$ and $S$ waves, and find the corresponding wave speeds $c_{P}$ and $c_{S}$.

Such an elastic solid occupies the half-space $z>0$, and the boundary $z=0$ is clamped rigidly so that $\mathbf{u}(x, y, 0, t)=\mathbf{0}$. A plane $S V$-wave with frequency $\omega$ and wavenumber $(k, 0,-m)$ is incident on the boundary. At some angles of incidence, there results both a reflected $S V$-wave with frequency $\omega^{\prime}$ and wavenumber $\left(k^{\prime}, 0, m^{\prime}\right)$ and a reflected $P$-wave with frequency $\omega^{\prime \prime}$ and wavenumber $\left(k^{\prime \prime}, 0, m^{\prime \prime}\right)$. Relate the frequencies and wavenumbers of the reflected waves to those of the incident wave. At what angles of incidence will there be a reflected $P$-wave?

Find the amplitudes of the reflected waves as multiples of the amplitude of the incident wave. Confirm that these amplitudes give the sum of the time-averaged vertical fluxes of energy of the reflected waves equal to the time-averaged vertical flux of energy of the incident wave.

[Results concerning the energy flux, energy density and kinetic energy density in a plane elastic wave may be quoted without proof.]

comment

• # B1.26

Derive Riemann's equations for finite amplitude, one-dimensional sound waves in a perfect gas with ratio of specific heats $\gamma$.

At time $t=0$ the gas is at rest and has uniform density $\rho_{0}$, pressure $p_{0}$ and sound speed $c_{0}$. A piston initially at $x=0$ starts moving backwards at time $t=0$ with displacement $x=-a \sin \omega t$, where $a$ and $\omega$ are positive constants. Explain briefly how to find the resulting disturbance using a graphical construction in the $x t$-plane, and show that prior to any shock forming $c=c_{0}+\frac{1}{2}(\gamma-1) u$.

For small amplitude $a$, show that the excess pressure $\Delta p=p-p_{0}$ and the excess sound speed $\Delta c=c-c_{0}$ are related by

$\frac{\Delta p}{p_{0}}=\frac{2 \gamma}{\gamma-1} \frac{\Delta c}{c_{0}}+\frac{\gamma(\gamma+1)}{(\gamma-1)^{2}}\left(\frac{\Delta c}{c_{0}}\right)^{2}+O\left(\left(\frac{\Delta c}{c_{0}}\right)^{3}\right)$

Deduce that the time-averaged pressure on the face of the piston exceeds $p_{0}$ by

$\frac{1}{8} \rho_{0} a^{2} \omega^{2}(\gamma+1)+O\left(a^{3}\right)$

comment
• # B2.25

A semi-infinite elastic medium with shear modulus $\mu_{1}$ and shear-wave speed $c_{1}$ lies in $y<0$. Above it there is a layer $0 \leq y \leqslant h$ of a second elastic medium with shear modulus $\mu_{2}$ and shear-wave speed $c_{2}\left(. The top boundary $y=h$ is stress-free. Consider a monochromatic shear wave propagating at speed $c$ with wavenumber $k$ in the $x$-direction and with displacements only in the $z$-direction.

Obtain the dispersion relation

$\tan k h \theta=\frac{\mu_{1} c_{2}}{\mu_{2} c_{1}} \frac{1}{\theta}\left(\frac{c_{1}^{2}}{c_{2}^{2}}-1-\theta^{2}\right)^{1 / 2}, \quad \text { where } \quad \theta=\sqrt{\frac{c^{2}}{c_{2}^{2}}-1} .$

Deduce that the modes have a cut-off frequency $\pi n c_{1} c_{2} / h \sqrt{c_{1}^{2}-c_{2}^{2}}$ where they propagate at speed $c=c_{1}$.

comment
• # B3.25

Consider the equation

$\phi_{t t}+\alpha^{2} \phi_{x x x x}+\beta^{2} \phi=0,$

where $\alpha$ and $\beta$ are real constants. Find the dispersion relation for waves of frequency $\omega$ and wavenumber $k$. Find the phase velocity $c(k)$ and the group velocity $c_{g}(k)$ and sketch graphs of these functions.

Multiplying equation $(*)$ by $\phi_{t}$, obtain an equation of the form

$\frac{\partial A}{\partial t}+\frac{\partial B}{\partial x}=0$

where $A$ and $B$ are expressions involving $\phi$ and its derivatives. Give a physical interpretation of this equation.

Evaluate the time-averaged energy $\langle E\rangle$ and energy flux $\langle I\rangle$ of a monochromatic wave $\phi=\cos (k x-w t)$, and show that

$\langle I\rangle=c_{g}\langle E\rangle .$

comment
• # B4.27

Derive the ray-tracing equations governing the evolution of a wave packet $\phi(\mathbf{x}, t)=$ $A(\mathbf{x}, t) \exp \{i \psi(\mathbf{x}, t)\}$ in a slowly varying medium, stating the conditions under which the equations are valid.

Consider now a stationary obstacle in a steadily moving homogeneous two-dimensional medium which has the dispersion relation

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

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

$\kappa=\frac{\alpha^{2}}{V^{2} \cos ^{2} \phi}$

Show that the group velocity of these waves can be expressed as

$\mathbf{c}_{g}=V\left(\frac{1}{2} \cos ^{2} \phi-1, \frac{1}{2} \cos \phi \sin \phi\right) .$

Deduce that the waves occupy a wedge of semi-angle $\sin ^{-1} \frac{1}{3}$ about the negative $x_{1}$-axis.

comment