Chapter 7, The Mathematics of Real Waves Video Solutions, Mathematics for Physics. 1

Problem 1

Find the expression for the sine-Gordon soliton, by first showing that the static sine-Gordon equation

$$
-\frac{\partial^2 \varphi}{\partial x^2}+\frac{m^2}{\beta} \sin \beta \varphi=0
$$

implies that

$$
\frac{1}{2} \varphi^{\prime 2}+\frac{m^2}{\beta^2} \cos \beta \varphi=\text { const. }
$$

and solving this equation (for a suitable choice of the constant) by separation of variables. Next, show that if $f(x)$ is solution of the static equation, then $f(\gamma(x-U t)), \gamma=\left(1-U^2\right)^{-1 / 2},|U|<1$ is a solution of the time-dependent equation.

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Problem 2

Lax pair for the non-linear Schrödinger equation. Let $L$ be the matrix differential operator

$$
L=\left[\begin{array}{cc}
i \partial_x & \chi^* \\
\chi & i \partial_x
\end{array}\right]
$$

and let $P$ the matrix

$$
P=\left[\begin{array}{cc}
i|\chi|^2 & \chi^{\prime *} \\
-\chi^{\prime} & -i|\chi|^2
\end{array}\right]
$$

Show that the equation

$$
\dot{L}=[L, P]
$$

is equivalent to the non-linear Shrödinger equation

$$
i \dot{\chi}=-\chi^{\prime \prime}-2|\chi|^2 \chi
$$

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01:06

Problem 3

Pantograph Drag. A high-speed train picks up its electrical power via a pantograph from an overhead line. The locomotive travels at speed $U$ and the pantograph exerts a constant vertical force $F$ on the power line.

We make the usual small amplitude approximations and assume (not unrealistically) that the line is supported in such a way that its vertical displacement obeys an inhomogeneous Klein-Gordon equation

$$
\rho \ddot{y}-T y^{\prime \prime}+\rho \Omega^2 y=F \delta(x-U t),
$$

with $c=\sqrt{T / \rho}$, the velocity of propagation of short-wavelength transverse waves on the overhead cable.
a) Assume that $U<c$ and solve for the steady state displacement of the cable about the pickup point. (Hint: the disturbance is time-independent when viewed from the train.)
b) Now assume that $U>c$. Again find an expression for the displacement of the cable. (The same hint applies, but the physically appropriate boundary conditions are very different!)
c) By equating the rate at which wave-energy

$$
E=\int\left\{\frac{1}{2} \rho \dot{y}^2+\frac{1}{2} T y^{\prime 2}+\frac{1}{2} \rho \Omega^2 y^2\right\} d x
$$

is being created to rate at the which the locomotive is doing work, calculate the wave-drag on the train. In particular, show that there is no drag at all until $U$ exceeds $c$. (Hint: While the front end of the wake is moving at speed $U$, the trailing end of the wake is moving forward at the group velocity of the wave-train.)
d) By carefully considering the force the pantograph exerts on the overhead cable, again calculate the induced drag. You should get the same answer as in part c) (Hint: The tension in the cable is the same before and after the train has passed, but the direction in which the tension acts is different. The force $F$ is therefore not exactly vertical, but has a small forward component. Don't forget that the resultant of the forces is accelerating the cable.)

Manik Pulyani

Numerade Educator

Problem 4

Inertial waves. A rotating tank of incompressible ( $\rho \equiv 1$ ) fluid can host waves whose restoring force is provided by angular momentum conservation. Suppose the fluid velocity at the point $\mathbf{r}$ is given by

$$
\mathbf{v}(\mathbf{r}, t)=\mathbf{u}(\mathbf{r}, t)+\boldsymbol{\Omega} \times \mathbf{r}
$$

where $\mathbf{u}$ is a perturbation imposed on the rigid rotation of the fluid at angular velocity $\boldsymbol{\Omega}$.
a) Show that when viewed from a co-ordinate frame rotating with the fluid we have

$$
\frac{\partial \mathbf{u}}{\partial t}=\left(\frac{\partial \mathbf{u}}{\partial t}-\boldsymbol{\Omega} \times \mathbf{u}+((\boldsymbol{\Omega} \times \mathbf{r}) \cdot \nabla) \mathbf{u}\right)_{\mathrm{lab}}
$$

Deduce that the lab-frame Euler equation

$$
\frac{\partial \mathbf{v}}{\partial t}+(\mathbf{v} \cdot \nabla) \mathbf{v}=-\nabla P
$$

becomes, in the rotating frame,

$$
\frac{\partial \mathbf{u}}{\partial t}+2(\boldsymbol{\Omega} \times \mathbf{u})+(\mathbf{u} \cdot \nabla) \mathbf{u}=-\nabla\left(P-\frac{1}{2}|\boldsymbol{\Omega} \times \mathbf{r}|^2\right)
$$

We see that in the non-inertial rotating frame the fluid experiences a $-2(\boldsymbol{\Omega} \times \mathbf{u})$ Coriolis and a $\nabla|\boldsymbol{\Omega} \times \mathbf{r}|^2 / 2$ centrifugal force. By linearizing the rotating-frame Euler equation, show that for small $\mathbf{u}$ we have

$$
\frac{\partial \omega}{\partial t}-2(\boldsymbol{\Omega} \cdot \nabla) \mathbf{u}=0
$$

where $\boldsymbol{\omega}=$ curl $\mathbf{u}$.
b) Take $\boldsymbol{\Omega}$ to be directed along the $z$ axis. Seek plane-wave solutions to $\star$ in the form

$$
\mathbf{u}(\mathbf{r}, t)=\mathbf{u}_o e^{i(\mathbf{k} \cdot \mathbf{r}-\omega t)}
$$
where $\mathbf{u}_0$ is a constant, and show that the dispersion equation for these small amplitude inertial waves is

$$
\omega=2 \Omega \sqrt{\frac{k_z^2}{k_x^2+k_y^2+k_z^2}}
$$

Deduce that the group velocity is directed perpendicular to $\mathbf{k}$-i.e. at right-angles to the phase velocity. Conclude also that any slow flow that is steady (time independent) when viewed from the rotating frame is necessarily independent of the co-ordinate $z$. (This is the origin of the phenomenon of Taylor columns, which are columns of stagnant fluid lying above and below any obstacle immersed in such a flow.)

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Problem 5

Non-linear Waves. In this problem we will explore the Riemann invariants for a fluid with $P=\lambda^2 \rho^3 / 3$. This is the equation of state of onedimensional non-interacting Fermi gas.
a) From the continuity equation

$$
\partial_t \rho+\partial_x \rho v=0
$$

and Euler's equation of motion

$$
\rho\left(\partial_t v+v \partial_x v\right)=-\partial_x P
$$

deduce that

$$
\begin{aligned}
\left(\frac{\partial}{\partial t}+(\lambda \rho+v) \frac{\partial}{\partial x}\right)(\lambda \rho+v) & =0 \\
\left(\frac{\partial}{\partial t}+(-\lambda \rho+v) \frac{\partial}{\partial x}\right)(-\lambda \rho+v) & =0
\end{aligned}
$$

In what limit do these equations become equivalent to the wave equation for one-dimensional sound? What is the sound speed in this case?
b) Show that the Riemann invariants $v \pm \lambda \rho$ are constant on suitably defined characteristic curves. What is the local speed of propagation of the waves moving to the right or left?
c) The fluid starts from rest, $v=0$, but with a region where the density is higher than elsewhere. Show that that the Riemann equations will inevitably break down at some later time due to the formation of shock waves.

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04:58

Problem 6

Burgers Shocks. As simple mathematical model for the formation and decay of a shock wave consider Burgers' Equation:

$$
\partial_t u+u \partial_x u=\nu \partial_x^2 u
$$

Note its similarity to the Riemann equations of the previous exercise. The additional term on the right-hand side introduces dissipation and prevents the solution becoming multi-valued.
a) Show that if $\nu=0$ any solution of Burgers' equation having a region where $u$ decreases to the right will always eventually become multivalued.
b) Show that the Hopf-Cole transformation, $u=-2 \nu \partial_x \ln \psi$, leads to $\psi$ obeying a heat diffusion equation

$$
\partial_t \psi=\nu \partial_x^2 \psi
$$

c) Show that

$$
\psi(x, t)=A e^{\nu a^2 t-a x}+B e^{\nu b^2 t-b x}
$$

is a solution of the heat equation, and so deduce that Burgers' equation has a shock-wave-like solution which travels to the right at speed $C=$ $\nu(a+b)=\frac{1}{2}\left(u_L+u_R\right)$, the mean of the wave speeds to the left and right of the shock. Show that the width of the shock is $\approx 4 \nu /\left|u_L-u_R\right|$.

Saman Zulfiqar

Numerade Educator