Exercises

1. Demonstrate that the phase velocity of traveling waves on an infinitely long beaded string is
   $$v_p = v_0\,\frac{\sin(k\,a/2)}{(k\,a/2)},$$
   where $v_0= \sqrt{T\,a/m}$, $T$ is the tension in the string, $a$ the spacing between the beads, $m$ the mass of the beads, and $k$ the wavenumber of the wave. What is the group velocity?

2. A uniform rope of mass per unit length $\rho$ and length $L$ hangs vertically. Determine the tension $T$ in the rope as a function of height from the bottom of the rope. Show that the time required for a transverse wave pulse to travel from the bottom to the top of the rope is $2\sqrt{L/g}$.

3. Derive expressions (9.27) and (9.29) for propagating electromagnetic waves in a plasma from Equations (9.19) and (9.23)–(9.26).

4. Derive expressions (9.39) and (9.40) for evanescent electromagnetic waves in a plasma from Equations (9.24), (9.25), and (9.36)–(9.38).

5. Derive Equations (9.49)–(9.52) from Equations (9.47) and (9.48).

6. Derive Equations (9.60) and (9.61) from Equations (9.54)–(9.59).

7. Derive Equations (9.207) and (9.208) from Equations (9.203)–(9.206).

8. Derive Equations (9.216)–(9.219) from Equations (9.214) and (9.215), in the limit $\alpha\ll 1$.

9. A medium is such that the product of the phase and group velocities of electromagnetic waves is equal to $c^{\,2}$ at all wave frequencies, where $c$ is the velocity of light in vacuum. Demonstrate that the dispersion relation for electromagnetic waves takes the form
   $$\omega^{\,2} = k^{\,2}\,c^{\,2}\pm\omega_0^{\,2},$$
   where $\omega_0$ is a constant.

10. The number density of free electrons in the ionosphere, $n_e$, as a function of vertical height, $z$, is measured by timing how long it takes a radio pulse launched vertically upward from the ground ($z=0$) to return to ground level again, after reflection by the ionosphere, as a function of the pulse frequency, $\omega$. It is conventional to define the equivalent height, $h(\omega)$, of the reflection layer as the height it would need to have above the ground if the pulse always traveled at the velocity of light in vacuum. Demonstrate that
    $$h(\omega) = \int_0^{z_0}\frac{dz}{[1-\omega_{p\,e}^{\,2}(z)/\omega^{\,2}]^{1/2}},$$
    where $\omega_{p\,e}^{\,2}(z)= n_e(z)\,e^{\,2}/(\epsilon_0\,m_e)$, and $\omega_{p\,e}^{\,2}(z_0)=\omega^{\,2}$. Show that if $n_e\propto z^{\,p}$ then $h\propto \omega^{\,2/p}$.

11. Show that the general eigenmode equation, (9.168), yields the following dispersion relation for electromagnetic waves propagating through a magnetized plasma:
    $$A\,n^{\,4}-B\,n^{\,2}+C =0,$$
    where

    $$A = S\,\sin^2\theta + P\,\cos^2\theta,$$

    $$B = R\,L\,\sin^2\theta + P\,S\,(1+\cos^2\theta),$$

    $$C = P\,R\,L.$$

12. Show that the solution to the dispersion relation derived in the previous exercise can be written
    $$n^{\,2} = \frac{B\pm F}{2\,A},$$
    where
    $$F= (R\,L-P\,S)^2\,\sin^4\theta + 4\,P^{\,2}\,D^{\,2}\,\cos^2\theta.$$
    Hence, deduce that $n^{\,2}$ is always real, and that $n$ is either purely real, or purely imaginary. This implies that electromagnetic waves in (cold) magnetized plasmas either propagate without evanescence, or decay without spatial oscillation.

13. Show that the dispersion relation derived in Exercise 11 can also be written in the form
    $$\tan^2\theta = - \frac{P\,(n^{\,2}-R)\,(n^{\,2}-L)}{(S\,n^{\,2}-R\,L)\,(n^{\,2}-P)}.$$

14. 1. Show that the cutoff frequencies for an electromagnetic wave propagating in a general direction through a magnetized plasma occur when
       $$P=0,$$
       or
       $$R=0,$$
       or
       $$L=0.$$
       Hence, deduce that the cutoff frequencies are independent of the direction of wave propagation.

    2. Show that the resonant frequencies for an electromagnetic wave propagating in a general direction through a magnetized plasma occur when
       $$\tan^2\theta = -\frac{P}{S}.$$

15. Consider an electromagnetic wave propagating in the positive $z$-direction through a conducting medium of conductivity $\sigma$. Suppose that the wave electric field is
    $$E_x(z,t) = E_0\,{\rm e}^{-z/d}\,\cos(\omega\,t-z/d),$$
    where $d$ is the skin-depth. Demonstrate that the mean electromagnetic energy flux across the plane $z=0$ matches the mean rate at which electromagnetic energy is dissipated, per unit area, due to Joule heating in the region $z>0$. [The rate of Joule heating per unit volume is $\sigma\,E_x^{\,2}$ (Fitzpatrick 2008).]

16. The aluminum foil used in cooking has an electrical conductivity $\sigma=3.5\times 10^{\,7}\,(\Omega\,{\rm m})^{-1}$, and a typical thickness $\delta=2\times 10^{\,-4}\,{\rm m}$ (Wikipedia contributors 2018). Show that such foil can be used to shield a region from electromagnetic waves of a given frequency, provided that the skin-depth of the waves in the foil is less than about a third of its thickness. Because skin-depth increases as frequency decreases, it follows that the foil can only shield waves whose frequency exceeds a critical value. Estimate this critical frequency (in hertz). What is the corresponding wavelength?

17. Consider a hollow, vacuum-filled, rectangular waveguide that runs parallel to the $z$-axis, and has perfectly conducting walls located at $x=0,\,a$ and $y=0,\,b$. Maxwell's equations for a TE mode (which is characterized by $E_z=0$) are (see Appendix C)

    $$\frac{\partial E_x}{\partial t} =-\frac{1}{\epsilon_0}\left(\frac{\partial H_y}{\partial z}-\frac{\partial H_z}{\partial y}\right),$$

    $$\frac{\partial E_y}{\partial t} = -\frac{1}{\epsilon_0}\left(\frac{\partial H_z}{\partial x}-\frac{\partial H_x}{\partial z}\right),$$

    $$\frac{\partial H_x}{\partial t} =\frac{1}{\mu_0}\,\frac{\partial E_y}{\partial z},$$

    $$\frac{\partial H_y}{\partial t} = - \frac{1}{\mu_0}\,\frac{\partial E_x}{\partial z},$$

    $$\frac{\partial H_z}{\partial t} = \frac{1}{\mu_0}\left(\frac{\partial E_x}{\partial y}-\frac{\partial E_y}{\partial x}\right),$$

    subject to the boundary conditions $E_y=0$ at $x=0,\,a$ and $E_x=0$ at $y=0,\,b$. Show that the problem reduces to solving
    $$\frac{\partial^{\,2} H_z}{\partial t^{\,2}} = c^{\,2}\left(\frac{... ...al^{\,2}}{\partial y^{\,2}}+\frac{\partial^{\,2}}{\partial z^{\,2}}\right)H_z,$$
    subject to the boundary conditions $\partial H_z/\partial x=0$ at $x=0,\,a$, and $\partial H_z/\partial y=0$ at $y=0,\,b$. Demonstrate that the various TE modes satisfy the dispersion relation
    $$\omega^{\,2} = k^{\,2}\,c^{\,2} + \omega_{m,n}^{\,2},$$
    where $k$ is the $z$-component of the wavevector,
    $$\omega_{m,n} = \pi\,c\left(\frac{m^{\,2}}{a^{\,2}}+\frac{n^{\,2}}{b^{\,2}}\right)^{1/2},$$
    and $m,\,n$ are non-negative integers, one of which must be non-zero.

18. Consider a hollow, vacuum-filled, rectangular waveguide that runs parallel to the $z$-axis, and has perfectly conducting walls located at $x=0,\,a$ and $y=0,\,b$. Maxwell's equations for a TM mode (which is characterized by $H_z=0$) are (see Appendix C)

    $$\frac{\partial H_x}{\partial t} =\frac{1}{\mu_0}\left(\frac{\partial E_y}{\partial z}-\frac{\partial E_z}{\partial y}\right),$$

    $$\frac{\partial H_y}{\partial t} = \frac{1}{\mu_0}\left(\frac{\partial E_z}{\partial x}-\frac{\partial E_x}{\partial z}\right),$$

    $$\frac{\partial E_x}{\partial t} =-\frac{1}{\epsilon_0}\, \frac{\partial H_y}{\partial z},$$

    $$\frac{\partial E_y}{\partial t} =\frac{1}{\epsilon_0}\,\frac{\partial H_x}{\partial z},$$

    $$\frac{\partial E_z}{\partial t} = -\frac{1}{\epsilon_0}\left(\frac{\partial H_x}{\partial y}-\frac{\partial H_y}{\partial x}\right),$$

    subject to the boundary conditions $E_y=0$ at $x=0,\,a$, $E_x=0$ at $y=0,\,b$, and $E_z=0$ at $x=0,\,a$ and $y=0,\,b$. Show that the problem reduces to solving
    $$\frac{\partial^{\,2} E_z}{\partial t^{\,2}} = c^{\,2}\left(\frac{... ...al^{\,2}}{\partial y^{\,2}}+\frac{\partial^{\,2}}{\partial z^{\,2}}\right)E_z,$$
    subject to the boundary conditions $E_z=0$ at $x=0,\,a$ and $y=0,\,b$. Demonstrate that the various TM modes satisfy the dispersion relation
    $$\omega^{\,2} = k^{\,2}\,c^{\,2} + \omega_{m,n}^{\,2},$$
    where $k$ is the $z$-component of the wavevector,
    $$\omega_{m,n} = \pi\,c\left(\frac{m^{\,2}}{a^{\,2}}+\frac{n^{\,2}}{b^{\,2}}\right)^{1/2},$$
    and $m,\,n$ are positive integers.

19. Deduce that, for a hollow, vacuum-filled, rectangular waveguide, the mode with the lowest cutoff frequency is a TE mode.

20. Consider a vacuum-filled rectangular waveguide of internal dimensions $5\times 10\,{\rm cm}$. What is the frequency (in MHz) of the lowest frequency TE mode that will propagate along the waveguide without attenuation? What are the phase and group velocities (expressed as multiples of $c$) of this mode when its frequency is $5/4$ times the cutoff frequency?

21. A wave pulse propagates in the $x$-$z$ plane through an inhomogeneous medium with the linear dispersion relation
    $$\omega = k\,v,$$
    where
    $$v(z) = v_0-v_1\,z.$$
    Here, $v_0$ and $v_1$ are positive constants. Show that if $k_z=0$ at $z=0$ then the equations of motion of the pulse can be written

    $$\frac{dx}{ds} = \frac{v}{v_0},$$

    $$\frac{dz}{ds} =\left(1-\frac{v^{\,2}}{v_0^{\,2}}\right),$$

    $$\frac{d^{\,2} x}{ds^{\,2}} =- \frac{v_1}{v_0}\,\frac{dz}{ds},$$

    $$\frac{d^{\,2} z}{ds^{\,2}} =\frac{v_1}{v_0}\,\frac{dx}{ds},$$

    where $s=v\,t$ denotes path-length. Hence, deduce that the pulse travels in the arc of a circle, of radius
    $$R = \frac{v_0}{v_1},$$
    whose center lies at $z=R$.

22. The speed of sound in the atmosphere decreases approximately linearly with increasing altitude (at relatively low altitude) due to an approximately linear decrease in the temperature of the atmosphere with height. Assuming that the sound speed varies with altitude, $z$, above the Earth's surface as
    $$v(z)=v_0-v_1\,z,$$
    where $v_0$ and $v_1$ are positive constants, show that sound generated by a source located a height $h\ll v_0/v_1$ above the ground is refracted upward by the atmosphere such that it never reaches ground level at points whose radial distances from the point lying directly beneath the source exceed the value
    $$r = \left(\frac{2\,v_0\,h}{v_1}\right)^{1/2}.$$
    This effect is known as acoustic shadowing.

23. A low amplitude sinusoidal gravity wave travels through shallow water of gradually decreasing depth $d$ toward the shore. Assuming that the wave travels at right-angles to the shoreline, show that its wavelength and vertical amplitude vary as $\lambda\propto d^{\,1/2}$ and $a\propto d^{\,-1/4}$, respectively.

24. Demonstrate that a small amplitude gravity wave, of angular frequency $\omega$ and wavenumber $k$, traveling over the surface of a lake of uniform depth $d$ causes an individual water volume element located at a depth $h$ below the surface to execute a non-propagating elliptical orbit whose major and minor axes are horizontal and vertical, respectively. Show that the variation of the major and minor radii of the orbit with depth is $A\,\cosh[k\,(d-h)]$ and $A\,\sinh[k\,(d-h)]$, respectively, where $A$ is a constant. Demonstrate that the volume elements are moving horizontally in the same direction as the wave at the top of their orbits, and in the opposite direction at the bottom. Show that a gravity wave traveling over the surface of a very deep lake causes water volume elements to execute non-propagating circular orbits whose radii decrease exponentially with depth.

25. Water fills a rectangular tank of length $l$ and breadth $b$ to a depth $d$. Show that the resonant frequencies of the water are
    $$\omega_{m,n}=\left[g\,k_{m,n}\,\tanh(k_{m,n}\,d)\right]^{1/2}$$
    where
    $$k_{m,n} = \pi\left(\frac{m^{\,2}}{l^{\,2}}+\frac{n^{\,2}}{b^{\,2}}\right)^{1/2},$$
    and $n$, $m$ are non-negative integers that are not both zero. Neglect surface tension.

26. Derive the dispersion relation (9.328), and show that it generalizes to
    $$\omega^{\,2} = \left(g\,k + \frac{T}{\rho}\,k^{\,3}\right)\tanh(k\,d)$$
    in water of arbitrary depth.

27. Show that in water of uniform depth $d$ the phase velocity of surface waves can only attain a stationary (i.e., maximum or minimum) value as a function of wavenumber, $k$, when
    $$k = \left[\frac{\sinh(2\,k\,d)-2\,k\,d}{\sinh(2\,k\,d)+2\,k\,d}\right]^{1/2} k_c,$$
    where $k_c= (\rho\,g/T)^{1/2}$. Hence, deduce that the phase velocity has just one stationary value (a minimum) for any depth greater than $3^{1/2}\,k_c^{-1}\simeq 4.8\,{\rm mm}$, but no stationary values for lesser depths.

28. Unlike gravity waves in deep water, whose group velocities are half their phase velocities, the group velocities of capillary waves are $3/2$ times their phase velocities. Adapt the analysis of Section 9.14 to investigate the generation of capillary waves by a very small object traveling across the surface of the water at the constant speed $V$. Suppose that the unperturbed surface corresponds to the $x$-$y$ plane. Let the object travel in the minus $x$-direction, such that it is instantaneously found at the origin. Find the present position of waves that were emitted with wavefronts traveling at an angle $\beta$ to the object's direction of motion (see Figure 9.6), when it was located at $(X$, $0)$. Show that along a given interference maximum the quantities $X$ and $\beta$ vary in such a manner that $X\,\sin^3\beta$ takes a constant value, $X_0$ (say). Deduce that the interference maximum is given parametrically by the equations

    $$x = \frac{X_0}{\sin^3\beta}\left(1-\frac{3}{2}\,\sin^2\beta\right),$$

    $$y = \frac{3}{2}\,X_0\,\frac{\cos\beta}{\sin^2\beta}.$$

    Sketch the pattern of capillary waves generated by the object. [Modified from Lighthill (1978).]