Exercises

1. Write the traveling wave $\psi(x,t)= A\,\cos(\omega\,t-k\,x)$ as a superposition of two standing waves. Write the standing wave $\psi(x,t)=A\,\cos(k\,x)\,\cos(\omega\,t)$ as a superposition of two traveling waves propagating in opposite directions. Show that the following superposition of traveling waves,
   $$\psi(x,t)= A\,\cos(\omega\,t-k\,x)+ A\,R\,\cos(\omega\,t+k\,x),$$
   can be written as the following superposition of standing waves,
   $$\psi(x,t) = A (1+R) \cos(k x) \cos(\omega t) + A (1-R) \sin(k x) \sin(\omega t).$$

2. Show that the solution of the wave equation,
   $$\frac{\partial^2 y}{\partial t^2} = v^2\frac{\partial^2 y}{\partial x^2},$$
   subject to the initial conditions

   $$y(x,0) = F(x),$$

   $$\dot{y}(x,0) =G(x),$$

   
   
   for $-\infty\leq x\leq \infty$ , can be written
   $$y(x,t)=\frac{1}{2}\left[F(x-v t)+F(x+v t) + \frac{1}{v}\int_{x-v t}^{x+v t}G(x') dx'\right].$$
   This is known as the d'Alembert solution.

3. Demonstrate that for a transverse traveling wave propagating on a stretched string,
   $$\langle {\cal I}\rangle = v \langle {\cal E}\rangle,$$
   where $\langle {\cal I}\rangle$ is the mean energy flux along the string due to the wave, $\langle {\cal E}\rangle$ is the mean wave energy per unit length, and $v$ is the phase velocity of the wave.

4. A transmission line of characteristic impedance $Z$ occupies the region $x<0$ , and is terminated at $x=0$ . Suppose that the current carried by the line takes the form
   $$I(x,t) = I_i\,\cos(\omega\,t-k\,x)+ I_r\,\cos(\omega\,t+k\,x)$$
   for $x\leq 0$ , where $I_i$ is the amplitude of the incident signal, and $I_r$ the amplitude of the signal reflected at the end of the line. Let the end of the line be open circuited, such that the line is effectively terminated by an infinite resistance. Find the relationship between $I_r$ and $I_i$ . Show that the current and voltage oscillate $\pi/2$ radians out of phase everywhere along the line. Demonstrate that there is zero net flux of electromagnetic energy along the line.

5. Suppose that the transmission line in the previous exercise is short circuited, such that the line is effectively terminated by a negligible resistance. Find the relationship between $I_r$ and $I_i$ . Show that the current and voltage oscillate $\pi/2$ radians out of phase everywhere along the line. Demonstrate that there is zero net flux of electromagnetic energy along the line.

6. A lossy transmission line has a resistance per unit length ${\cal R}$ , in addition to an inductance per unit length ${\cal L}$ , and a capacitance per unit length ${\cal C}$ . The resistance can be considered to be in series with the inductance. Demonstrate that the Telegrapher's equations generalize to

   $$\frac{\partial V}{\partial t} =-\frac{1}{{\cal C}} \frac{\partial I}{\partial x},\nonumber$$

   $$\frac{\partial I}{\partial t} =-\frac{{\cal R}}{{\cal L}} I - \frac{1}{\cal L} \frac{\partial V}{\partial x}\nonumber,$$

   
   
   where $I(x,t)$ and $V(x,t)$ are the voltage and current along the line. Derive an energy conservation equation of the form
   $$\frac{\partial{\cal E}}{\partial t} + \frac{\partial {\cal I}}{\partial x} =- {\cal R}\,I^{\,2},$$
   where ${\cal E}$ is the energy per unit length along the line, and ${\cal I}$ the energy flux. Give expressions for ${\cal E}$ and ${\cal I}$ . What does the right-hand side of the previous equation represent? Show that the current obeys the wave-diffusion equation
   $$\frac{\partial^2 I}{\partial t^2}+ \frac{{\cal R}}{{\cal L}}\,\fr... ...\partial t} = \frac{1}{{\cal L}\,{\cal C}}\,\frac{\partial^2 I}{\partial x^2}.$$
   Consider the low resistance, high frequency, limit $\omega\gg {\cal R}/{\cal L}$ . Demonstrate that a signal propagating down the line varies as

   $$I(x,t) \simeq I_0 \cos[k (v t-x)] {\rm e}^{-x/\delta},\nonumber$$

   $$V(x,t) \simeq Z I_0 \cos[k (v t-x)-1/(k \delta)] {\rm e}^{-x/\delta},\nonumber$$

   
   
   where $k=\omega/v$ , $v=1/\sqrt{{\cal L} {\cal C}}$ , $\delta = 2 Z/{\cal R}$ , and $Z=\sqrt{{\cal L}/{\cal C}}$ . Show that $k\,\delta \gg 1$ : that is, the decay length of the signal is much longer than its wavelength. Estimate the maximum useful length of a low resistance, high frequency, lossy transmission line.

7. Suppose that a transmission line consisting of two uniform parallel conducting strips of width $w$ and perpendicular distance apart $d$ , where $d\ll w$ , is terminated by a strip of material of uniform resistance per square meter $\sqrt{\mu_0/\epsilon_0} =376.73 \Omega$ . Such material is known as spacecloth. Demonstrate that a signal sent down the line is completely absorbed, with no reflection, by the spacecloth. Incidentally, the resistance of a uniform strip of material is proportional to its length, and inversely proportional to its cross-sectional area.

8. At normal incidence, the mean radiant power from the Sun illuminating one square meter of the Earth's surface is $1.35$ kW. Show that the amplitude of the electric component of solar electromagnetic radiation at the Earth's surface is $1010\,{\rm V}\,{\rm m}^{-1}$ . Demonstrate that the corresponding amplitude of the magnetic component is $2.7\,{\rm A}\,{\rm m}^{-1}$ . [From Pain 1999.]

9. According to Einstein's famous formula, $E=m\,c^2$ , where $E$ is energy, $m$ is mass, and $c$ is the velocity of light in vacuum. This formula implies that anything that possesses energy also has an effective mass. Use this idea to show that an electromagnetic wave of mean intensity (energy per unit time per unit area) $\langle {\cal I}\rangle$ has an associated mean pressure (momentum per unit time per unit area) $\langle {\cal P}\rangle = \langle {\cal I}\rangle/c$ . Hence, estimate the pressure due to sunlight at the Earth's surface (assuming that the sunlight is completely absorbed).

10. A glass lens is coated with a non-reflecting coating of thickness one quarter of a wavelength (in the coating) of light whose wavelength in air is $\lambda_0$ . The index of refraction of the glass is $n$ , and that of the coating is $\sqrt{n}$ . The refractive index of air can be taken to be unity. Show that the coefficient of reflection for light normally incident on the lens from air is
    $$R = \frac{(n-1)^2\,\cos^2[(\pi/2)\,(\lambda_0/\lambda)]}{4\,n+(n-1)^2\, \cos^2[(\pi/2)\,(\lambda_0/\lambda)]},$$
    where $\lambda$ is the wavelength of the incident light in air. Assume that $n=1.5$ , and that this value remains approximately constant for light whose wavelengths lie in the visible band. Suppose that $\lambda_0 = 550 {\rm nm}$ , which corresponds to green light. It follows that $R=0$ for green light. What is $R$ for blue light of wavelength $\lambda=450 {\rm nm}$ , and for red light of wavelength $650 {\rm nm}$ ? Comment on how effective the coating is at suppressing unwanted reflection of visible light incident on the lens. [From Crawford 1968.]

11. A glass lens is coated with a non-reflective coating whose thickness is one quarter of a wavelength (in the coating) of light whose frequency is $f_0$ . Demonstrate that the coating also suppresses reflection from light whose frequency is $3\,f_0$ , $5 f_0$ , et cetera, assuming that the refractive index of the coating and the glass is frequency independent.

12. A plane electromagnetic wave, linearly polarized in the $x$ -direction, and propagating in the $z$ -direction through an electrical conducting medium of conductivity $\sigma$ , is governed by

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

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

    
    
    where $E_x(z,t)$ and $H_y(z,t)$ are the electric and magnetic components of the wave. (See Appendix C.) Derive an energy conservation equation of the form
    $$\frac{\partial{\cal E}}{\partial t} + \frac{\partial {\cal I}}{\partial z} =- \sigma\,E_x^{\,2},$$
    where ${\cal E}$ is the electromagnetic energy per unit volume, and ${\cal I}$ the electromagnetic energy flux. Give expressions for ${\cal E}$ and ${\cal I}$ . What does the right-hand side of the previous equation represent? Demonstrate that $E_x$ obeys the wave-diffusion equation
    $$\frac{\partial^2 E_x}{\partial t^2} + \frac{\sigma}{\epsilon_0}\,\frac{\partial E_x}{\partial t}= c^2\,\frac{\partial^2 E_x}{\partial z^2},$$
    where $c=1/\sqrt{\epsilon_0 \mu_0}$ . Consider the high frequency, low conductivity, limit $\omega\gg \sigma/\epsilon_0$ . Show that a wave propagating into the medium varies as

    $$E_x(z,t) \simeq E_0 \cos[k (v t-z)] {\rm e}^{-z/\delta},$$

    $$H_y(z,t) \simeq Z_0^{ -1} E_0 \cos[k (v t-z)-1/(k \delta)] {\rm e}^{-z/\delta},$$

    
    
    where $k=\omega/c$ , $\delta = 2/(Z_0\,\sigma)$ , and $Z_0=\sqrt{\mu_0/\epsilon_0}$ . Demonstrate that $k\,\delta \ll 1$ : that is, the wave penetrates many wavelengths into the medium. Estimate how far a high frequency electromagnetic wave penetrates into a low conductivity conducting medium.

13. Sound waves travel horizontally from a source to a receiver. Let the source have the speed $u$ , and the receiver the speed $v$ (in the same direction). In addition, suppose that a wind of speed $w$ (in the same direction) is blowing from the source to the receiver. Show that if the source emits sound whose frequency is $f_0$ in still air then the frequency recorded by the receiver is
    $$f =\left(\frac{V-v+w}{V-u+w}\right)f_0,$$
    where $V$ is the speed of sound in still air. Note that if the velocities of the source and receiver are the same then the wind makes no difference to the frequency of the recorded signal. [Modified from French 1971.]