Exercises

1. 1. Write the traveling wave $\psi(x,t)= A\,\cos(\omega\,t-k\,x)$ as a superposition of two standing waves.
   2. 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.
   3. 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.

   Figure 6.2: Figure for Exercise 6.

   

6. Two co-axial transmission lines of impedances $Z_1$ and $Z_2$ are connected as indicated in Figure 6.2. That is, the outer conductors are continuous, whereas the inner wires are connected to either side of a resistor of resistance $R_L$. The length of the resistor is negligible compared to the wavelengths of the signals propagating down the line. Suppose that $Z_1>Z_2$. Suppose, further, that a signal is incident on the junction along the line whose impedance is $Z_1$.
   1. Show that the coefficients of reflection and transmission are

      $$R =\left(\frac{R_L-Z_1+Z_2}{R_L+Z_1+Z_2}\right)^2,$$

      $$T = \frac{4\,Z_1\,Z_1}{(R_L+Z_1+Z_2)^2},$$

      respectively.
   2. Hence, deduce that the choice
      $$R_L= Z_1-Z_2$$
      suppresses reflection at the junction. Demonstrate that, in this case, the fraction of the incident power absorbed by the resistor is
      $$A=1-\frac{Z_2}{Z_1}.$$

   3. Show, finally, that if the signal is, instead, incident along the line with impedance $Z_2$ (and $R_L=Z_1-Z_2$) then the coefficient of reflection is
      $$R=\left(1-\frac{Z_2}{Z_1}\right)^2,$$
      and the fraction of the incident power absorbed by the resistor is
      $$A = \left(1-\frac{Z_2}{Z_1}\right)\frac{Z_2}{Z_1}.$$

   This analysis suggests that a resistor can be used to suppress reflection at a junction between two transmission lines, but that some of the incident power is absorbed by the resistor, and the suppression of the reflected signal only works if the signal is incident on the junction from one particular direction.

   Figure 6.3: Figure for Exercise 7.

   

7. Consider a junction of three co-axial transmission lines of impedances $Z_1$, $Z_2$, and $Z_3$ whose inner and outer conductors are connected as shown in Figure 6.3. Suppose that a signal is incident on the junction along the line whose impedance is $Z_1$.
   1. Show that the coefficient of reflection is
      $$R = \left[\frac{Z_2\,Z_3-Z_1\,(Z_2+Z_3)}{Z_2\,Z_3+Z_1\,(Z_2+Z_3)}\right]^{\,2},$$
      and that the fractions of the incident power that are transmitted down the lines with impedances $Z_2$ and $Z_3$ are

      $$T_2 = \frac{4\,Z_1\,Z_2\,Z_3^{\,2}}{[Z_2\,Z_3+Z_1\,(Z_2+Z_3)]^{\,2}},$$

      $$T_3 = \frac{4\,Z_1\,Z_2^{\,2}\,Z_3}{[Z_2\,Z_3+Z_1\,(Z_2+Z_3)]^{\,2}},$$

      respectively.
   2. Hence, deduce that if the lines all have the same impedance then $R=1/9$ and $T_2=T_3=4/9$.
   3. Demonstrate, further, that if
      $$\frac{1}{Z_1}=\frac{1}{Z_2}+\frac{1}{Z_3}$$
      then there is no reflection, and
      $$\frac{T_2}{T_3} = \frac{Z_3}{Z_2}.$$

   This analysis suggests how one might construct a non-reflecting junction between three transmission lines that diverts a given fraction of the incident power into one of the outgoing lines, and the remainder of the power into the other outgoing line.

8. Consider the problem investigated in the previous question. Suppose that
   $$\frac{1}{Z_1}=\frac{1}{Z_2}+\frac{1}{Z_3},$$
   which implies that there is no reflection when a signal is incident on the junction along the transmission line whose impedance is $Z_1$. Suppose, however, that the signal is incident along the transmission line whose impedance is $Z_2$.
   1. Show that, in this case, the coefficient of reflection is
      $$R=\left(\frac{Z_2}{Z_2+Z_3}\right)^{\,2}.$$

   2. Likewise, show that the coefficient of reflection is
      $$R=\left(\frac{Z_3}{Z_2+Z_3}\right)^{\,2}$$
      if the signal is incident along the transmission line whose coefficient of reflection is $Z_3$.
   This analysis indicates that the lossless junction considered in the previous question can only be made non-reflecting when the signal is incident along one particular transmission line.

   Figure 6.4: Figure for Exercise 9.

   

9. Consider a junction of three identical co-axial transmission lines of impedance $Z$ that are connected in the manner shown in Figure 6.4.That is, the outer conductors are continuous, whereas the inner wires are connected via three identical resistors of resistance $R_L$. The lengths of the resistors are negligible compared to the wavelengths of the signals propagating down the lines.
   1. Show that if a signal is incident on the junction along a particular line then a fraction
      $$R =\left(\frac{R_L-Z/3}{R_L+Z}\right)^2$$
      of the incident power is reflected, and a fraction
      $$T = 4\left(\frac{Z/3}{R_L+Z}\right)^2$$
      is transmitted along each outgoing line.

   2. Hence, deduce that the choice
      $$R_L = \frac{Z}{3}$$
      suppresses reflection at the junction. Show that, in this case, half of the incident power is absorbed by the resistors.
   This analysis indicates how one might construct a lossy junction between three identical transmission lines which is such that there is zero reflection no matter along which line the signal is incident.

10. Consider a generalization of the problem considered in the previous question in which $N>1$ identical transmission lines of impedance $Z$ meet at a common junction, and are connected via identical resistors of resistance $R_L$. Show that the choice
    $$R_L = \left(1-\frac{2}{N}\right)Z$$
    suppresses reflection at the junction. Show, further, that, in this case, a fraction
    $$A = \frac{N-2}{N-1}$$
    of the incident power is absorbed by the resistors.

11. 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.
    1. 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.

    2. 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?
    3. Show that the current obeys the wave-diffusion equation
       $$\frac{\partial^{\,2} I}{\partial t^{\,2}}+ \frac{{\cal R}}{{\cal ... ... t} = \frac{1}{{\cal L}\,{\cal C}}\,\frac{\partial^{\,2} I}{\partial x^{\,2}}.$$

    4. 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.

12. 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.

13. 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 peak 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 peak amplitude of the magnetic component is $2.7\,{\rm A}\,{\rm m}^{-1}$. [From Pain 1999.]

14. 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 mean pressure due to sunlight at the Earth's surface (assuming that the sunlight is completely absorbed).

15. 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.
    1. 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.

    2. 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.]

16. 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.

17. 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.)
    1. 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?
    2. Demonstrate that $E_x$ obeys the wave-diffusion equation
       $$\frac{\partial^{\,2} E_x}{\partial t^{\,2}} + \frac{\sigma}{\epsi... ...artial E_x}{\partial t}= c^{\,2}\,\frac{\partial^{\,2} E_x}{\partial z^{\,2}},$$
       where $c=1/\sqrt{\epsilon_0\,\mu_0}$.
    3. 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.

18. 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.]