Exercises

1. 1. Show that the one-dimensional plane wave, (7.1), is a solution of the one-dimensional wave equation, (7.8), provided that
      $$\omega = k\,v.$$

   2. Demonstrate that the three-dimensional plane wave, (7.5), is a solution of the three-dimensional wave equation, (7.9), as long as
      $$\omega = \vert{\bf k}\vert\,v.$$

2. 1. Demonstrate that for a cylindrically symmetric wavefunction $\psi(\rho,t)$, where $\rho= (x^{\,2}+y^{\,2})^{1/2}$, the three-dimensional wave equation (7.9) can be rewritten
      $$\frac{\partial^{\,2}\psi}{\partial t^{\,2}} = v^{\,2}\left(\frac{... ...artial \rho^{\,2}} + \frac{1}{\rho}\,\frac{\partial\psi}{\partial\rho}\right).$$

   2. Show that
      $$\psi(\rho,t) \simeq \frac{\psi_0}{\rho^{1/2}}\,\cos(\omega\,t-k\,\rho-\phi)$$
      is an approximate solution of this equation in the limit $k\,\rho\gg 1$, where $\omega/k=v$.

3. 1. Demonstrate that for a spherically symmetric wavefunction $\psi(r,t)$, where $r=(x^{\,2}+y^{\,2}+z^{\,2})^{1/2}$, the three-dimensional wave equation (7.9) can be rewritten
      $$\frac{\partial^{\,2}\psi}{\partial t^{\,2}} = v^{\,2}\left(\frac{... ...\psi}{\partial r^{\,2}} + \frac{2}{r}\,\frac{\partial\psi}{\partial r}\right).$$

   2. Show that
      $$\psi(r,t) =\frac{ \psi_0}{r}\,\cos(\omega\,t-k\,r-\phi)$$
      is a solution of this equation, where $\omega/k=v$.

4. Consider an elastic sheet stretched over a rectangular frame that extends from $x=0$ to $x=a$, and from $y=0$ to $y=b$. Suppose that

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

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

   Show that the amplitudes and phase angles in the normal mode expansion (7.28) are given by

   $$A_{m,n} = \left(I_{m,n}^{\,2}+J_{m,n}^{\,2}\right)^{1/2},$$

   $$\phi_{m,n} =\tan^{-1}\left(\frac{J_{m,n}}{I_{m,n}}\right),$$

   where

   $$I_{m,n} = \frac{4}{a\,b}\int_0^a\int_0^b F(x,y)\,\sin\left(m\,\pi\,\frac{x}{a}\right)\sin\left(n\,\pi\,\frac{y}{b}\right)\, dx\,dy,$$

   $$J_{m,n} = \frac{4}{a\,b\,\omega_{m,n}}\int_0^a\int_0^b G(x,y)\,\sin\left(m\,\pi\,\frac{x}{a}\right)\sin\left(n\,\pi\,\frac{y}{b}\right)\, dx\,dy.$$

5. The radial oscillations of an ideal gas in a spherical cavity of radius $a$ are governed by the spherical wave equation
   $$\frac{\partial^{\,2}\psi}{\partial t^{\,2}} = v^{\,2}\left(\frac{... ...}\psi}{\partial r^{\,2}} + \frac{2}{r}\,\frac{\partial\psi}{\partial r}\right),$$
   subject to the boundary condition $\psi(a,t)=0$. Here, $r=(x^{\,2}+y^{\,2}+z^{\,2})^{1/2}$ is a spherical coordinate, $\psi(r,t)$ is the radial displacement, and $v$ is the speed of sound. Show that the general solution of this equation is written
   $$\psi(r,t)=\sum_{j=1,\infty} A_j\,{\rm sinc}\left(j\,\pi\,\frac{r}{a}\right)\,\cos(\omega_j\,t-\phi_j),$$
   where ${\rm sinc}(x)\equiv \sin x/x$,
   $$\omega_j = j\,\pi\,\frac{v}{a},$$
   and $A_j$, $\phi_j$ are arbitrary constants.

6. Show that a light-ray entering a planar transparent plate of thickness $d$ and refractive index $n$ emerges parallel to its original direction. Show that the lateral displacement of the ray is
   $$s = \frac{d\,\sin(\theta_1-\theta_2)}{\cos\theta_2},$$
   where $\theta_1$ and $\theta_2$ are the angles of incidence and refraction, respectively, at the front side of the plate.

7. Suppose that a light-ray is incident on the front (air/glass) interface of a uniform pane of glass of refractive index $n$ at the Brewster angle. Demonstrate that the refracted ray is also incident on the rear (glass/air) interface of the pane at the Brewster angle. [From Fitzpatrick 2008.]

8. 1. Show that the Fresnel relations, (7.75) and (7.76), for the polarization in which the magnetic intensities of all three waves are parallel to the interface can be written

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

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

      where $Z=Z_0/n$ represents impedance. (Here, $Z_0$ is the impedance of free space, and $n$ the refractive index.)
   2. Demonstrate that the Fresnel relations, (7.97) and (7.98), for the other polarization take the form

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

      $$T = \frac{4\,Z_2\,Z_1\,\cos\theta_i\,\cos\theta_t}{(Z_2\,\cos\theta_i+Z_1\,\cos\theta_t)^2}.$$

9. 1. Show that the expressions, (7.222) and (7.223), for the coefficients of reflection and transmission for a sound wave obliquely incident at an interface between two immiscible fluids can be written

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

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

      where $Z_1=\rho_1\,v_1$ and $Z_2=\rho_2\,v_2$ are the acoustic impedances of the two fluids.
   2. Show that the expression, (7.224), for the angle of intromission can be written
      $$\tan^2\theta_i = \frac{(Z_2/Z_1)^2-1}{1-(v_2/v_1)^2}.$$