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Next: Wave Pulses Up: Multi-Dimensional Waves Previous: Sound Waves in Fluids

Exercises

  1. Show that the one-dimensional plane wave, (529), is a solution of the one-dimensional wave equation, (536), provided that

    $\displaystyle \omega = k\,v.$

    Likewise, demonstrate that the three-dimensional plane wave, (533), is a solution of the three-dimensional wave equation, (537), as long as

    $\displaystyle \omega = \vert{\bf k}\vert v.
$

  2. Demonstrate that for a cylindrically symmetric wavefunction $ \psi(\rho,t)$ , where $ \rho= \sqrt{x^2+y^2}$ , the three-dimensional wave equation (537) can be re-written

    $\displaystyle \frac{\partial^2\psi}{\partial t^2} = v^2\left(\frac{\partial^2\psi}{\partial \rho^2} + \frac{1}{\rho}\,\frac{\partial\psi}{\partial\rho}\right).
$

    Show that

    $\displaystyle \psi(\rho,t) \simeq \frac{\psi_0}{\rho^{1/2}}\,\cos(\omega\,t-k\,\rho-\phi)$

    is an approximate solution of this equation in the limit $ k\,\rho\gg 1$ , where $ \omega/k=v$ .

  3. Demonstrate that for a spherically symmetric wavefunction $ \psi(r,t)$ , where $ r=\sqrt{x^2+y^2+z^2}$ , the three-dimensional wave equation (537) can be re-written

    $\displaystyle \frac{\partial^2\psi}{\partial t^2} = v^2\left(\frac{\partial^2\psi}{\partial r^2} + \frac{2}{r}\,\frac{\partial\psi}{\partial r}\right).
$

    Show that

    $\displaystyle \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

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

    Show that the amplitudes and phase angles in the normal mode expansion (556) are given by

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

    where

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

  5. The radial oscillations of an ideal gas in a spherical cavity of radius $ a$ are governed by the spherical wave equation

    $\displaystyle \frac{\partial^2\psi}{\partial t^2} = v^2\left(\frac{\partial^2\psi}{\partial r^2} + \frac{2}{r}\,\frac{\partial\psi}{\partial r}\right),$

    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

    $\displaystyle \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$ ,

    $\displaystyle \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

    $\displaystyle 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. Show that the Fresnel relations, (603) and (604), for the polarization in which the magnetic intensities of all three waves are parallel to the interface can be written

    $\displaystyle R$ $\displaystyle = \left(\frac{Z_1 \cos\theta_i-Z_2 \cos\theta_t}{Z_1 \cos\theta_i+Z_2 \cos\theta_t}\right)^2,$    
    $\displaystyle T$ $\displaystyle = \frac{4 Z_1 Z_2 \cos\theta_i \cos\theta_t}{(Z_1 \cos\theta_i+Z_2 \cos\theta_t)^2},$    

    where $ Z=Z_0/n$ represents impedance. (Here, $ Z_0$ is the impedance of free space, and $ n$ the refractive index.) Demonstrate that the Fresnel relations, (625) and (626), for the other polarization take the form

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

  9. Show that the expressions, (695) and (696), for the coefficients of reflection and transmission for a sound wave obliquely incident at an interface between two immiscible fluids can be written

    $\displaystyle R$ $\displaystyle = \left(\frac{Z_2 \cos\theta_i-Z_1 \cos\theta_t}{Z_2 \cos\theta_i+Z_1 \cos\theta_t}\right)^2,$    
    $\displaystyle T$ $\displaystyle = \frac{4 Z_2 Z_1 \cos\theta_i \cos\theta_t}{(Z_2 \cos\theta_i+Z_1 \cos\theta_t)^2},$    

    where $ Z_1=\rho_1\,v_1$ and $ Z_2=\rho_2\,v_2$ are the acoustic impedances of the two fluids. Show that the expression, (697), for the angle of intromission can be written

    $\displaystyle \tan^2\theta_i = \frac{(Z_2/Z_1)^2-1}{1-(v_2/v_1)^2}.
$


next up previous
Next: Wave Pulses Up: Multi-Dimensional Waves Previous: Sound Waves in Fluids
Richard Fitzpatrick 2013-04-08