PHYS212 : Lectures

Subsections

• 4.3.1 Plane Waves in Three Dimensions
• 4.3.2 Planar Boundaries

4.3 Reflection, Refraction, and Transmission

4.3.1 Plane Waves in Three Dimensions

• Acoustic plane waves in three dimensions are described by
  

  $$p(x,t) = p_o e^{i(\vec{k}\cdot\vec{r} - \omega t + \phi)}$$ (4.39)

  
  
  with a wavenumber vector
  

  $$\vec{k} = k_x \hat{\imath} + k_y \hat{\jmath} + k_z \hat{k}$$ (4.40)

  
  

• In three dimensions, the displacement of the medium by a plane wave is a vector function $\vec{\delta}$ with components given by
  

  $$\delta_x = \delta_{xo} e^{i(k_x x - \omega t + \phi)}$$

  $$\delta_y = \delta_{yo} e^{i(k_y y - \omega t + \phi)}$$

  $$\delta_z = \delta_{zo} e^{i(k_z z - \omega t + \phi)}$$ (4.41)

  
  

• The wavenumber vector $\vec{k}$ and the displacement vector $\vec{\delta}$ are collinear. $\vec{k}$ is always parallel to the velocity of the wave, while $\vec{\delta}$ oscillates along the velocity vector.

4.3.2 Planar Boundaries

Figure 4.2: Wave fronts of plane waves incident on, reflected from, and refracted by a planar interface between media.

Snell's Law

• Consider a planar boundary at $x = 0$ between two materials with $\rho_1$, $B_1$ and $\rho_2$, $B_2$.

• The frequency is the same on both sides of the boundary, since the transmitted wave is generated by the oscillations of the incident and reflected waves at the boundary.

• With identical frequencies and different phase velocities $v_1 = \sqrt{\frac{B_1}{\rho_1}}$ and $v_2 = \sqrt{\frac{B_2}{\rho_2}}$, we expect different wavenumbers in the two media.

• The reflected and incident waves have wavenumbers of the same magnitude,
  

  $$k_r = k$$ (4.42)

  
  
  because they have the same frequency and are in the same medium. Reflection means
  

  $$k_{rx} = -k_x$$ (4.43)

  
  
  and since nothing about the medium changes along $\hat{\jmath}$
  

  $$k_{ry} = k_y$$ (4.44)

  
  

• Eq.'s 4.43 and 4.44 give the law of reflection - that the angle of incidence equals the angle of reflection, as shown in Fig. 4.2.

• The pressure and displacement wave functions must be continuous in order to satisfy the wave equation. Along the boundary, at $t = 0$, and choosing $\phi = 0$, this constraint is expressed
  $$\begin{eqnarray*} p(0,y,0,0) + p_r(0,y,0,0,t) &=& p_t(0,y,0,0,t)\\ p_o e^{i k_y y} + p_{ro} e^{i k_{ry} y} &=& p_{to} e^{i k_{ty} y} \end{eqnarray*}$$
  
  
  which with Eq. 4.44 simplifies to
  

  $$(p_o + p_{ro}) e^{ik_y y} = p_{to} e^{ik_{ty} y}$$ (4.45)

  
  

• Eq. 4.45 can only hold for all $y$ if the exponents are equal. That is,
  
  $$k_y = k_{ty}$$
  
  
  or
  
  $$k \sin \theta_1 = k_t \sin \theta_2$$
  
  
  which can be written in terms of the phase velocities in the two media as
  

  $$\frac{\sin \theta_1}{v_1} = \frac{\sin \theta_2}{v_2}$$ (4.46)

  
  
  which is known as Snell's law.

Reflection and Transmission Coefficients

• The equality of the exponents in Eq. 4.45 relates the pressure amplitudes across the boundary
  

  $$p_o + p_{ro} = p_{to}$$ (4.47)

  
  

• An identical argument yields
  

  $$\delta_{xo} + \delta_{rxo} = \delta_{txo}$$ (4.48)

  
  
  for the component of the displacement normal to the boundary.

• Generalizing Eq. 4.4 to three dimensions, we can make the following connection between the displacement component normal to the boundary and the pressure,
  
  $$\delta_x(\vec{r},t) = \frac{ik_x}{\omega^2 \rho_o} p(\vec{r},t)$$
  
  
  and therefore
  
  $$\delta_{xo} = \frac{k_x}{\omega^2 \rho_o} p_o$$
  
  
  which with Eq.'s 4.48 and Eq. 4.43 yields
  

  $$\frac{k \cos \theta_1}{\omega^2 \rho_1} (p_o - p_{ro}) = \frac{k_t \cos \theta_2}{\omega^2 \rho_2} p_{to}$$ (4.49)

  
  

• Using the expression of Eq. 4.14 of the phase velocity, and defining the acoustic impedance
  

  $$Z \equiv \rho v_p = \sqrt{\rho B}$$ (4.50)

  
  
  Eq. 4.49 can be written in the form
  

  $$p_o - p_{ro} = \frac{Z_1 \cos \theta_2}{Z_2 \cos \theta_1} p_{to} = \frac{Z_1 \sec \theta_2}{Z_2 \sec \theta_2} p_{to}$$ (4.51)

  
  

• The sum of Eq.'s 4.47 and 4.51 gives the transmission coefficient
  

  $$T = \frac{p_{to}}{p_o} = \frac{2 Z_2 \sec \theta_2} {Z_1 \sec \theta_1 + Z_2 \sec \theta_2}$$ (4.52)

  
  

• Dividing Eq. 4.47 by $p_{o}$ gives an expression relating the transmission coefficient to the reflection coefficient
  

  $$R = \frac{p_{ro}}{p_o} = T - 1 = \frac{Z_2 \sec \theta_2 - Z_1 \sec \theta_1} {Z_1 \sec \theta_1 + Z_2 \sec \theta_2}$$ (4.53)

  
  

• Although $R$ and $T$ depend on both $\theta_1$ and $\theta_2$, Snell's law (Eq. 4.46) can be used to relate them to a single angle.

• There are two characteristic angles associated with a planar interface, Brewster's angle, the angle at which no reflection occurs, and the critical angle, the angle above which total internal reflection occurs. You will consider these for homework.

• For normal incidence, $\theta_1 = \theta_2 = 0$, and the incident, reflected, and transmitted waves lie along a single axis. Eq.'s 4.52 and 4.53 become
  

  $$T = \frac{2 Z_2}{Z_1 + Z_2}$$ (4.54)

  
  
  and
  

  $$R = \frac{Z_2 - Z_1}{Z_1 + Z_2}$$ (4.55)

  
  

• Caution! The reflection and transmission coefficients $T$ and $R$ above describe amplitudes, not intensities. You will use these coefficients to derive similar coefficients for intensity for homework.

Copyright © 2003-2008, Lewis A. Riley

Updated Wed Jan 18 09:51:28 2006