PHYS212 : Homework


Homework Assignment 7 (due 3/30)

Reflection and Refraction

  1. (a) Use Snell's law and the expression for the reflection coefficient at a planar interface

    \begin{displaymath}
R = \frac{Z_2 \sec \theta_2 - Z_1 \sec \theta_1}
{Z_1 \sec \theta_1 + Z_2 \sec \theta_2}
\end{displaymath}

    to show that Brewster's angle, the angle of incidence at which no reflection occurs, is given by

    \begin{displaymath}
\tan^2 \theta_b = \frac{\left( \frac{Z_2}{Z_1} \right)^2 - 1}
{1 - \left( \frac{v_2}{v_1} \right)^2}
\end{displaymath}

    (b) Total internal reflection can occur at a planar interface under certain circumstances. It occurs for angles of incidence greater than the critical angle $\theta_c$ for which $\theta_2 =
\frac{\pi}{2}$. Use Snell's law to find an expression for $\theta_c$.

  2. We can reduce the reflections from an interface between two media with impedances $Z_1$ and $Z_2$ by inserting a layer of a third material with impedance $Z_o$ such that $Z_1 < Z_o < Z_2$. For normal incidence, find an expression for the optimal value of $Z_o$ (i.e., the value giving minimal reflection) in terms of $Z_1$ and $Z_2$.

  3. The average intensity of an acoustic wave is given by

    \begin{displaymath}
\bar{I} = \frac{p_o^2}{2 \sqrt{\rho_o B}} = \frac{p_o^2}{2 Z}
\end{displaymath}

    where $Z$ is the acoustic impedance of the medium.

    (a) Use this expression and the reflection and transmission coefficients describing amplitudes we derived in class to find intensity transmission and reflection coefficients

    \begin{displaymath}
\mathcal{T} \equiv \frac{I_t}{I} \hspace{2cm}
\mathcal{R} \equiv \frac{I_r}{I}
\end{displaymath}

    for normal incidence.

    (b) Verify that these expressions conserve energy at the boundary. That is, show that

    \begin{displaymath}
\mathcal{T} + \mathcal{R} = 1
\end{displaymath}


Copyright © 2003-2009, Lewis A. Riley Updated Tue Apr 14 02:04:45 2009