PHYS212 : Homework


Homework Assignment 4 (due 2/16)

The Wave Equation, Fourier Series

  1. Show that the separate position and time dependent solutions of the wave equation

    \begin{eqnarray*}
\nonumber
f(x) &=& D \cos(kx) + E \sin(kx) \\
g(t) &=& F \cos(\omega t) + G \sin{\omega t}
\end{eqnarray*}

    can be combined to give product solutions

    \begin{displaymath}
y(x,t) = A \cos(kx -\omega t) + B \sin(kx - \omega t).
\end{displaymath}

    What relationships between the constants $A, B,...,G$ does this require?

  2. Using the complex exponential forms of the sine and cosine functions, show that

    \begin{eqnarray*}
\int_0^{\lambda} \cos \left( \frac{2 \pi n x}{\lambda} \right)...
...} \right)
\sin \left( \frac{2 \pi m x}{\lambda} \right) dx &=& 0
\end{eqnarray*}

    where the Kronecker delta $\delta_{nm}$ is

    \begin{displaymath}
\delta_{nm} = \left\{
\begin{array}{lc}
1 & (n = m)\\
0 & (n \neq m)
\end{array}\right.
\end{displaymath}

    and $n$ and $m$ are integers.

  3. A single wavelength $\lambda$, centered on the origin, of a triangular wave form with amplitude $A$ is described by

    \begin{displaymath}
f(x) =
\left\{
\begin{array}{lc}
A(1 + \frac{4x}{\lambda}) ...
...4x}{\lambda}) & (0 < x < \frac{\lambda}{2})
\end{array}\right.
\end{displaymath}

    (a) Find the coefficients of the sine/cosine Fourier series representing the wave form. Instead of integrating from $x = 0$ to $x
= \lambda$, shift your limits to $x = -\frac{\lambda}{2}$ and $x =
\frac{\lambda}{2}$.

    (b) Find an expression for a standing triangle wave on a string of length $L = \frac{\lambda}{2}$. Consider the endpoints of the string to be fixed at $x = -\frac{L}{2}$ and $x = \frac{L}{2}$

    (c) Use Maple to produce an animation of a standing triangle wave, including a reasonable number of terms in the Fourier series. To help you with Maple syntax, here is a Maple worksheet with an animation of a standing square wave: standingSquare.mws. You will need to change the a_o, a_n, and b_n calculations and shift the $x$ range of the animation to x = -0.5..0.5. (Since we aren't considering a particular string, I have set $A$, $L$ and $v$ to 1.)

Lab 2: The Hanging Chain


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