PHYS212 : Homework


Homework Assignment 5 (due 3/2)

Pulses, Fourier Transformations, Bandwidth Limits

  1. (a) Find the wavenumber spectrum (the Fourier transformation) of a triangular pulse described by

    \begin{displaymath}
f(x) = \left\{
\begin{array}{cc}
A(1+\frac{2x}{L}) & (\frac{...
...{L}{2})\\
0 & (\vert x\vert > \frac{L}{2})
\end{array}\right.
\end{displaymath}

    (b) Use Maple to produce a plot comparing the wavenumber spectrum for $L=1$ and $L=2$. (Use $A=1$ for both plots.) Do your plots demonstrate bandwidth limiting behavior similar to that we observed with the square and Gaussian pulses? Find a formal bandwidth limit expression for a triangle pulse ( $\Delta k \Delta x = ?$). As we did with the square pulse, use the big central peak in your wavenumber spectrum to evaluate $\Delta k$. Be sure to explain how you define the width $\Delta x$ of the pulse. (There are various methods and no right answer.)

    (c) Use Maple to plot a comparison between the inverse Fourier transformation (which should look like a triangle pulse) and a partial inverse Fourier transformation of your spectrum from part (a) which only includes the big peak in the spectrum centered around $k=0$. Maple will not be able to evaluate this integral analytically, so it will integrate it numerically to produce the plot. This means that the plot may take a while to produce, so don't worry that Maple is broken. This is what a triangle pulse would look like if it entered a medium in which waves of higher $k$ could not propagate. It also demonstrates importance (or lack thereof) of the portion of the spectrum to either side of the dominant peak.

  2. Fill in a gap in the notes by showing that the inverse Fourier transformation (including plane wave time dependence) of the Gaussian pulse

    \begin{displaymath}
y(x,t) = \frac{\sigma A}{\sqrt{2 \pi}}
\int_{-\infty}^{\infty}
e^{-\frac{\sigma^2 k^2}{2} + i k(x-v_p t)} dk
\end{displaymath}

    gives the original pulse including time dependence,

    \begin{displaymath}
y(x,t) = A e^{-\frac{(x - v_p t)^2}{2\sigma^2}}
\end{displaymath}

    Evaluating the integral involves completing the square in the exponent as we did in lecture in taking the Fourier transformation.

  3. Imagine we stretch the 60-link heavy chain we worked with in Lab 2 horizontally between two fixed points with a tension of 100 N. The linear mass density $\mu$ of the chain is 0.1456 kg/m, and the links are 2.8 cm long. Ignore the influence of gravity (i.e. sagging and nonuniform tension).

    (a) Find expressions for the phase velocity

    \begin{displaymath}
v_p = \frac{\omega}{k}
\end{displaymath}

    and group velocity

    \begin{displaymath}
v_g = \frac{d \omega}{d k}
\end{displaymath}

    of waves on the chain. To get you started, we have the normal mode frequencies from lecture

    \begin{displaymath}
\omega_M = \frac{2}{\ell} \sqrt{\frac{T_o}{\mu}}
\sin \left( \frac{M \pi}{2(n+1)} \right)
\end{displaymath}

    The wavelengths of the modes are described by

    \begin{displaymath}
M\frac{\lambda}{2} = L = (n+1) \ell
\end{displaymath}

    so

    \begin{displaymath}
k = \frac{M \pi}{(n+1) \ell}
\end{displaymath}

    and hence,

    \begin{displaymath}
\omega(k) = \frac{2}{\ell} \sqrt{\frac{T_o}{\mu}}
\sin \left( \frac{k \ell}{2} \right)
\end{displaymath}

    We have a non-linear $\omega(k)$, so it appears that waves on a finite chain are dispersive.

    (b) Show that the phase velocity and group velocity are approximately the same for small $k$.

    (c) At what $k$ value do the group and phase velocities differ by 1%? What mode number $M$ has a $k$ value closest to this limit?


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