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

(b) Use Maple to produce a plot comparing the wavenumber spectrum for and . (Use 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 ( ). As we did with the square pulse, use the big central peak in your wavenumber spectrum to evaluate . Be sure to explain how you define the width 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 . 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 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

gives the original pulse including time dependence,

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 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

and group velocity

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

The wavelengths of the modes are described by

so

and hence,

We have a non-linear , 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 .

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

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