PHYS212 : Homework |
Pulses, Fourier Transformations, Bandwidth Limits
(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.
(a) Find expressions for the phase velocity
(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 |