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

can be combined to give product solutions

What relationships between the constants does this require?

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

where the Kronecker delta is

and and are integers.

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

(a) Find the coefficients of the sine/cosine Fourier series representing the wave form. Instead of integrating from to , shift your limits to and .

(b) Find an expression for a standing triangle wave on a string of length . Consider the endpoints of the string to be fixed at and

(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 range of the animation to x = -0.5..0.5. (Since we aren't considering a particular string, I have set , and to 1.)

Lab 2: The Hanging Chain

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