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Boundary conditions and Fourier superposition: Instructor's Guide

Main Ideas

1. Fourier Series
2. Boundary Conditions

Estimated Time: 30 minutes, or as long as you'll let it go

In groups of three, students must decide which harmonics are present in a wave form and decide how the wave form develops in time.

Prerequisite Knowledge

This exercise is similar to one in the Oscillations class, and several homeowrk examples, where Fourier components of a an oscillatory function were found. This activity also repeats tasks from the Initial Conditions activity earlier in this course.

Activity: Introduction

Discussion until this point has been about single frequency waves. The solution to the non-dispersive wave equation are valid for any frequency, and nothing specifies a particular frequency. In this example we will see how shapes other than sinusoids are allowed, and how to deconstruct the waveform to find the component harmonic waves. The constraints are:

• $\psi \left( x,t=0 \right)=A\sin \left( \frac{\pi x}{L} \right)\left( 1+\cos \left( \frac{\pi x}{L} \right) \right);\left. \frac{\partial \psi \left( x,t \right)}{\partial t} \right|_{t=0}=0$
• $\psi (x=0,t)=0\text{ }\psi (x=L,t)=0\text{ }$

Activity: Student Conversations

• In very rare cases, a group may immediately see the the expression given can be expanded to $\psi \left( x,t=0 \right)=A\sin \left( \frac{\pi x}{L} \right)+\frac{A}{2}\sin \left( \frac{2\pi x}{L} \right)$, which immediately identifies the two components. Even if they do, there is sometimes some discussion about whether or not another frequency component could “appear later”.
• Some groups simply multiply the $\psi \left( x,t=0 \right)$ function by $cos \left( \omega t \right)$ claiming that this satisfies the initial condition. Usually someone asks what $\omega$ means, and this may lead the group to realize that there is a different frequency associated with each wavelength component. Sometimes, the group needs prompting: if all waves of different wavelengths travel with the same velocity, then what does it imply about the frequency of oscillation of each component?
• We use the language of dispersion explicitly, so that students are able to say and understand, “The rope is a non-dispersive medium.” But they need practice with this language, so it helps to have the group articulate that “non-dispersive” means that a waveform retains its shape, which means that all the sinusoidal components travel at the same velocity.
• Sometimes, students forget that superposition is the key idea. They understand that they can build a waveform by adding waves of different wavelengths, but it is harder for some students to understand that a given wave form can be deconstructed into such components, and furthermore, why this is a useful thing to do.

Activity: Wrap-up

It is helpful to perform several mini wrap-ups.

• Call on a group that recognized the simple expansion of the waveform yielded the sinusoidal components to present its results. Then call on a group that explicitly used the Fourier decomposition. This can be done before the groups have completed the time dependence part of the activity.
• Once the groups have worked through the time development of the waveform, ask group representatives to present results. End with a discussion and presentation of the correct results to reassure the students that they have worked through the problem correctly, or, for those who did not complete the activity or made a mistake, that they have the correct answer on hand.
• Direct the students to a homework problem that gives practice with this Fourier decomposition skill.
• The idea that different wave length components of the wave form have different frequencies will come back in the quantum part of the course when they study the time dependent Schroedinger equation, and it is a good idea to make this point. In fact, the students have already seen time evolution of the quantum state in the Spins course, so it is a good connecting point.

Extensions

This activity describes a 2-component waveform. A more challenging problem is a triangle waveform, which has an infinite number of components. I began with this activity, but realized that when I gave a picture of the triangle waveform, the additional step of recognizing that it must be written in algebraic form (and, surprisingly, how to write it in a piecewise form!) interfered with the Fourier decomposition exercise.

I therefore switched to using the simpler 2-component simpler waveform, presented algebraically, and the students found it just as challenging. One might

• try both wave forms in a single activity,
• present the triangle wave form as an algebraic expression
• leave the triangle from for homework

In any event, there should be follow-up homework problems. A skewed triangle wave, which is drawn on $\left( \psi ,x \right)$ axes, rather than being written in functional form, is a good one. The students really need practice with translating drawn functions into mathematical form, especially if the function is piecewise, like the skewed triangle form.

Any simple sum of two or three harmonics of the same base sinusoid can be easily disguised by using trigonometric sum formulae. Assign one such for homework.

Once Fourier analysis is mastered, Mathematica can be brought in - at first just to help solve the integrals, and finally, the students will discover the canned functions.

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