Peer Review

Fourier: Making Waves

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Reviewed:: May 31, 2006 by Physics

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Ratings

Overall Rating:

Content Quality:

Effectiveness:

Ease of Use:

Overview:: This simulation shows how generic waveforms are represented by Fourier series. The number of harmonics, either sine or cosine terms, and their relative amplitudes can be varied by the user. Graphs display both the individual harmonics and the composite waveform. Audio output for the composite is also available. Some standard wave types (e.g. square wave, sawtooth wave, etc.) are available as presets. There is also an option for Fourier decomposition of a Gaussian wave packet where the width of the packet and the spacing of the expansion components is adjustable. Finally, this applet includes an interactive wave game where users try to match waveforms by combining harmonics.
Type of Material:: Simulation
Recommended Uses:: In class demo or homework assignments.
Technical Requirements:: Java Application
Identify Major Learning Goals:: The primary goal is to develop a qualitative understanding of the effects of
adding harmonics in a Fourier expansion and to help students visualize the resultant waveforms.
Target Student Population:: Lower and upper division undergraduate science and engineering majors.
Prerequisite Knowledge or Skills:: Students should have a working knowledge of basic wave parameters (wavelength, frequency, amplitude, etc.) and trigonometric functions.

Content Quality

Rating:

Strengths:

This simulation gives a complete graphical representation of Fourier series. The component amplitudes, the individual harmonics, and the resulting sum are all displayed. The ability to listen to the wave output adds another qualitative aspect to the display.

Options are available to display static graphs of either the spatial or time dependence only, as well as dynamical displays of the resulting wave.

There are presets for standard waveforms (square, sawtooth, triangular) and one for a Gaussian wave packet, as well user-created waveforms. A separate module illustrates the transition from discrete to continuous transforms for a wave packet.

Concerns:

Gibb's phenomena is not directly addressed in this applet. This will probably need to be pointed out by the instructor.

It is not possible to include arbitrary relative phases to the harmonics; only positive or negative amplitudes are available. The maximum number of harmonics available is eleven.

Potential Effectiveness as a Teaching Tool

Rating:

Strengths:

The applet communicates the concept of frequency components in a Fourier spectrum very effectively. It is controlled directly by the user, allowing students to explore and develop a qualitative understanding of the concepts.

Concerns:

This applet is less useful for quantitative work. The wavelength and period tools are merely draggable markers that show peak-to-peak distance. Students can not use these to make measurements.

The "discrete to continuous" tab illustrates the results of decreasing the spacing in the wave number domain. The choices for component spacing are limited and the wave packet must be a Gaussian distribution. It is unclear if the purpose is to illustrate the approach to the continuous Fourier transform or to show how a simple wave packet is formed from Fourier components.

The reviewers had a divergent opinion as to the value of the "wave game" activity. One felt it could be used to encourage students to explore the physics while the other felt that it lacked the quantitative aspects necessary for students to make and test predictions.

This applet will require some background and explaining by the instructor for students to learn from it.

Ease of Use for Both Students and Faculty

Rating:
Strengths:: The controls for the simulation are generally intuitive and easy to use.
Concerns:: The help available in this program consists of only a few screen annotations.

Most students will need some additional guidance from instructors to make use of this program. The purpose for the "discrete to continuous" tab in particular is likely to be unclear.

Other Issues and Comments:: The easy-to-use interface makes this a worthwhile learning object. However, because the applet covers Fourier series, Fourier transforms, and superposition of traveling waves without much distinction, students may miss some of the topics.

This simulation is part of a growing collection created by the Physics Education Technology Group. The materials in this collection are designed based on the latest research in physics education and testing of student impact. Research results and instructor-submitted lesson plans are available on the web site.
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