This java applet presents the concepts of Fourier Series through an interactive wave generator. The applet supports inclusion of sine, cosine, and the constant term with different applitudes with an adjustable fundamental frequency. Summations of Fourier series through the 15'th harmonic are supported. The result is displayed graphically; sound output is also available. Coefficients may be input either graphically using slider controls or explicitly via a text box.
Understanding superposition of waves and Fourier analysis of a periodic signal.
Target Student Population:
Upper-division undergraduate (Junior level Waves or Mathematical Physics), can also be used as a lecture demo in Introductory Physics or Physics of Sound.
Prerequisite Knowledge or Skills:
Experience with waves and the concept of superposition. More advanced use of the applet will require understanding the concepts of Fourier analysis.
Type of Material:
Simulation suitable for lecture demos, homework assignments, or laboratory exercises.
Evaluation and Observation
The user-interface is highly intuitive and easy to use. Combining audio and visual feedback makes the results of manipulating amplitudes and frequencies very immediate and compelling. Students can both see and hear the results of mixing frequencies. The user can control the amplitudes of the frequencies using sliders or by direct numeric input. Also, the amplitudes for any wave can be obtained. There is a link to a page with the definition of Fourier series, the formulae for the amplitudes, and three example waves (square wave, rectified wave, and sawtooth wave). Users can also select these examples to play. There are also links to other, similar applets and material.
Potential Effectiveness as a Teaching Tool
Recommended Uses for Material: Lecture demostrations, homework assignments, or as a component of a laboratory exercise.
The applet includes all terms through the 15'th harmonic. This is sufficient to provide highly accurate reconstructions of standard Fourier series examples. This applet is flexible in the sorts of assignments that could be attached to it. Students can be asked to compute a Fourier series, then build the wave on the sythesizer, or vice versa. There could be competitions to generate different sorts of sounds. It could also be used as a tool in a laboratory to analyze recorded sounds.
As currently constructed, instructors will need to develop the additional assignments and other material necessary to help guide students throught the applet.
Ease of Use for Both Students and Faculty
The GUI is very well laid out and intuitive. The ability to input coefficents explicitly via a text box makes the applet better suited for quantitative exploration of specific examples.
There is no means of saving a particular configuration for later use. When the fundamental frequency is changed,
the amplitude of the waves is set to zero. To restore the waves with the new frequency, the user must adjust one of the amplitude controls.
The instructions for using the applet are brief and only minimally explain the features of the applet, including the ability to input frequency amplitudes numerically.