These applets provide the user with a graphical and audible demonstration of the Fourier series approximation. A user can set values for the first few Fourier coefficients of a function and the applet draws the resulting waveform. The enhanced version can also play the resulting sound. The original version of the applet is available in German and English; the source code for both versions is available under the GNU public license.
Type of Material:
It requires a "Java-enabled" browser and a sound card.
Identify Major Learning Goals:
To illustrate the approximation process and develop intuition for Fourier series, by showing the effect of individual terms on the total waveform.
Target Student Population:
Mostly Real Analysis, ODE, and PDE students. Also students encountering Fourier series for the first time, whether in mathematics, science or engineering courses.
Prerequisite Knowledge or Skills:
Basic knowledge of Real Analysis and Fourier series.
This applet allows users to change 13 (27 in the enhanced version of the applet) Fourier coefficients as well as argument of the sin and cos parts (in the enhanced version). The result of the Fourier approximation with the above-mentioned number of terms is given as a graph. In the enhanced version, a sound signal corresponding to a tone for the waveform produced is generated. The applet can store up to three different waveforms which is helpful for comparing different sequences or different numbers of terms. The extended version of the applet has simple editing capabilities, making it easy to enter the coefficients for square or saw-tooth waves, or to reset all coefficients to 0.
The site could use a more extended introduction section on Fourier series and Fourier approximation. Some Analysis and/or Physics textbook examples where this tool can be used would be a plus. The original version of the applet only allows the user to choose 13 coefficients in the Fourier series (a_0 through a_6, b_1 through b_6), and to increment their values in steps of 0.1. This is not really enough to demonstrate the examples given, such as the square wave with coefficient b_3 = 1/3. (This problem is largely remedied in the extended version, which allows the user to choose a_n and b_n for n <= 13, in steps of 0.02.)
Potential Effectiveness as a Teaching Tool
The applet (particularly the extended version) does one thing, and does it well. As a learning tool it can help students establish a relation between a set of Fourier coefficients in an approximating Fourier polynomial and a graph of a periodic function that is the result of this approximation. Students should acquire a good understanding of the effect of adding different terms into a Fourier series, and of the convergence of the Fourier series to the function.
The effectiveness of the original version of the applet is somewhat lessened by the small number of terms allowed. An opportunity to see all three graphs in different color at the same time in the enhanced version would be a plus. The enhanced version of the page should permanently replace the basic one.
Ease of Use for Both Students and Faculty
This is a very straightforward applet to use. The enhanced version function syntax is similar to most graphing calculators and most computer algebra systems and is clearly explained in the notes that accompany the applet. The interface is very intuitive, and the feedback (displayed waveform) from the applet is immediate, so that the effect of changing a coefficient is easy to follow in real time.
The applet definitely needs a stand-alone reset button that would reset the values of all the coefficients to zero at once. The coefficients are generally entered by positioning a scroll bar, either by dragging the scroll bar directly or by clicking on direction buttons. These buttons occasionally stick so that the scroll bar overshoots; while not serious, this can be frustrating since the buttons are most likely to be used for fine-tuning the value of the coefficient.
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