FOURIER SOUND

This mathlet generates tones from some standard waveforms and their Fourier reconstructions. It also shows the graph of the waveform and its Fourier reconstruction. The user can adjust the Fourier coefficient to see the tonal and visual effect on the waveform.

Graphs
The mathlet is color coded. Depending on the mode, the original waveform is shown in one color and the Fourier reconstruction in a different color. All the buttons, checkboxes and sliders use the same color coding.

Sounds Menu

This lets the user choose from several standard periodic waveforms (square, triangular, etc.)
There are two other waveforms in the menu:
- Impulse train. This is approximated as a periodic sequence of short duration high amplitude pulses. It's Fourier series is a sum of equal amplitude cosine terms.
- White noise. This is a random sequence of independently chosen samples --as such it doesn't have a fundamental frequency or Fourier series in the sense of this applet. The Fourier coefficients are set to 0. Any change of Fourier coefficient causes a new random sequence to be chosen, but otherwise has no effect on the sound. Likewise for frequency, gain etc.

Scale Menu This has the 12 notes of the chromatic scale starting at middle C (the frequency of middle C is chosen so A = 440 hz). The notes are equally spaced on a logarithmic scale. When chosen, the original waveform is set to be a sine wave (pure tone) of the given frequency.

Buttons

Play/Stop: Play or stop the currently selected sound(s).
Zero phase: Sets the phase angles of all the Fourier coefficients to zero.

Checkboxes

Show: Show the associated waveform.

Radio buttons

Play: Play the associated waveform.
a,b/amplitude phase: Display the Fourier coefficients in rectangular or amplitude phase form.

Sliders

Frequency slider: (labeled 'f') Set the frequency of the associated waveform. Finer control: (see below) shift changes the increment to 1 and alt (option) changes it to 10.
Gain (DB) sliders: Add a gain to the associated waveform.
Zoom slider: Change the scale on the t (horizontal) axis.
Keyboard control Once any slider is selected you can increment/decrement it with the arrow keys (up or right to increment, left or down to decrement). In case arrow keys are not available, you can also use d or w to increment and s or a to decrement. (This is the standard WASD control from the gaming world.)
Finer control If you have shift and alt (or option) keys, then, for some of the sliders, pressing shift or alt (option) while using the arrows or the a,w,s,d keys will change the increment. This allows for finer control of the slider value.

Fourier coefficients The Fourier coefficients are computed from the known waveforms theoretically

The Fourier coefficients are shown in the columns in the rectangular on the lower right of the window.
If Rectangular is selected the values are the the usual coefficients, a_n and b_n, of cosine and sine.
If Amplitude-phase is selected they are the polar form of a_n and b_n.
The relationship between the types of coefficients is
a_ncos(ωt) + b_nsin(ωt) = A_ncos(ωt - φ_n)
The checkboxes next to the coefficients determine whether or not they are included in the Fourier series used in the reconstructed waveform.
You can select or deselect a range of coefficients by shift-clicking a checkbox. Shift-clicking affects all the boxes below the checked box.
You can change the reconstructed waveform by using the sliders to the right of each pair of coefficients. Finer control: (see above) shift changes the increment to 0.01 and alt (option) changes it to 0.001.