The app controls are labelled in Italian: below, each label is given as it appears in the app, followed by its English meaning.

Analysis of an irregularly shaped wave

1. Instructions
   • Click here to start the app and in the Trasformazioni (transformations) window select Cancella tutto (clear all)
   • Select Suono (sound) with the mouse;
   • by dragging the mouse over the yellow horizontal line (which at some point will turn green), build a waveform of any shape; note that the program automatically makes the wave you have drawn periodic;
   • Using the corresponding slider, vary the "Numero di armoniche" (number of harmonics) until it reaches zero (obviously, in this case the sound will stop);
   • Now start, again using the slider, to insert one harmonic at a time, starting from the lowest-frequency one.
2. Observations
   • The progressive insertion of the new harmonics has an effect that is
     • "graphical": the program shows in green the shape of the wave formed by the superposition of the harmonics of increasing frequency; as we add contributions, the green waveform begins to "trace" more and more closely the irregular (but periodic) signal built "by hand" with the mouse. This is nothing but Fourier's theorem at work: any periodic waveform (even a very irregular one like ours) can be obtained as a sum of sinusoids with frequencies that are multiples of the frequency of the wave and with suitable phases;
     • "acoustic": as we add harmonic contributions, the sound changes its timbre. Note that the lower-frequency harmonic contributions in any case produce "pleasant" sounds (this is because the default fundamental frequency of 262 Hz is chosen in a range of frequencies that our ear perceives as musical).
3. Other experiments
   • Now try, without changing the waveform you have already created, to lower the "frequenza della fondamentale" (fundamental frequency) with the corresponding slider, and to set the number of harmonics to zero.
   • Insert one harmonic at a time. For a while you will hear nothing, because the first harmonics have a frequency below the threshold of hearing;
   • At some point you will begin to hear an "intermittent" sound (a succession of pulses, one for each period of the fundamental: it is the beating between neighbouring harmonics); if you speed up the insertion of the harmonics you will obtain a sound effect worthy of an experienced film sound engineer!! Repeat the experiment with an even more irregular waveform and the effect will be even more spectacular.

Fourier analysis of a pulse

1. Instructions
   • Click here to start the app and in the Trasformazioni (transformations) window select Cancella tutto (clear all)
   • Select Suono (sound) with the mouse;
   • by dragging the mouse over the yellow horizontal line (which at some point will turn green), build an extremely narrow waveform; note that the program automatically makes the wave you have drawn periodic, so we cannot speak of a pulse in the strict sense, if by pulse we mean a single disturbance of very short duration and large amplitude.
   • Using the corresponding slider, vary the "Numero di armoniche" (number of harmonics) until it reaches zero (obviously, in this case the sound will stop);
   • Now start, again using the slider, to insert one harmonic at a time, starting from the lowest-frequency one.
2. Observations
   • The experiment is entirely analogous to the previous one: after all, the pulse is a particular irregular waveform, in the sense that it does not appear in the "repertoire" of default waves (sine, triangle, square wave, etc.). The interest lies less in the sound than in the graphical aspect of the contribution of the amplitudes of the various harmonics: the narrower the pulse, the more significant the amplitudes of the high-order harmonics become. The pulse is a wave whose shape differs so much from the sine wave that a very large number of harmonics is needed to reconstruct it with sinusoidal functions alone.