Sound theory: Combination of sinusoids

Sinusoids are the simplest waveforms, but also the least interesting in terms of sound, aesthetically speaking.

Let's spice things up and make them a little more complicated. We have said how any waveform can be boiled down to a combination (sum) of sinusoids with the right amplitude and phase. This is the extraordinary discovery the French mathematician Jean Baptiste Fourier (1768-1830) made. Anyhow, let's start from the beginning, taking two in-phase waveforms. If we return to our example of the the dot circumnavigating the circumference anti clockwise, let's now imagine two waveforms generated from two points and setting off at the same time and travelling at the exact same speed:

Sum and difference of sinusoids

We can see that the sum of the two sinusoids is a single sinusoid measuring double the size of the former two. What happens sound-wise? We hear a sound that has the same frequency as the two single sinusoids but with a doubled amplitude, resulting in an increase in volume. How much has the volume increased by? Not quite double the volume, but a little less in fact (we'll fully explain why later).

What happens if we sum two waveforms at the same frequency that are out of phase (imagine the same two dots, one going anti clockwise and the other clockwise)? You don't need me to answer this question... Do you?

Too simple? Ok, then let's take two waveforms that are out of phase by 90 degrees but with different frequencies (one double the other). The following sound is a sinusoid with a frequency of 1 KHz and a phase of 0^o:

Sinusoid [f=1 KHz, φ=0^o]  [Track 1]

The following sound is a sinusoid with a 2 KHz frequency that is double the previous one and with an initial phase of 90^o:

Sinusoid [f=2 KHz, φ=90^o]  [Track 2]

The graphs of the two waveforms are compared in the following figure:

Comparison between sinusoids

As already mentioned, one peculiar characteristic of sounds is that thay can be summed up without them interfering with each other. By summing up the two previous sounds we get a new one. If we listen to it we can clearly discern the two added components. Give it a try:

Sinusoid sum of 1 KHz (0^o)+ 2 KHz (90^o)  [Track 3]

This new waveform has the rate shown on the following figure, obtained as a point by point sum of the two sinusoids:

Point by point sum of two sinusoids