Sound theory: Combination of sinusoids

Sinusoids are the simplest imaginable wave form, and as such is also the least interesting from a sound aesthetics perspective. Let us then make things more interesting for ourselves and complicate them a little. We have said how any wave form is conducible to a combination (sum) of sinusoids with an adequate amplitude and phase. This is the extraordinary discovery the French mathematician Jean Baptiste Fourier (1768-1830) made. Anyhow, let's start from the beginning. Let's take two in-phase waveforms. Returning to our example of the the dot circumnavigating the circumference anti-clockwise, now imagine two wave forms generated from two points and setting off at the same instant and going 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 audio-wise? We hear a sound that is on the same frequency as the two single sinusoids but the doubled amplitude makes the volume increase. By how much? Not quite double the volume, a little less infact, but we'll speak about this a little later.

What happens if we sum two wave forms out of phase (imagine the same two dots, one going anti-clockwise, the other clock-wise)? You don't need me to answer this question...

Too simple for you? Ok then let's take two out of phase by 90 degrees with different frequencies (one is 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 frequency that is double that of the previous one: 2KHz and with an initial phase of 90^o:

Sinusoid [f=2 KHz, φ=90^o]  [Track 2]

The graphs of the two wave forms are compared in the following figure:

Comparison between sinusoids

As already mentioned, a particular 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 sound within which we can clearly discern the two added components:

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

This new wave form has the rhythmic flow shown on the following figure, obtained as a sum of the two sinusoids:

Sum of two sinusoids