Fourier's Theorem

In my initial post about the sine wave and all of its magical implications on sound, I mentioned that the sine wave is the building block of sound, meaning that every waveform can be produced by combinations of sine waves. I did not go into very much depth in this claim, however. The scientific explanation for this is given by Fourier's Theorem: "any periodic function that is reasonably continuous can be expressed as the sum of series of sine or cosine terms."
http://www.sfu.ca/sonic-studio/handbook/Fourier_Theorem.html

Initially this may be hard to grasp. How can every imaginable periodic function just be sine waves? I like to think of it by beginning with the concept of wave interference:

http://www.gwoptics.org/images/ebook/interference-explain.png

Imagine waves that aren't offset by those intervals, and you start to get more complicated waves. This is a good demo that illustrates this simply with three harmonic waves that you can combine:
http://phet.colorado.edu/sims/normal-modes/normal-modes_en.html

Once you reach a basic understanding of this concept, you can use this resource:
http://phet.colorado.edu/en/simulation/fourier
It provides you with the tools to make some very complicated waves, but it also gives you presets with different basic wave types to show how they are sums of sine functions.

An interesting extension of this is to look at the app's square wave:

What is interesting to me, is that the square wave is composed of only odd harmonics, which is identical to what happens to sound in a tube that is closed at one end. A clarinet is similar in that it is closed at one end. It is a bit more complicated though, because clarinets have a bell-shaped opening. Regardless, the waveform of sound from a clarinet looks similar to this square wave:

http://hyperphysics.phy-astr.gsu.edu/hbase/music/imgmus/claw2.gif