[abstrakraft]

There are a couple of incorrect assertions near the beginning that should be addressed:

1. "That is, sound waves are not made out of sines as real-word phenomena."

Sinusoids are the solutions to the harmonic oscillator differential equation (\ddot{x} = -w^2 x). The harmonic oscillator is an excellent model, or a component of excellent models, for many natural phenomena, including vibrating strings and columns of air, pressure waves in fluids, the list goes on. The utility of sinusoids is not due simply to their convenient mathematical properties, it's because many natural signals are sparse in the frequency domain, precisely because sinusoids do describe real world phenomena.

2. "A piston on a crankshaft produces "pure" sinusoidal up-and-down motion when operating at constant angular velocity."

Uh, no. The motion of a piston is approximately sinusoidal as you decrease the radius of the crank, but in no case is it a "pure" sinusoid.

[abstrakraft]

Another way to phrase my response to #1 above is that sinusoids are the natural modes of many systems. That is, you put a sinusoid in, and you get a sinusoid of the same frequency out. It may change in amplitude and/or phase, but it still has the same fundamental shape. This is again because these systems act like harmonic oscillators or systems of harmonic oscillators.

So why harmonic oscillators? The same analysis shows up any time you have a restoring force proportional to, and in the opposite direction of, the displacement. This shows up in springy things (recall the old mechanical engineer's adage: everything's a spring until it breaks), and fields of all sorts (electric, gravitational, magnetic, pressure).