[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).