Mathematical equation for the sound wave that a piano makes
Piano tones are actually quite complicated.  The initial strike is full of non-harmonic tones that quickly dampen down.  After that, the harmonics vary from note to note, piano to piano.  I suggest you look at some actual sample files, capture a whole number of somewhat consistent waveforms, do a DFT, and read the coefficients from there.

As for the other answer that was given, some one may find this interesting:

$$ \sin^3\left(\pi x\right) + \sin \left( \pi \left( x + \frac{2}{3} \right) \right) $$

$$ = \left( \frac{ e^{i\pi x} - e^{-i\pi x} }{2i}  \right)^3 + \sin\left( \pi x+ \frac{2}{3}\pi \right) $$

$$ = \frac{ e^{i3\pi x} - 3 e^{i\pi x} + 3 e^{-i\pi x} - e^{-i3\pi x} }{-8i} + \sin( \pi x ) \cos \left(\frac{2}{3}\pi \right) + \cos( \pi x ) \sin \left(\frac{2}{3}\pi \right) $$

$$ = -\frac{ 1 }{4} \sin(3\pi x) + \frac{ 3 }{4} \sin(\pi x)  + \sin( \pi x ) \left(\frac{-1}{2} \right) + \cos( \pi x ) \left(\frac{\sqrt{3}}{2} \right) $$

$$ = -\frac{ 1 }{4} \sin(3\pi x) + \frac{ 1 }{4} \sin(\pi x)   + \frac{\sqrt{3}}{2} \cos( \pi x )  $$

Hope this helps,

Ced