I'm learning about phonons and this is really interesting. Why can't I describe two phonons colliding as a harmonic oscillation? Why does it have to be anharmonic for the phonon model to work?
The basic reason is that the potential energy between atoms in a real crystal lattice isn't solely quadratic in the lattice spacing - there are higher order terms, so the model of a lattice as coupled harmonic oscillators breaks down.
Modeling the lattice as a series of coupled harmonic oscillators can't account for a number of observed phenomena. Thermal conductivity and thermal expansion are just two of the most familiar properties. If crystals behaved like coupled harmonic oscillators, they would have infinite thermal conductivity, and the lattice spacing would not depend on temperature - i.e. there would be no thermal expansion or contraction. Obviously that's not what is observed, and these effects are explained by the higher order (anharmonic) terms in the lattice potential energy.
There's a good additional discussion about phonons at the earlier post :
If the potentials are all quadratic, then the force varies linearly with displacement. That means the superposition principle holds. (Don't confuse that with quantum superposition!) In other words the lattice waves then behave like electromagnetic waves: you put two of them together and they simply add together but don't otherwise affect one another; they don't scatter.
If the potentials are higher order, then the force does not vary linearly with position, and the superposition principle does not hold. (It holds only for linear systems.)