Harmonic vs Anharmonic Interactions in Lattice

[YAHA]

I am currently working my way through Kitel's Solid State Physics book. When discussing the consequences of the harmonic assumption (quadratic degree of freedom for interatomic lattice interactions), he states that

1) the lattice waves do not interact
2) a single wave does not change form or decay over time.

Why is that?

 

[Jano L.]

It is basic property of the model. Coupled harmonic oscillators move in such a way that their motion can be written as sum of motions due to standing waves; these standing waves behave as independent harmonic oscillators.