In order to see how phonons should be affected by disorder, I've been playing around with a model involving a 1D chain of masses linked by springs, where the spring strengths are all the same but the masses have a 50% chance of having one of two values. By solving the eigenvalue problem, I get a dispersion such as the one plotted below. (It has k=1 and m=23 or 209, for 400 masses.) There seems to be a well-behaved acoustic region for low energy/wavevector, followed by more randomly-located energies at higher wavevector.
My question is, what dispersion should we expect for $N \rightarrow \infty$, and what methods exist to determine it?