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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. dispersion of phonons with randomly varying masses

My question is, what dispersion should we expect for $N \rightarrow \infty$, and what methods exist to determine it?

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    $\begingroup$ There's a lot of literature on various types of waves in "random media". You'll find answers if you search around. I think that what you're seeing looks pretty typical. Long wavelength modes aren't much affected by the randomness, because their long wavelength means they have many masses per wavelength and kind of average them together. The shorter wavelength modes are more affected by the randomness and can become localized. $\endgroup$ – lnmaurer Jul 14 '15 at 20:27
  • $\begingroup$ It turns out that Freeman Dyson figured out an exact solution to this problem in 1953 in this paper, specifically, expressions (54)-(56). It's a bit abstract, though, and I'll wait until I get around to doing the calculations to see qualitatively what kind of spectrum I should get before posting an answer. $\endgroup$ – user1704042 Jul 21 '15 at 17:50

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