We consider the localization properties of a lattice of coupled masses and springs with random mass and spring constant values. We establish the full phase diagrams of the system for pure mass and pure spring disorder. The phase diagrams exhibit regions of stable as well as unstable wave modes. The latter are of interest for the instantaneous-normal-mode spectra of liquids and the nascent field of acoustic metamaterials. We show the existence of delocalization-localization transitions throughout the phase diagram and establish, by high-precision numerical studies, that the universality of these transitions is of the Anderson type.

We study numerically the disorder-induced localization-delocalization phase transitions that occur for mass and spring constant disorder in a three-dimensional cubic lattice with harmonic couplings. We show that, while the phase diagrams exhibit regions of stable and unstable waves, the universality of the transitions is the same for mass and spring constant disorder throughout all the phase boundaries. The combined value for the critical exponent of the localization lengths of ν = 1.550(-0.017)(+0.020) confirms the agreement with the universality class of the standard electronic Anderson model of localization. We further support our investigation with studies of the density of states, the participation numbers and wave function statistics.

We use a transfer-matrix method to study the localization properties of vibrations in a 'mass and spring' model with simple cubic lattice structure. Disorder is applied as a boxdistribution to the force-constants k of the springs. We obtain the reduced localization lengths ΛM from calculated Lyapunov exponents for different system widths to roughly locate the squared critical transition frequency ω 2 c . The data is finite-size scaled to acquire the squared critical transition frequency of ω 2 c = 12.54 ± 0.03 and a critical exponent of ν = 1.55 ± 0.002.

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