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 regi
ons 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 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 box-distribution to the force-constants $k$ of the springs. We obtain the red
uced localization lengths $Lambda_M$ from calculated Lyapunov exponents for different system widths to roughly locate the squared critical transition frequency $omega_{text{c}}^2$. The data is finite-size scaled to acquire the squared critical transition frequency of $omega_{text{c}}^2 = 12.54 pm 0.03$ and a critical exponent of $ u = 1.55 pm 0.002$.
