Attenuation of shear sound waves in jammed solids

Abstract

We study the attenuation of long-wavelength shear sound waves propagating through model jammed packings of frictionless soft spheres interacting with repulsive springs. The elastic attenuation coefficient, α(ω), of transverse phonons of low frequency, ω, exhibits power law scaling as the packing fraction ϕ is lowered towards ϕ_c, the critical packing fraction below which rigidity is lost. The elastic attenuation coefficient is inversely proportional to the scattering mean free path and follows Rayleigh’s law with α(ω) ∼ ω⁴(ϕ − ϕ_c)^−5/2 for ω much less than ω^* ∼ (ϕ − ϕ_c)^1/2, the characteristic frequency scale above which the energy diffusivity and density of states plateau. This scaling of the attenuation coefficient, consistent with numerics, is obtained by assuming that a jammed packing can be viewed as a mosaic composed of domains whose characteristic size ^* ∼ (ϕ − ϕ_c)^−1/2 diverges at the transition.