Paper detail

Attenuation of shear sound waves in jammed solids

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 law with $α(ω)\sim ω^4 (ϕ- ϕ_c)^{-5/2}$ for $ω$ much less than $ω^* \sim (ϕ- ϕ_c)^{1/2}$, 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 $\ell^ * \sim (ϕ-ϕ_c) ^{-1/2}$ diverges at the transition.