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, $\alpha (\omega )$, of transverse phonons of low frequency, $\omega$, exhibits power law scaling as the packing fraction $\varphi$ is lowered towards ${\varphi }_c$, 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 $\alpha (\omega )\sim {\omega }^4(\varphi -{\varphi }_c{)}^{-5\mathrm{/}2}$ for $\omega$ much less than ${\omega }^{\ast }\sim (\varphi -{\varphi }_c{)}^{1\mathrm{/}2}$, 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 ${\mathrm{\ell }}^{\ast }\sim (\varphi -{\varphi }_c{)}^{-1\mathrm{/}2}$ diverges at the transition.