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 phi is lowered towards phi(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 alpha(omega) similar to omega(4)(phi - phi(c))(-5/2) for omega much less than omega* similar to (phi - phi(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 l* similar to (phi - phi(c))(-1/2) diverges at the transition.