  The damping or attenuation coefficient of sound waves in solids due to
impurities scales with the wavevector to the fourth power, also known as
Rayleigh scattering. In amorphous solids, Rayleigh scattering may be enhanced
by a logarithmic factor although computer simulations offer conflicting
conclusions regarding this enhancement and its microscopic origin. We present a
tensorial replica field-theoretic derivation based on heterogeneous or
fluctuating elasticity (HE), which shows that long-range (power-law) spatial
correlations of the elastic constants, is the origin of the logarithmic
enhancement to Rayleigh scattering of phonons in amorphous solids. We also
consider the case of zero spatial fluctuations in the elastic constants, and of
power-law decaying fluctuations in the internal stresses. Also in this case the
logarithmic enhancement to the Rayleigh scattering law can be derived from the
proposed tensorial HE framework.
