Damping of Long-Wavelength Acoustic Phonons in a Disordered Anomalous Solid with Parabolic Phonon Dispersion

We propose a new anomalous soft solid with parabolic acoustic phonon dispersion relation, which yields a strong damping of long-wavelength acoustic phonons (sonic or super-sonic wave) under an induced weak disorder. The proposal is confirmed by calculating the damping rate of the acoustic phonons numerically. The damping rate (Lyapunov exponent) corresponding to the inverse of localization length has a remarkable peak at very low frequency. Both the peak value and corresponding characteristic frequency are power-law increasing functions of the strength of disorder. When we derive the peak value of the Lyapunov exponent as a function of the characteristic frequency, the peak value numerically appears proportional to the root of the characteristic frequency in our one- and two-dimensional cases. The numerical result is enough for us to confirm a strong localization of the acoustic phonons in the frequency region of sound wave of the order of 10(3)Hz in one and two dimensions.