Dispersion and Self-Modulation of Waves Propagating in a Solid with Dislocations

In the paper, the basic equations describing ultrasonic wave propagation in a medium with dislocations are derived. Dispersion relations are given under the assumption that dislocations oscillate without damping, i.e., the dislocation component of the general system is conservative. It is shown that ultrasonic wave propagation is characterized by two dispersion branches (acoustic and optical). As the wave number increases, the phase velocity of the wave belonging to the acoustic branch decreases asymptotically from a finite value to zero, while the velocity of the wave belonging to the optical branch decreases asymptotically from infinity to a finite value corresponding to the longitudinal wave velocity. The Nighthill criterion is applied to study modulation instability. The form of the wave packets into which a quasi-harmonic wave is divided due to modulation instability is determined. There can be both periodic stationary wave envelopes and a solitary stationary wave envelope. It is found how the height and width of the wave packet formed due to self-modulation of a quasi-harmonic wave correlate with the basic characteristics of the dislocation structure.