Acoustic-wave propagation in quasiperiodic, incommensurate, and random systems.

Abstract

Acoustic‐wave propagation in one‐dimensional systems with quasiperiodic, incommensurate, and random modulation is studied [J.P. Lu and J.L. Bitman, Phys. Rev. B 38, 8067 (1988)]. In the short‐wavelength limit it was found that if the initial pulse is narrow (with a spatial extension of a few lattice spacings), the pulse is localized in a quasiperiodic system, as in the case of a random system. This indicates that at short length scale a quasiperiodic system is similar to a random system. On the other hand, if the initial injected wave has a wavelength much larger than the lattice spacing, it is found that there is resonance for quasiperiodic and incommensurate systems when the wave vector satisfies the Bragg conditions. In this case the propagation appears to be diffusive rather than propagative; namely, the total energy of the initial wave does not propagate along the chain as it does otherwise, but is homogeneously distributed over the region of space where the wave front passed. The problem is solved analytically in the long‐wavelength limit in terms of two‐mode‐coupling theory and the Fourier spectrum of the quasiperiodic systems. Analytical results are in full agreement with numerical simulations.