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

We have studied acoustic-wave propagation in one-dimensional systems with quasiperiodic, incommensurate, and random modulation. In the short-wavelength limit we found that if the initial pulse is narrow (with spatial extension of the order of a few lattice spacings), the pulse is localized in a quasiperiodic system as in the case of a random system. This result shows that at short length scales a quasiperiodic system is similar to a random system. Therefore, those transport properties that do not depend on long-range order should behave similarly in quasicrystals and in an amorphous metallic glass, in agreement with experimental findings. On the other hand, if the injected wave has a wavelength much larger than the lattice spacing, we found that there are resonances for quasiperiodic and incommensurate systems when the wave vector satisfies the Bragg conditions. In this case 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. We solved the problem analytically in the long-wavelength limit in terms of two-mode-coupling theory and the Fourier spectrum of the quasiperiodic system. Analytical results are in full agreement with numerical simulations. These indicate that on a long-range scale quasiperiodic order is important not only for the crystal structure but also for those physical properties which depend on long-range order, such as acoustic-wave and microwave propagation