Quasiperiodic Bloch-Like States in a Surface-Wave Experiment

Bloch-like surface waves associated with a quasiperiodic structure are observed in a classic wave propagation experiment which consists of pulse propagation with a shallow fluid covering a quasiperiodically drilled bottom. We show that a transversal pulse propagates as a plane wave with quasiperiodic modulation, displaying the characteristic undulatory propagation in this quasiperiodic system and reinforcing the idea that analogous concepts to Bloch functions can be applied to quasicrystals under certain circumstances.

Figure 1

Snapshot of the system vibrating vertically at a frequency of $35\mathrm{H}\mathrm{z}$, under conditions of continuous wave excitation. The inset shows its Fourier spectrum with a well defined and discrete set of relevant components.

Figure 2

The solid line is the Fourier transform of the impulsive signal as picked up by an accelerometer at the center of the vessel. It shows clear resonances at approximately 20, 30, and $50\mathrm{H}\mathrm{z}$. Such resonances are specific of the system. The first peak at very low frequency, close to the origin, corresponds to the Fourier transform of the square pulse that excites the system. The signal in the time domain is presented in the inset. The dashed line is the gray scale of the subpattern along the $\Gamma \mathrm{\text{−}}X$ direction as represented in Fig. 4 versus frequency. The gray scale (between $0$ and $1$) was rescaled to match the peak of the Fourier transform at 30 Hz. Three standing waves indicating narrow band gaps appear at approximately 20, 32, and 50 Hz.

Figure 3

Temporal sequence of patterns observed when the system is disturbed with a transverse pulse. (a) Quasicrystalline pattern observed at the start of each pulse. (b) A transitory weak turbulence is observed after $0.04\mathrm{s}$. (c) Standing waves draw clear quasiperiodic grids between $0.08$ and $0.24\mathrm{s}$.

Figure 4

Experimental (a) and theoretical (b) Fourier transform of the pattern shown in Fig. 3c. (c) Inverse Fourier transform of a subpattern along the $\Gamma \mathrm{\text{−}}X$ is shown. The Octonacci sequence is clearly recovered. (d) Inverse Fourier transform of pattern shown in Fig. 3b. The structure of the octagonal tiling underlying on the well quasiperiodic arrangement of the vessel bottom is recovered.