Extremal Transmission and Beating Effect of Acoustic Waves in Two-Dimensional Sonic Crystals

The extremal transmission of the acoustic wave near the Dirac point in two-dimensional (2D) sonic crystals, being inversely proportional to the thickness of sample, has been demonstrated experimentally for the first time. Some unusual beating effects have been observed experimentally, when the acoustic pulse transports through the 2D sonic crystal slabs. Such phenomena are completely different from the oscillations of the wave in a slab or cavity originating from the interface reflection or the Fabry-Perot effect. They can be regarded as an acoustic analogue effect to Zitterbewegung of the relativistic electron. The physical origination for the phenomenon has been analyzed.

Figure 1

(a) Calculated photonic band structure of acoustic wave for a triangular lattice of steel cylinder with $R/a=1/3$ in a water background. (b) The measured transmission coefficient of acoustic wave along the $\Gamma K$ direction for a 2D sonic crystal corresponding to (a). The solid line corresponds to the result with the sample thickness $L=10a$ and the dashed line to that with $L=20a$.

Figure 2

(a) Schematic picture depicting measure processes. (b) The product of $T$ and $L$ as a function of sample thickness at different frequencies. Dark circle dots (experimental data) and the red line (numerical results) correspond to the case at $f=0.55\mathrm{MHz}$. The dark square and triangular dots correspond to the experimental results at $f=0.535$ and 0.45 MHz, respectively. The green lines (solid, dashed, and dotted) are drawn only for view. The green solid line can be regarded as corresponding to the theoretical result $T\propto 1/L$. The other parameters are identical to those in Fig. 1.

Figure 3

The field energy distribution at (a) $f=0.45\mathrm{MHz}$ and (b) $f=0.55\mathrm{MHz}$ along the $\Gamma K$ direction and (c) at $f=0.55\mathrm{MHz}$ along the $\Gamma M$ direction. The structure and parameter are the same as Fig. 2.

Figure 4

Experimental data of the transmission after the injection of a Gaussian pulse with center frequency $f=0.55\mathrm{MHz}$ into the samples. Dark, red, and green correspond to the cases with $L=14a$, $19a$, and $24a$, respectively. (a), (b), and (c) correspond to the cases with the filter frequency widths $\Delta f=0.01$, 0.02, and 0.03 MHz, respectively.

Figure 5

(a) The phase of transmission wave as a function of frequency for the sample with $L=24a$. The solid line and dark circle dots correspond to the numerical results and experimental data, respectively. (b) Comparison between the theoretical result (solid line) for velocity [$\overline{v}(t)$] of pulse evolution and experimental measurements (circle, triangular dots) for the rate of change of the energy flux $j(t)$. Triangular and circle dots correspond to the cases with $L=14a$ and $24a$ for the filter frequency widths $\Delta f=0.02\mathrm{MHz}$, respectively. Here the dashed line (red) and the dotted line (green) are the corresponding numerical results for two kinds of cases, respectively.