Frequency-dependent surface wave suppression at the Dirac point of an acoustic graphene analog

Dirac points in the band structure of acoustic systems are essential features affording classical analogs of quantum condensed matter states. We show that measured dispersion curves near and at the Dirac point of an acoustic graphene analog can be suppressed by strong variations in the impedance boundary between free field and surface wave regimes under certain conditions. Increased Rayleigh scattering and diffractive excitation are shown to increase the dispersed surface wave pressure amplitude, circumventing the impedance-based wave suppression. The improved excitation and scattering conditions for observing acoustic Dirac points for two samples with two distinct operational frequency ranges are reported.

Figure 1

Sample, experimental setup, and excitation details. (a) Photograph of the 3D printed hexagonal lattice. The lattice unit cell is indicated by the dotted yellow lines and the lattice vectors $a_1$ and $a_2$. The direction of microphone displacement used for the line scan measurement technique is indicated by the unit vector $\widehat{x}$ and the dashed yellow arrow. The dark, circular regions are the half-open cylindrical cavities. (b) Photograph of the sample with a cavity resonance frequency $f_r$ = 20.9 kHz. The measurement microphone is suspended vertically above the sample. The source, configured for the grazing incidence excitation method, is shown behind the sample. (c) Photograph of the 3D printed cone (white) coupled to the source cavity used for the diffractively coupled excitation method as depicted in the cross-sectional image shown in (d). (d) Cross-sectional schematic of the diffractively coupled excitation method. Here sample dimensions are provided for the sample with $f_r$ = 20.9 kHz. The source cavity ($L$ = 7.06 mm) is open on both ends and has a resonance frequency ($f_r$ = 20.9 kHz) equal to the resonance frequency of the half-open cylindrical cavities ($L$ = 2.83 mm) comprising the hexagonal lattice.

Figure 2

FEM-predicted band structures and sound lines for the two samples with cavity resonance frequencies $f_r$ = 20.9 kHz (solid black line) and $f_r$ = 12.4 kHz (dotted blue line), which correspond to Dirac frequencies $f_D$ = 22.8 kHz and $f_D$ = 13.0 kHz (dot-dashed purple lines), respectively. The regions of each band structure highlighted in magenta are the regions targeted by our experiments; these highlighted predications are shown as magenta dots in Figs. 3, 5, 8 for comparison to the experimental data. The dashed green lines are the predicted sound lines (${\mathrm{SL}}_P$) using $c$ = 342 m/s as the air wave speed [note that the measured sound lines (${\mathrm{SL}}_M$) are shown as solid green lines in Figs. 3, 5, 8]. Inset: The red arrows indicate the path around the irreducible Brillouin zone taken by sweeping the values of wave-number pairs $k_x, k_y$ during the band-structure eigenfrequency computation.

Figure 3

Frequency-dependent surface wave suppression near and at the Dirac point. All dispersion curves (grayscale) are obtained via the line scan measurement technique. The green lines represent the measured sound line (${\mathrm{SL}}_M$) and the magenta dots are the FEM predicted band structure along the $\widehat{x}$ direction (a) Cavity resonance $f_r$ = 20.9 kHz excited with the grazing incidence excitation method. (b) Cavity resonance $f_r$ = 20.9 kHz excited with the diffractively coupled excitation method. (c) Cavity resonance $f_r$ = 12.4 kHz excited with the grazing incidence excitation method. (d) Cavity resonance $f_r$ = 12.4 kHz excited with the diffractively coupled excitation method.

Figure 4

FEM predictions of the real part of the normalized specific acoustic impedance $Z_N=Z/Z_0$ (colorscale) in and above unit cell cavities. (a) Impedance prediction for cavity resonance $f_r$ = 20.9 kHz at $f$ = 20.2 kHz. (b) Impedance prediction for cavity resonance $f_r$ = 20.9 kHz at the Dirac frequency $f_D$ = 22.8 kHz. (c) Impedance prediction for cavity resonance $f_r$ = 12.4 kHz at $f$ = 12.1 kHz. (d) Impedance prediction for cavity resonance $f_r$ = 12.4 kHz at the Dirac frequency $f_D$ = 13.0 kHz.

Figure 5

Comparison of the line scan and phase delay measurement techniques and higher Brillouin zone FEM predictions. The diffractively coupled excitation method is used in each case. Dispersion curves obtained with a line scan are shown in grayscale, and these are the same datasets shown in Figs. 3 and 3. Dispersion curves obtained with the two point phase delay measurement technique are indicated by the dotted orange line. The green lines represent the measured sound line (${\mathrm{SL}}_M$). The magenta dots are the FEM predicted band structure along the $\widehat{x}$ direction ($\theta =0^{\circ }$), and these are the same predictions shown in Figs. 3 and 3. The yellow squares are the FEM predicted band structure for $\theta =\pm {60}^{\circ }$ with respect to $\widehat{x}$. (a) Dispersion curves obtained for the $f_r$ = 20.9 kHz sample. A phase shift of $-6\pi$ is required for the phase delay measurement. (b) Dispersion curves obtained for the $f_r$ = 12.4 kHz sample. A phase shift of $-4\pi$ is required for the phase delay measurement.

Figure 6

Measured spectral content of Gaussian first derivative waveforms used to excite the ASWs. (a) Waveform centered at $f_c$ = 24 kHz. Inset: Amplitude normalized time series of the waveform where $P_N=P/P_{\max }$. (b) Waveform centered at $f_c$ = 16 kHz. Inset: Measured amplitude normalized time series of the waveform where $P_N=P/P_{\max }$.

Figure 7

FEM predictions of the imaginary part of the normalized specific acoustic impedance $Z_N=Z/Z_0$ (colorscale) in and above unit cell cavities. (a) Impedance prediction for cavity resonance $f_r$ = 20.9 kHz at $f$ = 20.2 kHz. (b) Impedance prediction for cavity resonance $f_r$ = 20.9 kHz at the Dirac frequency $f_D$ = 22.8 kHz. (c) Impedance prediction for cavity resonance $f_r$ = 12.4 kHz at $f$ = 12.1 kHz. (d) Impedance prediction for cavity resonance $f_r$ = 12.4 kHz at the Dirac frequency $f_D$ = 13.0 kHz.

Figure 8

Line scan with the diffractively coupled excitation method versus phase delay measurement with grazing incidence excitation method. The line scan dispersion curves are shown in grayscale, the phase delay dispersion curves are shown with a cyan dotted line, the green lines represent the measured sound line, the magenta dots are the FEM predicted band structure along the $\widehat{x}$ direction ($\theta =0^{\circ }$). (a) Dispersion curves obtained for the $f_r$ = 20.9 kHz sample. No phase shift is required for this particular phase delay measurement. (b) Dispersion curves obtained for the $f_r$ = 12.4 kHz sample. A phase shift of $-2\pi$ is required for the phase delay measurement.