Directional asymmetry of the nonlinear wave phenomena in a three-dimensional granular phononic crystal under gravity

We report the experimental observation of the gravity-induced asymmetry for the nonlinear transformation of acoustic waves in a noncohesive granular phononic crystal. Because of the gravity, the contact precompression increases with depth inducing space variations of not only the linear and nonlinear elastic moduli but also of the acoustic wave dissipation. We show experimentally and explain theoretically that, in contrast to symmetric propagation of linear waves, the amplitude of the nonlinearly self-demodulated wave depends on whether the propagation of the waves is in the direction of the gravity or in the opposite direction. Among the observed nonlinear processes, we report frequency mixing of the two transverse-rotational modes belonging to the optical band of vibrations and propagating with negative phase velocities, which results in the excitation of a longitudinal wave belonging to the acoustic band of vibrations and propagating with positive phase velocity. We show that the measurements of the gravity-induced asymmetry in the nonlinear acoustic phenomena can be used to compare the in-depth distributions of the contact nonlinearity and of acoustic absorption.

Figure 1

Sketch of the experimental setup.

Figure 2

Dispersion relations of the bulk modes in a hcp granular crystal along the $z$ direction. The cyclic frequencies are normalized with the cutoff frequency ${\omega }_L^c$ of the longitudinal mode $L$.

Figure 3

Transmission through an aluminum block with a shear transducer in excitation and either a shear (continuous line) or a longitudinal (dashed line) transducer in detection.

Figure 4

Examples of the amplitude of the spectra of the source signal (black continuous line) and transmitted signal (gray dashed line) with the identification of linear and nonlinear wave propagation in the case of an excitation with a central frequency at 60 kHz. The second harmonic component is here visible at 120 kHz. The maximum amplitude component of the demodulated wave is found around 3 kHz.

Figure 5

Transmitted amplitude of the linear (continuous curves and filled markers) and demodulated (dashed curves and empty markers) waves as a function of the central frequency of the excitation for a 15-mN static loading applied on the contacts at the top of the crystal. Propagation from the bottom to the top in black squares, from the top to the bottom in blue circles. The markers correspond to the experimental data, the curves correspond to a smoothed interpolation of the experimental results. The vertical curves represent the cutoff frequencies of the modes $L$ and $RT$ at the top and the bottom of the crystal.

Figure 6

Difference of the amplitudes of the demodulated waves from the two paths of propagation, from the bottom to the top minus from the top to the bottom, in function of the central frequency of the excitation and for different static loading applied on the top of the crystal. The squares represent $F_L^c$ at the top and the bottom of the crystal. The circles represent $F_{RT}^c$ at the top and the bottom of the crystal.

Figure 7

Dependence of the amplitude of the demodulated wave on the excitation amplitude with a central pump frequency of 120 kHz and a static loading of 20 mN per contact. The central frequency of the pump wave belongs to the $RT$ band.

Figure 8

Dependence of the amplitude of the demodulated wave on the excitation amplitude with a central pump frequency of 60 kHz and a static loading of 20 mN per contact. The central frequency of the pump wave belongs to the $L$ band.

Figure 9

Example of calculation of $\alpha \left(z\right)$, $\varepsilon \left(z\right)$, $e^{-{\int }_0^z\alpha \left(z^{\prime }\right)d{z}^{\prime }}$ (blue and red lines), and $\varepsilon \left(z\right)e^{-2{\int }_0^z\alpha \left(z^{\prime }\right)d{z}^{\prime }}$ (blue and red dashed lines) over $z$ to visualize the evolution of the functions inside Eq. (3) following the two paths of propagation.

Figure 10

$\psi =0$ with $L_{\mathrm{att}}$, $L_{NL}^{\mathrm{inh}}$, and the ratio $L_{\mathrm{att}}^{\mathrm{inh}}/L_{NL}^{\mathrm{inh}}$.