The Song of Dunes as a Wave-Particle Mode Locking

Singing dunes, which emit a loud sound as they avalanche, constitute a striking and poorly understood natural phenomenon. We show that, on the one hand, avalanches excite elastic waves at the surface of the dune, whose vibration produces the coherent acoustic emission in the air. The amplitude of the sound ($\simeq 105\mathrm{d}\mathrm{B}$) saturates exactly when the vibration makes the grains take off the flowing layer. On the other hand, we show that the sound frequency ($\simeq 100\mathrm{H}\mathrm{z}$) is controlled by the shear rate inside the sand avalanche, which for granular matter is equivalent to the mean rate at which grains make collisions. This proves the existence of a feedback of elastic waves on particle motion, leading to a partial synchronization of the avalanching sand grains. It suggests that the song of dunes results from a wave-particle mode locking.

Figure 1

(a) Measurement setup. (b) Samples of the sound (dotted line) and surface acceleration (solid line) signals, just before the avalanche front reaches the transducers (left) and just after (right). The dashed line is the same acceleration signal from which the Fourier components smaller than $500\mathrm{H}\mathrm{z}$ have been removed. It evidences the high frequency noise induced by avalanching grains (right), not present before the avalanche (left). The sound may be listened to at Ref. 1.

Figure 2

Root mean square amplitude of the pressure (dotted line) and acceleration (solid line) signals, measured on a $1\mathrm{s}$ moving window. The avalanche propagates at approximately $0.5\mathrm{m}{\mathrm{s}}^{-1}$ and reaches the transducers at $t=0$, as can be observed on the noise amplitude (dashed line, see Fig. 1). The scales have been chosen in order to make the curves collapse if the surface behaves as a loud speaker [Eq. (1)]. When the accelerometer is fixed in approximately the same place (right), it is engulfed inside the avalanche so that the signal gets reduced, compared to the measurements at the surface (left).

Figure 3

Dispersion relation of surface elastic waves, the sand being at rest [transducers distant by $5\mathrm{c}\mathrm{m}$ (down triangle), $15\mathrm{c}\mathrm{m}$ (up triangle), $25\mathrm{c}\mathrm{m}$ (square), and $42\mathrm{c}\mathrm{m}$ (circle)] or avalanching (star). The dotted line corresponds to a nondispersive relation $f=c{\lambda }^{-1}$, with $c=40\mathrm{m}{\mathrm{s}}^{-1}$ and the solid line to the Rayleigh-Hertz relation $f=c_0^{2/3}{g}^{1/6}{\lambda }^{-5/6}$, with $c_0=230\mathrm{m}{\mathrm{s}}^{-1}$.

Figure 4

Average velocity profile in a laboratory avalanche for different flow heights: $H=8d$ (triangle), $H=10d$ (square), $H=15d$ (circle). The solid line corresponds to the velocity profile $U(z)=fz$ with the sound frequency $f=100\mathrm{H}\mathrm{z}$. Inset: Autocorrelation of the microphone signal measured in the core of a steady and homogeneous avalanche (solid line) and a very intermittent one (dotted line).

Figure 5

Dynamics of a single grain rolling down an inclined array of grains of same diameter that oscillates at the frequency $f$ with an amplitude $ϵ$. $\Gamma$ is the frequency of collisions at $f=0$. The diagram shows the region of mode-locking, i.e., the region of parameters space—the rescaled amplitude $ϵ/d$ and the rescaled frequency $f/\Gamma$—where the grain collisions synchronize on the oscillation. The star indicates the parameters selected in booming avalanches ($ϵ\simeq 0.24d$ and $f=\Gamma$).