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| Fig. 1: General experimental mechanism. A slider sits at the top of the dune and slides down, creating a man-made avalanche. Geophones at different points along the dune measure the acoustic emissions. |
Since Marco Polo's time, desert travelers have heard noises they described as cannons, aircraft, trumpets, bells, and other sounds, yet invariably looked around to find absolutely nothing [1]. For hundreds of years, the mystery reigned. Finally, scientists say they know the answer; it lies in the shifting sands of the dunes.
The various noises the desert travelers heard correspond to what researchers now classify into three distinct categories: singing, burping, and booming sounds [2]. Each of these categories corresponds to increasingly lower frequencies. Singing sand is the most common, and occurs on beaches. The mechanism for these sounds are the most well understood - when the grains are mechanically sheared, they produce a high (greater than 500 Hz) frequency sound. Burping sounds are produced when sand is shaken back and forth in a jar. They are described as short bursts of lower (150-300 Hz) frequency sound. Booming sands can be heard during an avalanche, either natural or man-made, and correspond to low (70-105 Hz) frequency sound. These sounds are also extremely loud to the point of deafening, and the vibrations they produce are so intense that it is impossible to stand on the ground on which the sound is occurring. Due perhaps to their relative rarity, and perhaps to their mechanistic complexity, booming sands have been historically the least understood.
Most field experiments are conducted in the same way. Because the sound has been correlated to sand avalanching down the top of the dune, experimentalists go up to the top of a dune and simply slide down, inducing an avalanche. The avalanche is then recorded on geophones at different places along the dune (see Figure 1).
The first known explanation for booming was provided by Poynting and Thomson in 1909: their proposed model was
where f is the frequency and t is the time needed between successive grain collisions [3]. A later explanation, in 1954, was given by Bagnold, who said that based on the shear rate inside the avalanche,
where g represents the acceleration due to gravity and D is the average particle diameter [4]. Average grain diameter is a fairly precise standard for any specific batch of dune sand, because compared to other sand is it sorted with a small standard deviation.
Recently, three different research groups have argued both for and against this claim, each producing a different explanation for what causes the sound of the booming sand dunes.
Music is produced through a coupling of excitation and resonance in instruments. For example, the bow of a violin sticks against the strings, creating an excitation. The length of the string then specifies a particular resonance, and therefore frequency, of the sound that is produced. Another example is the vibration of a reed on a clarinet (or the vibration of lips on a brass instrument), which creates an excitation in the air that is then resonated along the length of the instrument. Sand dunes do not behave in either of these fashions. First, the sand grains do not stick to each other to create an excitation - on the contrary, researchers have found that humidity will prevent the booming sounds, and dry grains moving freely produce the sound. Second, there is no resonance inside the dune, both because the same dunes have been known to produce different frequencies, and because dunes of different sizes produce the same frequencies.
Another clue is that wind is not the cause of the sound, because the same sound can be reproduced by pushing the dune sand together using arms or legs. Douady et. al actually tested this in a laboratory setting, building a laboratory device that contains a rotating blade through some sand grains. They obtained different constant and well-defined frequencies, showing that the booming sound does not require the dune itself as a resonator, the differing continuous range of frequencies obtained proving that it is not a resonance within the experimental device. Rajchenbach found that in an avalanche, the grains must hit lower grains as they travel [6]. This means that instead of accelerating grains, the dunes have grains of constant velocity, because each collision imparts some force onto the next grain [7]. Putting this together with their experimental evidence, they found that neither velocity nor mass but only the mean shear applied to the grains controlled the frequency.
Their findings show that there is a coupling between each successive layer of grains, which propagates down the entire depth of the avalanche with a characteristic velocity. The frequency of sound produced is thus due to the collision rate of grains in the "shear" layer of the surface. This model has the grains bumping into each other, making a standing wave that in turn synchronizes the grains. This means the frequency is imposed by the granular motion rather than by a resonance, implying that the sound dunes are a completely new kind of musical instrument.
Andreotti found that the surface of the sound bed acts like a loudspeaker membrane and causes the acoustic emission [8]. The sound is only produced when layers of sand above a certain thickness slide over each other. The amplitude saturates when the surficial grains begin to move. The power P, emitted per unit surface area in the air was measured to be
where f is the frequency, and ρair and cair are the density of air and sound velocity in air, respectively. Andreotti found that the amplitude of the vibrations decreases greatly with depth so that the elastic waves generally appear close to the surface. Putting this all together, he concludes that the frequency of the sound emission is dependent on the collision rate inside the avalanche so that the collisions excite an elastic wave. That is, while Douady et al. argue that a sound arises from a resonance within the shear layer, Andreotti claims that the collisions between grains on the shear layer excite elastic waves outside the shear layer. Once the sand layer is oscillating with a frequency f, it induces another force of frequency f in a feedback mechanism similar to that of a phase-locked loop.
Andreotti concludes that the main acoustic emission of the booming avalanches is begun by coherent elastic waves, near the surface, elliptically polarized and with an ampliude of a quarter of a grain diameter. It starts out as an incoherent mechanical vibration, and in the dune becomes coherent acoustic radiation, albeit with low output and poor coherence.
Yet the controversy does not end here. Both of the previously mentioned groups accept that Bagnold's theory on the shear rate is correct [9]. Yet in a recent paper, Vriend et al. claim that the well-accepted theory that the frequency of the sound grains as a function of the average grain diameter is not accurate. A unique property of dunes is that they have subsurface layering so that they can act as seismic waveguides - the surficial layer sandwiched between two regions of higher compressional wave velocity: the atmosphere, and the substrate half-space. By taking measurements at existing dunes, the group found that for a frequency fn, if the phase difference between two consecutive descending waves is an integral multiple of 2π, there will be constructive interference when:
for a critical angle φ and phase changes ε. Because the surficial velocity is less than the velocities of the surrounding regions, each additional wavetrain will reinforce the next one , creating a coupling for horizontal transmission between the wave guide and the air. This means that the booming frequency is fixed due to the depth of the surficial layer, and not due to the diameter of the individual sound grains.
The mystery of the booming sand dunes is not yet solved, but with at least three groups currently working on it, the next decade promises to bring many answers to crack the code of the dunes.
© 2007 Limor S. Spector. The author grants permission to copy, distribute and display this work in unaltered form, with attribution to the author, for noncommercial purposes only. All other rights, including commercial rights, are reserved to the author.
[1] Marco Polo, in The Description of the World ed. A.C. Moule and Paul Pelliot (G. Routlege, London, 1938), Chap. LVII.
[2] J. F. Lindsay et al., Geog. Soc. of Am. Bull., 87 463 (1971).
[3] J. H. Poynting and J.J. Thomson, Textbook of Physics: Sound, (Charles Griffin, 1909).
[4] R. A. Bagnold, The Physics of Blown Sand and Desert Dunes (Methuen, 1954).
[5] S. Douady et al., Phys. Rev. Lett. 97, 018002 (2006).
[6] J. Rajchenbach, Phys. Rev. Lett. 90, 144302 (2003).
[7] L. Quartier et al., Phys. Rev. E 62, 8299 (2000).
[8] B. Andreotti, Phys. Rev. Lett. 93, 238001 (2004).
[9] Vriend et al., Geophys. Res. Lett. 34 L16306 (2007).