Desert booming can be heard after a natural slumping event or during a sand avalanche generated by humans sliding down the slip face of a large dune. The sound is remarkable because it is composed of one dominant audible frequency (70 to 105 Hz) plus several higher harmonics. This study challenges earlier reports that the dunes’ frequency is a function of average grain size by demonstrating through extensive field measurements that the booming frequency results from a natural waveguide associated with the dune. The booming frequency is fixed by the depth of the surficial layer of dry loose sand that is sandwhiched between two regions of higher compressional body wave velocity. This letter presents measurements of booming frequencies, compressional wave velocties, depth of surficial layer, along with an analytical prediction of the frequency based on constructive interference of propagating waves generated by avalanching along the dune surface.
Introduction
Explorers including Marco Polo [Polo, 1984] in the Gobi Desert, the Emperor Baber [Marquess Curzon of Kedleston, 1923] in Afghanistan, and Charles Darwin [Darwin, 1839] in Chile have been mystified by the booming sounds of the desert. Sustained booming is defined as the continuous, loud droning sound emitted from a large sand dune after inducing a sand avalanche on its leeward face [Criswell et al., 1975; Lindsay et al., 1976]. An avalanche of sand can be initiated naturally when sand exceeds its angle of repose or can be induced by a manmade slide. Booming is a seasonal phenomenon and investigators [Haff, 1986; Lewis, 1936] have noted that moisture in the sand can eliminate the booming sound completely. The booming sound differs fundamentally from the “squeaking” sound on sand beaches at frequencies around 1000 Hz [Humpries, 1966; Nori et al, 1997; Sholtz et al., 1997] and from “burping” sounds when sand is shaken back-and-forth in a jar [Goldsack et al., 1998; Haff, 1979]. The burping sounds consist of short (t < 0.25 s) bursts at frequencies (150-300 Hz) higher than booming sounds and with different spectral characteristics.
An explanation for the booming sound is found in Poynting and Thompson’s [1909] classic 1909 physics textbook, proposing that the frequency is related inversely to the time required to pass between successive collisions of individual grains. Bagnold [1954] provides a similar argument based on the shearing of dilation, and finds that the frequency should vary as (g/D)1/2, where g is the acceleration due to gravity and D is the average particle diameter. More recently, Andreotti [2004], Douady et al. [2006] and Bonneau et al. [2007] support the (g/D)1/2 scaling and argue that the frequency is controlled by the shear rate inside the avalanche. The dependence on granular properties alone suggests that booming should occur on all dunes, in contradiction to observations. The current work presents new experimental evidence that supports an alternative interpretation of the booming based on a resonating waveguide. This waveguide model explains why the booming phenomena only occur in certain locations and at certain times of the year. It also provides an explanation for the continuation of booming for up to a minute when all visible shearing has ceased (auxiliary material Animation SI1).
Method
At Dumont Dune, just south of Death Valley National Park, California, USA, measurements of the booming frequencies were made at two dunes on 11 and 12 September 2006. The elevation above the desert floor was approximately 45 m and 11 m for the large and small dune respectively. Both dunes had a slip face near the crest at an inclination of 30°. To initiate the booming sounds, human sliders descended the steep face at a constant speed of 1.1 m/s, creating a slide in the surrounding sand. Figure 88 shows the free-surface profiles of the large and small Dumont dune.
Figure 90 presents recordings of the sustained booming frequency created during the slide, measured with a microphone at location B and with an array of seismic geophones and positioned downhill from location A. The sound did not start immediately, varied somewhat during the slide and showed one dominant frequency with several higher harmonics (Figure 90a). The largest amplitude measured by the geophone signal was obtained around location B (Figure 90c). The booming sound dimensioned and disappeared as the sliders descended farther down the dune where the surface slope lessened. Visible surface avalanching occurred during the slide on the smaller dune (Figure 90f), but booming could not be initiated resulting in a broad band emission at low magnitude. When the experiment was repeated on the large Dumont Dune in the winter of December 2006 (not shown here), no sustained booming could be initiated, although faint, short squeaks were audible during the slide. These squeaks had a lower frequency (~65 Hz), a shorter duration (~0.2 seconds) and a lower amplitude than the booming emission. The definition of sustained booming sound does not apply here as the acoustic emission is short and not sustained.
Over the course of 5 summers, visits were made to Dumont Dunes and to 3 other booming locations: Big Dune near Beatty, NV; Eureka Dunes in Death Valley National Park, CA; and Kelso Dune in Mojave National Preserve, CA. At each location the dune had a clear slip face below the crest at the angle of repose of the sand. The sustained booming frequency was measured with either a microphone or with a single geophone during an induced avalanche of sand. Booming could never be initiated on faces that were below the angle of repose.
Sand obtained at each location was sieved in the laboratory to determine the average grain diameter and its standard deviation. The average grain diameter ranges from 0.18 to .31 mm. Compared with other sands, dune sand is well sorted with a relatively small standard deviation because of its acolian history. [Humpries, 1966; Lindsay et al., 1976]. The sustained booming frequency is presented in Figure 89 as a function of the average grain size and does not correlate with the particle diameter.
In addition to the booming frequencies, the geophones were used to determine the body wave velocities within the dune using a seismic refraction survey technique. An array of 96 geophones were positioned, beginning 8 m from the crest (location A), with a space of 1 meter. The geophones recorded the wave propagation initiated by the striking of a plate with a sledge hammer, as exemplified in Figure 143, for an impulse at A.
Results
The seismic records are particularly clean as the surface waves, which propagate at the speed of approximately 50m/s, are strongly attenuated. By analyzing the slopes in Figure 143, discrete velocity layering is apparent in the summer recording, while the velocity gradually increases without distinct layers for the same dune in winter. The first-arriving body waves for the large (shots A-E) and the small (shots F-H) dune at Dumont are used to determine the subsurface velocity distribution. The large dune (Figure 91a) has a large lateral velocity gradient and contains a low velocity layer to a depth of 1.5 meters that acts as a waveguide for acoustic energy. On the small dune (Figure 91b) the surficial velocities are similar in magnitude; however, the layering is less apparent and the first refraction velocity, 600 m/s, is higher, presumably because of the limited height of the dune and the relative proximity of the desert floor.
As suggested by Andreotti [2004], the wave velocities can also increase due to hydrostatic pressure within the dune. The standard scaling between velocity and pressure in granular materials states that c ~ P1/6. The relation predicts a 16% increase in velocity at a depth of 10 m for sand with a density of 1500 kg/m3, compared to a 250% increase observed in the data. Hence, the velocity increase is not explained by a simple increase due to hydrostatic pressure. The jumps in velocity cannot be explained by pressure increases and are instead a result of structural differences. These structural changes are due to a local high water content or chemically altered sand. Andreotti only considers low-speed surface waves of around 50 m/s as the speed of the booming sound. By cross-correlating the geophone signals, the phase speed of booming is measured at 200 m/s near the crest of the dune and increasing to 350 m/s further downhill. Hence, booming results from the propagation of body waves not surface waves.
The dune can act as a seismic waveguide [Ewing at al., 1957; Officer, 1958] because of the subsurface layering. The avalanching of the surface layer acts as its moving source of energy. Waves propagating at c1 in the surficial layer are reflected at the atmospheric boundary and the substrate half-space. The surficial layer of thickness H is sandwiched between the higher velocity atmosphere (c0) and substrate half-space (c2). For the frequency fn associated with mode n (where n = 1, 2, 3, …) for which the phase difference between two subsequent descending waves is an integral number of 2π, wavefronts interfere constructively when:
For the special case of incidence at the critical angle ø = øcr the phase changes ∑10 and ∑12, as defined by Officer [1958], are zero. No attenuation occurs in either the atmosphere, or the substrate half-space, resulting in the maximum excitation of the waveguide. For the condition where the velocities c0 and c2 are equal, the amplitude of the booming is at its maximum magnitude as experimentally observed in Figures 90b-90e and Figure 91a. The frequency is computed as,
Since the velocity c0 is larger than the surficial velocity c1, successive wavetrains will reinforce each other resulting in a coupling for the horizontal transmission between the waveguide and the upper medium. In practice, not all waves travel at the critical angle and some loss of energy will occur at the interface. The frequency predicted by equation is compared with experimental results from the July, August, and September 2006 data at Dumont Dunes (Table 1). For the 3 different dates, the agreement between the measurement and the calculated frequency is closest in the upper region of the dune where the maximum amplitude of the booming sound occurs and where the air velocity matches the substrate half-layer as assumed by equation. The booming sound cannot be generated where the velocity of the surficial layer of the dune approaches or exceeds the velocity of the air. The observed harmonics are explained by analyzing higher modes of the resonance at n = 2, 3,….
The effect of the avalanche speed was investigated by comparing two slides produced at different sliding speeds of V ≈ 1 m/s (auxiliary material Animation S3) and V ≈ 2 m/s (auxiliary material Animation S4) in August 2006. The slides occurred on two neighboring sections of the dune approximately 15 meters apart laterally. The frequency of the sustained sound was essentially the same: 83 ± 8 Hz for the slow slide (auxiliary material Audio S1) and 87 ± 5 Hz for the fast slide (auxiliary material Audio S2). The sustained tone and its harmonics are not influenced by the speed of the avalanche. However, the slower slide incorporated a greater surface area involved in the avalanche and the amplitude of the acoustic emission was a factor two higher. Hence, the amplitude of the booming increases with the amount of avalanching sand, as displayed for the large slide in auxiliary material Animation S2.
Conclusion
The avalanching sand acts as a source for the acoustic emission, and the waveguide sets the frequency. Waves interfere constructively and reinforce each other resulting in a loud audible emission. The sand surface interacts with the atmosphere and acts as a loudspeaker by propagating disturbances into the atmosphere. For slopes shallower than 30°, such as on the lower foothill or the windward face, booming could not be initiated. The December experiment on the large dune demonstrates that a continuous velocity distribution, without apparent layer, does not provide the conditions for sustained booming. Seasonal changes in environmental parameters like temperature, precipitation, irradiation, and wind direction contribute to the variations in subsurface velocities and dune features. Moisture that is not evaporated seeps down into the dune, increasing the velocity and eliminating the layering structure. Smaller dunes lack the required subsurface structure and sufficient length to create the waveguide.
Acknowledgments
The authors would like to thank the late Ron Scott, Norman Brooks, George Rossman, and Tom Heaton for their scientific suggestions and Steve Hostler and Gustavo Joseph for their guidance and help. The help of the undergraduate students Natalie Becerra, Patricio Romano-Pringles, Ransom Williams, Nora DeDontney, the late Steve Gao, and many others, was essential during the field experiments at various locations. Travel and equipment support for N. M. V. was provided through funding from the Pieter Langerhuizen Lambertuszoon Fonds.
[1]Auxiliary material data sets are available at ftp://ftp.agu.org/apend/gl/
We show here that the standard physical model used by Vriend et al. to analyse seismograph data, namely a non-dispersive bulk propagation, does not apply to the surface layer of sand dunes. According to several experimental, theoretical, and field results, the only possible propagation of sound waves in a dry sand bed under gravity is through an infinite, yet discrete, number of dispersive surface modes. Besides, we present a series of evidences, most of which have already been published in the literature, that the frequency of booming avalanches is not controlled by any resonance as argued in this article. In particular, plotting the data provided by Vriend et al. as a table, it turns out that they do not present any correlation between the booming frequency and their estimate of the resonant frequency.
ACOUSTICS IN SAND DUNES
It is a well-known fact that sand dunes present a layered structure due to successive avalanches and capillary water retention. We focus here on the acoustic propagation within the metre scale surface layer composed of dry sand. According to Vriend et al. [2007], in [Andreotti 2004; Bonneau et al. 2007], we “only consider low-speed surface waves of around 50 m/s” while “booming results from the propagation of body waves, not surface waves.” This statement reflects a misunderstanding of acoustic waves in granular media. In fact, we have shown that, contrarily to ordinary elastic solids, bulk modes (the ‘body waves’) do not exist at all [Bonneau et al. 2007]. Instead, there is an infinite yet discrete number of surface modes (not a single one) with a dispersive propagation (i.e. not a single wave speed independent of vibration frequencies). This has been directly evidenced in the field and in a lab experiment (Figure 144) and reproduced by an independent team [Jacob et al. 2007].
The explanation is simple. Sand is a divided medium that presents non-linearities of geometrical origin. As the grains do not have plane/plane contact, they behave like a spring whose stiffness depends on the normal force applied to put them into contact. As a consequence, the speed of sound c depends on confining pressure P – roughly as c∝Z1/3P1/6, where Z is the effective number of contacts per grain. A striking consequence is that the order of magnitude of the propagation speed c under a pressure of ~ 1 m of sand is lower than the sound of velocity in air (although density is 103 times larger). Therefore, the effective compressibility is extraordinarily small! This property cannot be explained without involving geometrical effects at the scale of a grain. This dependence on pressure has been proven several times experimentally and numerically for moderately large P (~ 10 – 100 m of sand in Jia et al.). Due to gravity-induced pressure (P∝pgz), the surface layer of a dune does not constitute a homogeneous system as c increases with depth. Thus, no plane wave Fourier mode can exist in such a medium; only an infinite number of surface modes guided by the sound speed gradient may propagate. Consquently, a mode labelled n is localized within a depth n ∧ below the surface (∧ is the wavelength) i.e., in a zone where the typical pressure is P ~ pgn∧. Thus, the surface propagation velocity increases with the mode number n in a similar way as c(P) (between n1/4 and n1/3 typically). So, even in the limit of an infinite depth (no finite-depth layering), gravity produces a wave-guide effect, but no resonance.
Considering now a layer of sand of finite depth H, a second wave-guide effect gets superimposed to the first one and resonant modes may appear. As the system is still not homogeneous due to gravity, these resonant modes are not Fourier modes. By definition, they do not propagate – i.e., they have a vanishing group of velocity – so that they correspond to cut-off frequencies of the system: no wave can propagate at a velocity smaller than the first resonant frequency fR. The influence of the finite depth H is in fact limited to a very narrow range of frequencies. As soon as the depth H is larger than the wavelength (in practice, for a frequency of f 25% above fR), it can be considered as infinite, and the gravity effect prevails [Bonneau et al. 2007].
Then, it is easy to realize that if one strikes such a gradient-index medium with a sledge hammer (basically, the procedure used by Vriend et al.), a series of wave packets corresponding to the different surface modes will be propagated that are related to the gravity induced index gradient and not to the effect of the finite depth H. Moreover, this procedure only gives access to the group velocity at the mean frequency of excitation – yet not controlled nor specified by Vriend et al. – and not at all at the frequency f of spontaneous booming. Consequently, from such a procedure, just by reading the multi-modal structure of a seismograph obtained using only surface transducers, one cannot conclude to the existence of multiple layers nor determine the resonant frequencies. Actually, if Vriend et al. were right, they could easily provide a crucial test by performing measurements with transducers buried in the bulk, as those reported in Andreotti [2004]. The vibration in the soil during booming avalanches is already strongly reduced at 60 mm below the surface, which evidences directly that the song of dunes is not related to a resonant mode at the meter scale but to surface propagation.
FIELD EVIDENCES AGAINST THE SELECTION OF THE FREQUENCY BY A RESONANCE
Vriend et al. have missed another very important step of the argument presented in Andreotti [2004]. The booming frequency f is controlled by the shear rate y (i.e. by the rate at which the grains collide with each other) in the shear zone separating the avalanche from the static part of the dune. Of course, y depends on the way the avalanche is forced. When a granular flow is driven by a pressure gradient, e.g. by pushing with the hand or the bottom, y can be varied and so does the emission frequency f. Besides, the reproduction of the phenomenon at small scale [Haff 1986, Douady et al. 2006; Bonneau et al. 2007] is a direct proof that the acoustic emission is not related to the dune itself. Note in particular that controlled lab experiments have allowed to produce sustained booming (not short squeaks), varying continuously from gravity to pressure gradient driving.
Spontaneous avalanches on the slip face of dunes are driven by gravity in a homogeneous and steady way. In that case, only, y and thus f are expected to scale as √g/d, with subdominant dependencies on the nature of the grains and the presence of cohesion. These conditions can be reproduced by a man-made slide only if the velocity of the body is constant. This is never the case in the movies provided by Vriend et al. to illustrate their measurements. In the auxiliary animation, we show that a pulsed driving leads to a low frequency squeak similar to that measured by Vriend et al. (e.g. point at 70 Hz in their Figure 92). It is pretty clear that such an inhomogeneous and unsteady driving does not resemble the spontaneous avalanching process and thus the avalanche has no reason to yield a constant booming frequency f. By contrast, during our 13 field trips in Morocco from 2002 up to now, we have recorded around 20 spontaneous avalanches and 100 man-triggered slides on barkhan dunes ranging from 4 to 42 m in height. This was done in different places and by different weather conditions. We always measured the same frequency f within a tone. In particular, during our field trip of April 2007, we have performed a series of experiments for different flow thicknesses H, measured coherently by two independent techniques. Again, provided the avalanches were homogeneous and steady, the frequency f turned out to be constant. This demonstrates that the frequency is not either selected by a resonance effect over the depth of the avalanching flow.
Figure 92 compares our predictions for the frequency f of homogeneous booming avalanches (f ~ 0.4√g/d) and that proposed by the Caltech team (f = fR). Once the data obtained by Vriend et al. in Dumont are plotted (Figure 92b) – and not anymore presented in a Table – no correlation can be observed between f and what they claim to be the resonant frequency fR. By contrast, with a choice of representation different from Vriend et al., the relation between f and √g/d appears beyond any doubt. This accumulation of evidences shows that the emission frequency f is neither controlled by a resonance effect at the scale of the avalanche as suggested by Douady et al. [2006] nor at the scale of a superficial layer as suggested by Vriend et al [2007].
The instability mechanism proposed in Andreotti [2004] is based on the interaction between the plastic deformations (shear band) and the elastic compression (acoustics) of the granular material. The collisions between grains induce a transfer of momentum from the translation to vibration modes. In turn, the surface waves tend to synchronize to collisions. Importantly, this phase synchronization linear instability does not depend on the fact that the vibration modes are propagating or resonant. Although some ‘mysteries’ about singing dunes still remain to be solved, our explanation remains the single one consistent with all the existing observations.
Figure 144: Field experiment: propagation of a synthetic gaussian wave-packet at 350 Hz at the surface of a 50 cm deep flat booming sand bed (proto-dune on a limestone plateau) i.e., without any layering effect. (a) Signal envelope (t) measured by correlation with the emitted signal at 5 cm (dotted line), 150 cm (solid line) from the source.
(b) Comparison with the model proposed Bonneau et al. 2007. The three wave-packets received are direct evidences of the existence of multiple propagative surface modes.
Table 02: Data points for which we do know that (i) avalanches were spontaneous or at least homogeneous and steady (ii) the grain diameter was determined from samples taken in the middle of the slip face, at the place where the avalanche was recorded. When possible, the frequency was averaged over several realisations.
Table 03: Average frequency f of the data points measured from Vriend et al. No description of the conditions of experiment has been provided by these authors. The six movies they provided show the same technique to create avalanches, which are pulsed and inhomogeneous. No description of the place where the grains were sampled has been provided by these authors. The data obtained in Dumont on December 2005 have not been taken into account for several reasons: no recording is provided, no movie is provided, the width of the frequency peak is extremely large (40% of the central frequency) which is the signature of a small squeak induced by an unsteady avalanche (see our Auxiliary movie). So it cannot be used in a plot aiming to show data obtained in homogeneous and steady conditions.
Figure 145: (a) The Makhnovist drum experiment: response of the booming dune to a normal tap containing constituting a broadband excitation. After the propagative modes excited have left, the resonant mode i.e. which does not propagate stays. The tail following the tap contains a well dened frequency that can be interpreted as the rst compression resonant mode. (b) Auto-correlation function of the signal shown in (a). The resonant frequency is around 73 Hz for these particular conditions of eld measurements. The wet sand layer is at a depth around H = 50 cm below the surface. [after Bonneau et al. 2006] (c) Amplitude of vibration at 5 m from the source as a function of the frequency f, for permanent sinusoidal signals emitted at the surface of a 4m high booming dune. The wet sand layer is at a depth H = 50 cm below the surface.