Sound-producing sand grains constitute one of nature's most
puzzling and least understood physical phenomena. They occur
naturally in two distinct types: booming and squeaking sands.
Although both varieties of sand produce unexpectedly pure acoustic
emissions when sheared, they differ in their frequency range and
duration of emission, as well as the environment in which they tend
to be found. Large-scale slumping events on dry booming dunes can
produce acoustic emissions that can be heard up to 10 km away and
which resemble hums, moans, drums, thunder, foghorns or the drone of
low-flying propeller aircraft. These analogies emphasize the
uniqueness of the phenomenon and the clarity of the produced sound.
Although reports of these sands have existed in the literature for
over one thousand years, a satisfactory explanation for either type
of acoustic emission is still unavailable.
There exist two distinct types of sand that are known to produce manifest acoustic emissions when sheared. The more common of the two, known colloquially as "squeaking" or "whistling" sand, produces a short (< 1/4 sec), high-frequency (500-2500 Hz) "squeak" when sheared or compressed. It is fairly common in occurrence, and can be found at numerous beaches, lake shores and riverbeds around the world. The other, rarer type of sound-producing sand occurs principally in large, isolated dunes deep in the desert (Nori et al.,1996; Criswell et al., 1975). The loud, low-frequency (typically 50-300 Hz) acoustic output of this "booming" sand, resultant upon avalanching, has been the subject of desert folklore and legend for centuries. Marco Polo (1295) wrote of evil desert spirits which "at times fill the air with the sounds of all kinds of musical instruments, and also of drums and the clash of arms." References can be found dating as far back as the Arabian Nights (Carus-Wilson, 1915), and as recently as the science fiction classic Dune (Herbert, 1984). Charles Darwin (1889) also makes mention of it in his classic Voyages of the Beagle . At least 31 desert and back-beach booming dunes have been located in North and South America, Africa, Asia, the Arabian Peninsula and the Hawaiian Islands (Lindsay et al., 1976; Miwa and Okazaki, 1995). Sharply contrasting differences between squeaking and booming sands have led to a consensus that although both types of sand produce manifest acoustic emissions, their respective sounding mechanisms must be substantially different. More recent laboratory production of "squeaks" in booming sand (Haff, 1979) has nonetheless suggested a closer connection between the two mechanisms. A satisfactory explanation for either type of acoustic activity is still unavailable.
This brief review is not a closed chapter in a well-understood
research area, but is a summary of the incomplete and unsatisfactory
proposals put forward to explain sound production in squeaking and,
especially, booming sand. Our review points out the serious problems,
unsatisfactory theories, and the severe limitations of different
approaches published so far.
Booming and squeaking sands each show a markedly different response to water exposure. Booming occurs best when the grains are very dry, preferably several weeks after the last rain. Small amounts of atmospheric humidity, which creates a fluid surface coating on the grains, effectively preclude booming emissions in these desert sands (Lewis, 1936). Even mixing as little as five drops of water into a 1-liter bag full of booming sand can silence the acoustic emissions (Haff, 1979). Similarly, squeaking sand that is visibly moist is not acoustically active either. However, sound is most easily produced from squeaking sand immediately after the grains have been "washed" in water and subsequently dried. It is not clear whether this is due to the washing away of fine impurities in the sample (Brown et al., 1961) or to the creation of a looser, more natural grain packing (Clarke, 1973), although it may explain why squeaking sand typically does not extend inland more than 30 m from the shore (Richardson, 1919). This process of cleaning can also "revive" squeaking sand that has lost its ability to squeak, a condition that often occurs after repeated compression (Hashimoto, 1951). Finally, squeaking sand can emit sound even when completely submerged in water (Lindsay et al., 1976; Brown et al., 1961), suggesting that intergranular cohesion in moist sand precludes acoustic output.
The mean grain size (diameter) of most sand, whether or not it is acoustically active, is roughly 300 µm. The frequency of emission generated by squeaking sand is thought to vary as the inverse square root of the mean grain size (Bagnold, 1954a, 1966), although mean grain size does not by itself determine the ability of sand to sound (Lindsay et al., 1976). It is unlikely that booming frequencies depend similarly on grain size alone, as a fairly wide range of fundamental frequencies is often generated in large-scale slumping events. Particle-size distributions (Folk and Ward, 1957) of sound-producing grains usually only extend over a narrow range, a condition that is called "being well-sorted." Also, booming and squeaking grains tend to be both spherical and lack abrupt surface asperities. The latter condition is called "being well-rounded" (Powers, 1953), a term not to be confused here by our use of the word "polished," which we take to mean granular surfaces which are smooth on the 1-mm length scale. Both types of sand further exhibit an unusually high shear strength, and in the case of squeaking sand, a decrease in shear strength has been shown to correspond to a decrease in sounding ability (Hashimoto, 1954; Humphries, 1966). Experiments in which spherical glass beads produce acoustic emissions similar to those of genuine squeaking sand when compressed, albeit under somewhat contrived experimental conditions (Brown et al., 1965), provide further support for the notion that squeaking is caused primarily by frictional effects. The combination of these four grain properties (a high degree of: size-sorting, sphericity, roundedness and resistance to shear) is thought to be critical for the onset of squeaking. Booming, on the other hand, is substantially less sensitive to differences in grain shape and sorting, and is likely governed by the unusually smooth and polished surface texture present in all booming grains (Lindsay et al., 1976).
Exactly what governs either sounding mechanism is still an open
question. Research has been hindered both by the rarity of the
phenomena and the difficulty in reproducing the sounds in a
laboratory environment. Moreover, the increasing traffic of vehicles
on dunes appears to suppress the natural sounds of sands.
Furthermore, many researchers had problems differentiating between
booming and squeaking sands, and the early literature on the topic is
often marred by inconsistencies as well as vague and imprecise terms
such as "musical" or "barking" sands. One of the first scientific
treatments of the subject attributed the squeaking sound to periodic
oscillations of air pockets located between the grains (Bolton,
1889a). Also, a preliminary explanation of booming relied on observed
electrical charging of the grains (Lewis, 1936). Such air-cushion and
electrical charge models were, however, based on misguided evidence
and have been discredited by more recent experiments. Most modern
theories stress the importance of intergranular friction. It has been
suggested that the unusually smooth and polished surfaces of booming
grains may allow for exaggerated vibration at the natural resonant
frequency of sand (Criswell et al., 1975; Lindsay et al., 1976).
Differences in grain packing have also been considered (Humphries,
1966). The most complete development is given by Bagnold (1954a,
1966), who argues that both types of emission result from non-linear
oscillations of dispersive stress in the layer of moving grains, or
shear layer. Even this analysis, however, fails to address key
aspects of the booming mechanism.
Booming emissions produce a wide variety of sounds that have been
compared to moans, hums, roars, drums, tambourines, thunder, cannon
fire, the rumble of distant carts, foghorns, the buzzing of telegraph
wires and the drone of low-flying propeller aircraft (Lindsay et al.,
1976; Curzon, 1923). These analogies all emphasize the distinct
quality of the booming sand phenomenon and the clarity with which
these sounds are produced. The comparisons to drums and tambourines
moreover illustrate how the booming mechanism can produce
characteristic beat frequencies (1 to 10 Hz), thought to result from
periodic amplitude modulation of the acoustic emissions which are
often observed in prolonged, large-scale flows. Perhaps most
illuminating, however, are those analogies which compare booming
emissions to musical instruments, such as trumpets, bells or
low-stringed instruments. Such clear emissions usually occur only in
small-scale flows, whereby only one fundamental frequency of
vibration is produced. Criswell et al. (1975) point out the
similarities in the acoustic amplitude trace of a small-scale, in
situ booming emission and that of a pipe organ. It is remarkable
that an avalanche of granular material could produce an acoustic
oscillation comparable in purity to an finely-crafted musical
instrument. Our 1994 observations of small, induced avalanches at
Sand Mountain, Nevada, during the second driest summer on record,
reveal emissions similar to a didjeridoo (an aboriginal instrument
from Australia) with its low, droning cadence.
In general, several fundamental frequencies of vibration are often
present, especially when large volumes of shearing sand are involved.
Exactly what precipitates the transition from one to several modes of
fundamental vibration is not completely understood. Each fundamental
frequency seems to exhibit its own rise and fall time, independent of
the others, and they are thought to result from the collective
vibration of sand grains which can occur along all directions
(Criswell et al., 1975). Taken together, these frequencies can cover
a fairly broad range, the width of which is determined by a variety
of factors which differ from dune to dune: 50-80 Hz at Sand Mountain,
Nevada (Criswell et al., 1975); 50-100 Hz at Korizo, Libya
(Humphries, 1966); 130-300 Hz in booming sand from the Kalahari
desert in South Africa (Lewis, 1936); and 300-770 Hz at Dunhuang,
China (Jianjun et al., 1995). Such broad-band output tends to be
muddy in quality, but by virtue of the larger volumes of shearing
sand, also loud. Comparisons to thunder or the drone of low-flying
propeller aircraft are common. Also, the terms "roaring" and
"humming" seem to have first been introduced rather loosely by Lewis
(1936), and their subsequent use by later authors has at times been
somewhat confusing. It seems that "humming" was the term used by
Lewis when only one fundamental frequency of emission was heard,
while in "roaring," two or more were present.
Fully developed avalanches, in which sliding plates of sand remain intact for most of their motion, exploit the shear (and hence acoustic) potential of booming dunes to the utmost extent. Soundings can quickly grow to near-deafening volumes, comparable in intensity to rumbling thunder (Curzon, 1923). Under the right conditions, emissions can be heard up to 10 km away and last as long as 15 minutes. Perhaps more surprisingly, the booming mechanism produces seismic ground vibrations roughly 200 to 400 times more efficiently than the coincident oscillations in air pressure (Criswell et al., 1975). These ground vibrations, thought to occur along all directions, have on occasion been reported as being so intense so as to make standing in the midst of a fully developed flow nearly impossible (Curzon, 1923). Of course, shearing that takes place over a much smaller area, like that caused by running one's hand through the sand, also creates acoustic emissions. In any case, a critical amount of booming sand must shear before acoustic emissions of any sort are produced. Bagnold (1954b) cites that running one's hand through the sand provides just about the minimum amount of displacement needed in order to produce soundings. Although decidedly lower in intensity, these small-scale slumpings, or sudden small slides, produce soundings that are often likened to the low notes of a cello or bass violin. The relationship between intergranular frictional effects to sound production in booming sand becomes clear upon actually sensing the tactile vibrations caused by the grains resonating in a coherent manner during a booming event. Tactile vibrations created by small-scale slumpings are often compared with a minor electric shock (Criswell et al., 1975).
This efficiency in converting mechanical shearing energy into
seismic vibrations suggests that booming may also be responsible for
the curious "moonquakes," thought to originate on the slopes of Cone
Crater, that have been recorded at the Apollo 11, 14, 15 and 17
landing sites (Criswell et al., 1975; Dunnebier et al., 1974). These
moonquakes begin abruptly two earth days after the lunar sunrise,
continue nearly uninterrupted throughout the lunar day (lunation),
and cease promptly at sunset. It is likely that these seismic events
are triggered by heat-induced slumping of lunar soil. However, if
this seismic activity were due to conventional conversion of shearing
energy into seismic energy, soil would have to shear in such large
volumes that the leading angle of Cone Crater would fall below the
static angle of repose in less than 100 years (Criswell et al.,
1975). This is clearly at odds with the fact that Cone Crater is at
least 30 million years old.
Much attention has been given to examining the morphology of
sound-producing grains and, in particular, to addressing the role
that exceptional granular polishing might play in both types
of sounding mechanisms. Figure 1 presents a comparison of the
morphological features of silent beach, squeaking beach and desert
booming sand grains using electron microscopy. Criswell et al. (1975)
and Lindsay et al. (1976) suggested that a high degree of polishing
alone is responsible for booming, and that all the other physical
parameters typically associated with booming sand simply serve to
enhance granular smoothing in desert environments. They argue that
very smooth granular surfaces will decrease the amount of energy lost
during shearing as a result of low mechanical coupling between the
grains. Sufficiently smooth surfaces would allow almost elastic
collisions, thus narrowing the resonant frequency of what is
otherwise ordinary dry sand [which is ~80 Hz (Ho, 1969), well within
the range of most booming dunes]. Although straightforward, this
analysis does not take into account the large amplitude of
oscillation that grains in the shear plane experience during a
booming event. These vibrations are nearly as large as the grains
themselves (Criswell et al., 1975), and lie well outside the
bounds of the linear analysis (Wu, 1971) used to model the elastic
deformation of grains. Moreover, it appears that this amplitude
represents only the lower bound of magnitudes attainable during a
booming event (Criswell et al., 1975). Any realistic model of booming
must be based on non-linear pressure vibrations.
Booming dunes are often found at the downwind end of large sand
sources, which is a likely consequence of the need to optimize
granular polishing. Specifically, wind-driven saltation of sand (the
bouncing of grains upon a granular surface) across the desert floor
increasingly rounds off granular asperities the further the sand is
blown (Sharp, 1966). Since some preliminary degree of rounding is
essential before a grain can be polished, one would expect the
probability of finding large accumulations of highly polished desert
grains to increase with distance downwind from large sources of sand.
Granular polishing can take place either during the long-range
saltation transport of desert sands, e.g., 35 miles in the case of
the booming Kelso dunes of California (Sharp, 1966), or by a
sufficiently long residence time within the dune itself (Lindsay et
al., 1976). Examples of the latter effect are the collections of
back-beach booming dunes of the Hawaiian islands of Kauai and Niihau,
located no more than 300 m inland from their source beach sand
(Lindsay et al., 1976; Bolton, 1889b). In this case, it is likely
that preliminary rounding is brought about by the relatively long
residence time of the grains on the beach itself, the result of
limited sediment supply and slow off-shore currents (Lindsay et al.,
1976). Substantial rounding is likely to take place before the grains
are blown into the dunes behind the beach, where subsequent polishing
presumably occurs. Some combination of the above arguments probably
accounts for the existence of the Jebel Nakus and Bedawin Ramadan
booming dunes of Egypt, located on the Sinai Peninsula 3 km inland
from the Gulf of Suez. We also suspect that a variable sand source
distance to Sand Mountain exists, as we observed that particular
booming sand to possess a spectrum of polishing histories.
Booming dunes usually form far enough downwind from large sand
sources to permit development of reasonably well-sorted grain-size
distributions (Lindsay et al., 1976; Sharp, 1966). This fact has
prompted much speculation as to how important certain grain-size
parameters are to the sounding mechanism. Highly sorted grain-size
distributions are in fact common, although by itself, this is not
likely to determine the ability of sand to boom. On the contrary, the
booming sands of Korizo and Gelf Kebib, both in Libya, have been
noted for their uncharacteristically broad range of particle sizes
(Humphries, 1966; Bagnold, 1954b). Moreover, silent dune sand is
often as well-sorted as nearby booming sand (Lindsay et al., 1976),
and monodisperse (i.e., identical-size) glass beads have never been
observed to boom. Booming is also largely independent of grain shape.
Close inspection of Sand Mountain (see Figure 1d) and Kalahari
booming sand (Fig. 7 in Lewis, 1936) reveals that not all grains are
highly spherical or rounded. Furthermore, quartz grains in Dunhuang,
China, have obtuse edges and irregularly shaped pits distributed on
their surfaces (Jianjun et al., 1995). Also, Lewis (1936) claims to
have produced booming in ordinary table salt, which has cubical
The role that other grain-size parameters might play in the
booming mechanism has also been the subject of considerable research.
For instance, booming sand usually contains an excess of slightly
finer-sized grains than its average grain size (Lindsay et al.,
1976). The asymmetry that this condition creates in the particle-size
distribution of the sand is called "fine-skewness." Humphries (1966),
in particular, gave substantial consideration to the role that
fine-skewness might play in the sounding phenomena. However, since
nearly any size-fraction of booming sand exhibits pronounced
acoustical activity (Haff, 1986; Leach and Rubin, 1990; Miwa, Okazaki
and Miura, 1995), it is unlikely that the presence of a finely-skewed
particle-size distribution alone directly affects the sounding
ability of the sand. Booming is also very sensitive to the addition
of very fine-sized fragments and grains (on the order of 1-mm in diameter), which seem to disrupt the
collective grain behavior (Haff, 1986).
Booming sand is often observed to boom best at or near the leeward
dune crest. Several factors may be responsible for this, including
the fact that crest sand tends to be better sorted and the grains are
more rounded and polished (Lindsay et al., 1976). Another important
factor is that sand around the crest tends to dry most quickly.
Although precipitation is rare in desert environments, when it does
occur, sand dunes retain the water they absorb with remarkable
efficiency. Sand near the dune surface (< 20 cm deep) dries off
fairly quickly, since water evaporates off the grains into the
interstitial air, which, when heated during the day, expands and
carries the water vapor out of the dune. At night, cold, dry air
fills the granular interstices and repeats the process until the
surface sand is completely dry. However, sand is a poor conductor of
heat, and this temperature gradient rarely reaches more than 20 cm
into the dune. Beyond that, with no mechanism to drive it anywhere,
interstitial air can become completely saturated with water vapor and
remain trapped for years (Bagnold, 1954c). Since booming sand tends
to shear off in plates that are roughly 4 inches deep (Humphries,
1966), sounds occur in those parts of the dune which dry off the
fastest. Near the leeward dune crest, the combination of smooth,
well-sorted grains and the constant recirculation of interstitial air
resulting from the flow of wind over the crest helps promote complete
granular drying deep into the dune. Note also that although
spontaneous acoustical activity normally only occurs on leeward
slopes, sand from the windward side usually possesses comparable
acoustical potential. Windward sand is usually not as loosely packed
or steeply inclined as leeward sand, and hence does not shear
spontaneously as easily, but when properly loosened up, it frequently
emits sound just as readily as leeward sand (Haff, 1979). For
completeness, we note that during the unusually dry conditions of
summer 1994, our tests at Sand Mountain revealed booming over much of
the dune's leeward sides, with the most robust emissions occurring
near its base.
Wind carrying airborne sand grains across the dune crest has a
greater tendency to deposit the grains closer to the top of the
leeward than near its bottom. Consequently, sand accumulates faster
in the upper portions of the leeward slope than in the lower
portions, slowly increasing the angle that the dune's leading edge
makes with the horizontal. General slumping occurs when this angle
reaches ~34°, the angle of dynamic repose for dry desert sand
(Bagnold, 1954d). Typically, large plate-like slabs of sand break off
along clearly defined cracks near the crest. Especially in booming
sand, it is possible that these cracks form in regions of more
symmetrical skewness, which are typically less resistant to shear
(Humphries, 1966). The plates themselves usually remain more or less
intact until they get near the base of the dune, where the change to
a gentler slope slows their slide (Haff, 1979).
An unusual, and as yet not fully understood aspect of booming sand
is the manner in which these plates subsequently break apart. Rather
than simply disintegrating into loose flow upon hitting the gentler
basal slopes, the upper (trailing) portions of booming sand plates
are at times seen collapsing, or telescoping, into the lower
(leading) portions. Haff (1979) compares the appearance of this
effect to that of a sheared deck of cards, and suggests that it may
result from distinct boundaries formed between shearing and
stationary grains within the plate itself. Furthermore, the free-flow
of sand that finally does result from the break-up is unusually
turbulent, resembling "a rush of water seen in slow motion" (Bagnold,
1954b). The connection between this flow phenomenon and the booming
mechanism is as yet not fully understood. It is not clear from the
literature whether this "stacking" effect and the subsequent
turbulent motion occurs only when the sand is booming, and hence is
the result of sound propagation through the sand, or whether it
always occurs, and is the result of some unique grain packing. On the
other hand, no such rippling motion has been reported by Criswell et
al. (1975) and Lindsay et al. (1976), suggesting that this effect may
be subtle or completely absent in some booming dunes. More detailed
field observations are needed.
The piezo-electric properties of quartz crystals were at one time
thought to play a significant role in the booming mechanism, although
as yet there has been no evidence to suggest that this may be the
case with squeaking sand. It is well known that electrical
polarization arises when pressure is applied to both ends of certain
axes within a quartz crystal (Cady, 1946). It has been proposed that
the way in which stress is applied to booming grains when sheared may
cause an accumulation of these tiny piezo-electric dipoles, which
would then somehow be responsible for the pronounced acoustic output
of the sand (Bagnold, 1954b). Such speculation began after Lewis
(1936) observed that upon slowly pouring Kalahari booming sand,
grains would occasionally adhere to one another so as to form
filaments as long as a half inch. An electroscope verified that these
filaments did indeed exhibit electrical charge. Furthermore, if
booming sand was shaken inside a glass jar, a significant number of
grains were observed to adhere to the sides of the glass, where they
remained for several days. The grains were also noted to cling most
densely at places where the temporary surface of the sand had rested
against the glass. However, this has also been observed in normal
One could even extend this reasoning to explain why booming is
precluded by the adsorption of very small amounts of water, which
could effectively interfere with the necessary polarization of
grains. Nonetheless, Lewis (1936) was able to demonstrate that
grounding the sand had no effect on its acoustic output. Moreover,
since booming occurs naturally in the calcium carbonate sand dunes of
Hawaii and Lewis claims to have produced booming in sodium chloride
crystals, this "electrical connection" should be considered tenuous
The earliest reports of booming sands were made by desert nomads,
who interpreted the noises as supernatural ghosts and demons (Curzon,
1923). Similarly unorthodox notions, such as that the soundings
result from the eruption of subterranean volcanoes, persisted well
into the late-nineteenth century (Carus-Wilson, 1888). At this time
more serious systematic investigations into the phenomena first
began. In 1889, Bolton (1889a), one of the first to extensively study
the phenomena, published his model of "air-cushion" theories. He
proposed that the sounds result from thin films of adsorbed gases
deposited on the grains by the gradual evaporation of water. The
acoustic emissions would arise from the vibration of elastic air
cushions, and the volume and pitch of the emissions would be modified
by the surface structure of the grains themselves and extinguished by
smaller fragments and debris in the sample. Given the importance
assigned to water in this model, the reader might assume that Bolton
was concerned exclusively with squeaking beach sands. Booming dunes
exist in extremely arid desert regions which receive essentially no
rainfall for years at a time. As it turns out, most of the "musical"
sand which Bolton sampled was actually squeaking beach sand (although
he did use this model to explain booming sands as well, and did not
seem to attach much importance to the differences between the two).
However, concrete empirical evidence in support of this theory has
never been produced, and the fact that tactile vibrations observed in
booming sand from Sand Mountain, Nevada, (shown on Fig. 2) at
atmospheric pressure are no different from those observed at 1.5
mm Hg air pressure (Criswell et al., 1975) effectively discredits an
air-cushion mechanism as the cause of booming emissions. To the
authors' knowledge, the last paper published in support of Bolton's
air-cushion model was by Takahara (1965). It is, however, merely a
reaffirmation that squeaking sand produces better acoustic emissions
immediately after washing.
As was stated at the outset of this paper, it is likely that the
cause of the acoustic emissions is closely related to the frictional
behavior between the grains during shearing. Carus-Wilson (1891), a
contemporary of Bolton, was the first to propose that intergranular
frictional effects may create sound in certain types of "musical" (in
fact squeaking) sands. He was, it seems, the first to correctly
conclude that the grains are in general highly spherical,
well-rounded, well-sorted and unusually smooth, and postulated that
acoustic emissions must result from collectively "rubbing" grains
which exhibit these four properties. He, nonetheless, drew criticism
for not elaborating on precisely how a frictional sounding mechanism
might work. Most notable among his critics was the physicist Poynting
(1908), who showed that if the soundings should result only from the
natural vibrations of the grains themselves, frequencies of no less
than 1 megahertz could be produced. Working together, Poynting and J.
J. Thomson (1922) made their own attempt to extend Carus-Wilson's
reasoning by coupling it with the principles of granular dilatancy,
as put forth by Reynolds (1885). Dilation is simply the expansion in
volume that a granular substance undergoes when it deforms under
applied shear. This principle can be rephrased by stating that fixing
the volume of a granular mass precludes its deformation. A familiar
example of this volume expansion is the drying out, when stepped on,
of wet sand around one's foot. The pressure applied by the foot
creates a deformation in the sand, causing expansion in the region
immediately surrounding the place of compression. This expansion, or
dilation, is large enough to cause temporary drainage from the
compressed to the expanded regions.
Poynting and Thomson (1922) reasoned that if a close-packed
granular substance consists of monodisperse, spherical particles,
then the dilation caused by shearing should be (roughly) periodic in
time. Such uniform variations in dilation, they conclude, are likely
related to the uniform oscillations which produce booming and
squeaking. Specifically, suppose a shearing stress is applied to one
layer of such grains, causing it to slide over another layer of
identical grains. Following Reynolds (1885), they reasoned that this
motion requires some grains to dilate, or rise out of
their interstices, move over neighboring grains, and then fall down
into adjacent interstices. So long as the shearing stress remains
constant, successive expansions and contractions should occur
periodically. Needless to say, real grain flow is far more complex
than specified in this simple model. It is unrealistic to
expect that the uniform size and spherical shape of the grains is
enough to ensure such highly-ordered behavior in loose aggregates. In
fact, sound-producing grains are not, in general, perfectly
monodisperse. Their model also implies that the frequency of
emissions should depend on grain size and the rate of shear alone.
This contradicts the fact that squeaking emissions are usually 5 to
10 times higher in frequency than booming emissions, even though both
types of grains are usually ~300 mm in
diameter. Also, if sand is already shearing in a thin layer (as in
most squeaking sand), the addition of shear stress at the surface is
more likely to create new shear planes parallel to the existing one
rather than increasing the rate of shear at the existing plane (and
hence, by the Poynting and Thomson model, increasing the frequency of
emission) (Ridgeway and Scotton, 1973). Of course, not all of the
added shear stress can be accommodated at new shear planes, and some
increase in frequency is bound to occur. But in neither booming nor
squeaking sand does frequency of emission vary linearly with rate of
shear (Lewis, 1936; Bagnold, 1954b). This argument would not apply to
booming sand, which shears in rather thick (~4 inches) thick layers.
Lewis (1936) nonetheless found that quadrupling the rate of shear
roughly doubled the frequency of emission and resulted in a
pronounced increase in volume.
In 1966 the British engineer and field commander R. A. Bagnold,
having already done a large amount of work on granular mechanics in
general, put forth the most complete attempt at explaining the
booming mechanism to date. According to Bagnold, both types of
acoustic emissions result from non-linear oscillations of dispersive
stress along the shear plane. The idea that the cause of the sound
may result from disturbances in the shear plane itself stems from
experiments performed by Bagnold (1954b) in which booming was
produced by shaking sand in a jar. He found that a critical amount of
downward force must be applied to each stroke in order to keep the
sand sounding and estimated the magnitude of this force to be
approximately equal to the weight of 8-10 cm of sand; roughly the
depth at which shearing planes form in booming sand. The weight of
the sliding grains alone, it seems, exerts just enough force on the
shear plane to sustain the oscillations.
Bagnold introduced the concept of "dilatation," 1/l, of a granular substance, and defined it to be
the ratio of the mean intergranular surface-to-surface separation, s,
to mean grain diameter, D. This ratio is qualitatively identical to
the "dilation" of Reynolds, but is easier to work with
mathematically. Since for most natural packing densities, dilatation,
as defined above, is far smaller than unity, its inverse, the linear
concentration l = D/s, is generally used.
In the limit of closest possible packing densities, the linear
concentration approaches infinity. Bagnold showed that most granular
materials remain fairly rigid for linear concentrations down to l = l2 Å17, the point at which dilatation becomes
just large enough to allow general slumping. Granular materials with
l < 17 begin to take on the properties
of a fluidized bed of particles. The system behaves as a
non-Newtonian fluid for small mean intergranular separation,
that is resistance to shear exists at zero shear rate. Below some
linear concentration, l3 Å 14, the grains are too disperse to
effectively transmit intergranular stress. The system then
becomes a Newtonian fluid, losing all resistance to shear.
Piling up sand to above its dynamic friction angle results in the shearing of a thin layer of grains near the surface of the sand. A repulsive stress in the plane of shear (l Å 17) results from the successive collisions of flowing grains upon stationary grains. Of particular interest is the stress component normal to the plane of shear, which is responsible for sustaining the dilation in the first place. In steady-state equilibrium flow, this component is equal and opposite to the normal component of the compressive stress due to the weight of the shearing grains under gravity. However, Bagnold argues, should the shearing grains attain a relative interfacial velocity in excess of the system's preferred velocity, without internal distortions occurring in the shearing layer, the shearing grains could begin vibrating collectively. He reasons that a large, sudden increase in dispersive stress must be created to compensate for the increase in velocity. This creates dilation throughout the entire sliding layer, raising it slightly up off the plane of shear. The dispersive stress itself in turn quickly decreases in magnitude as dilation increases, and the sliding grains soon collapse under their own weight back into the bulk sand. This compacts the original l Å 17 slip face into a denser (l > 17) packing, creating a new l Å 17 slip plane slightly closer to the surface of the sand. This process repeats so long as the grains are shearing. The expression
was derived for the frequency at which this saltation should
occur, where is the local
gravitational acceleration. The Appendix summarizes the derivation of
this relation for .
Although elegant, this analysis does not completely describe a
booming event. In the first place, a booming sample with a mean grain
size of 300 mm should, according to
Bagnold, boom at a frequency of ~ 240 Hz; well outside the range of
Korizo, Libya (50-100 Hz) and Sand Mountain (50-80 Hz). Equally
problematic is that only one frequency is predicted. It is not clear
how four or five separate modes of ground vibration, all with
different axes of vibration, could be created simultaneously from a
single, saltating layer of grains. Consideration of the low-frequency
(1-10 Hz) beats that typically accompany prolonged flows, and of the
frequency modulation that occurs by varying the rate of shear, is
also notably absent from Bagnold's analysis. Grain size may be one
component that fixes the frequency range for a given sample of
booming sand, but there must be other factors that Bagnold fails to
take into account. Furthermore, other experimental results (Leach and
Rubin, 1990; Leach and Chartrand, 1994) indicate that the frequency
of emission for a given size fraction of booming sand seems to
decrease linearly with increasing grain size, rather than decrease
with the square root of grain size as proposed by Bagnold.
Bagnold applies a similar line of reasoning in explaining
squeaking, the primary difference being that here a compressive
stress gives rise to the sound, rather than a general slumping shear.
This makes sense, since acoustic emissions in squeaking sand are most
naturally created by a quick, sharp compression, like when walking on
it, rather than by general shearing like avalanching in dune
environments. Bagnold proposes that dispersive stress should again be
created in the shear planes in a similar way as in booming sand.
However, since compression subjects the grains in the shear plane to
K times more acceleration than they would experience during slumping,
the emission frequency should be times higher. The constant K varies
among different types of granular systems and depends on, among other
things, the area of sand compressed and the maximal angle of repose
of the sand (Terzaghi, 1943). The frequencies emitted by squeaking
sand seem to fit Bagnold's model better than those of booming sand
(Lindsay et al., 1976).
There are many qualitative differences between booming and
squeaking emissions. For instance, squeaking emissions almost always
produce only a single fundamental frequency of vibration. The
multiple-band phenomena observed in large booming events almost never
occurs. Conversely, squeaking sand often produces four or five
harmonic overtones (Takahara, 1973), while at most only one harmonic
of the fundamental tone has been observed in booming sand (Criswell
et al., 1975). Understandably, a consensus that squeaking sand never
booms and that booming sand never squeaks has arisen in the
literature (Criswell et al., 1975; Lindsay et al., 1976; Bagnold,
1954b) We too were unsuccessful in getting booming sand to squeak. A
closer look at the older literature may nonetheless suggest
otherwise. For instance, in 1889 Bolton (1889b) writes:
"The sand of the Hawaiian islands possesses the acoustic
properties of both classes of places [beaches and deserts]; it gives
out the same notes as that of Jebel Nagous [an Egyptian booming dune]
when rolling down the slope, and it yields a peculiar hoot-like sound
when struck together in a bag, like the sands of Eigg, Manchester,
Mass., and other sea-beaches [squeaking sand]."
More recently, Haff (1979) has also been able to produce similar
high-frequency squeaking using booming sand from the Kelso dunes,
both in situ on the dune as well as in a laboratory. The
fundamental frequencies of these vibrations are close to 1200 Hz,
implying that compression of booming sand doesn't just amplify
high-order harmonic frequencies of low-frequency fundamental tones
(e.g., 50-300 Hz). This provides some support for Bagnold's theory,
which implies that the only difference between the two modes of
emission are compressional versus shear-induced slumping. Subtle
differences do exist between booming sand that squeaks and genuine
squeaking sand, however. For instance, frequency analysis by Haff
clearly shows that multiple fundamental frequencies are still present
in squeaking emissions from booming sand. It is further interesting
to note that all desert sands Haff sampled were able to
produce some type of squeaking emission when compressed vertically in
a container. Bolton (1889a), on the other hand, noted that the sand
of Jebel Nakus, an Egyptian booming dune, could not produce such
"squeaks" when compressed. It is possible that since Bolton was
working in situ, adequate pressure might not have been
applied. Moreover, Haff found that silent beach sand does not
necessarily squeak when compressed. Most likely, the occurrence of
squeaking in sand corresponds very directly with grain shape and
morphology. Most silent desert sand grains tend to be more
spherical, rounded and polished than silent beach sands (Lindsay et
al., 1976). It would be interesting to see if table salt, which Lewis
(1936) claims can be made to boom, could also be made to squeak. We
have observed some booming after sieving silent sand.
The back-beach booming dunes of Hawaii, which differ from most
other booming dunes in a number of fundamental ways, provide an
interesting case study. In the first place, they are composed
primarily of calcium carbonate grains from sea shells, and are
thought to be the only naturally-occurring booming sands not made out
of quartz. The grains themselves are unusually large (~460 µm)
and the frequency range of their emissions appears to be wider than
that of most other booming sands, although the latter point has yet
to be precisely determined (Lindsay et al., 1976). Also, they are the
only non-desert booming dunes and seem to exhibit a slightly higher
tolerance for water exposure than do other booming dunes (Haff,
1986). Nonetheless, the loose packing and the rough surface profile
of back beach dunes may be what keeps them from squeaking.
More comprehensive investigations of the Hawaiian dunes are
The speculative idea that booming and squeaking are produced by
the same mechanism, the output of which is modulated either by
certain intrinsic grain parameters or by the method of shearing used
to produce the sounds, is appealing and warrants further
investigation. We choose here to remain with convention and treat the
two types of emission as if they were produced by two distinct
It seems unlikely that booming and squeaking sand grains, which
both exhibit mean grain diameters of ~300 µm and appear somewhat
similar in electron micrographs, could produce such substantially
different modes of acoustic emission simply on the basis of some
intrinsic grain parameter. Instead, the way the sand is sheared
probably determines the frequency of emission, especially in booming
sands. Increasing the rate of shear seems to increase its frequency
of emission. Quickly compressing the booming sand vertically, thus
creating a very high rate of shear, reportedly produces emissions
which resemble genuine squeaking.
While the two modes of acoustic output from sand may be inherently
connected and mostly governed by the method of shearing, the actual
onset of acoustical activity in a sand sample probably is controlled
by certain intrinsic grain parameters. The most critical parameter
that governs the ability of sand to boom seems to be a high shear
resistance. Both varieties of sound-producing sand possess a large
fraction of smooth grains, and booming sand grains smaller than ~200
µm tend to be very smooth. In addition, squeaking sand is
typically well-sorted. Monodisperse glass spheres, which exhibit a
lower shear resistance than any type of sand, cannot boom, and squeak
only under somewhat contrived experimental conditions. A
highly-sorted collection of smooth, well-rounded spherical grains is
thus not a sufficient condition for sound-production in sand.
In summary, important physical properties that distinguish booming
sands from their silent counterparts are: high surface smoothness and
high-resistance to shear. Other factors that contribute are the
roundedness, the sphericity of the grains and differing roughness
between the larger grains in the sample. Booming occurs best at very
low humidity, since humidity creates a fluid surface coating that
acts like a lubricant and lowers the shear resistance. Humidity also
increases the cohesion between grains. Booming is enhanced when the
grains are loosely packed, since a very tight packing will preclude
shearing. Effects due to collective or resonant motion are also
considered to be important. However, these collective effects are
still not well understood, in spite of research work on collective
motion in granular materials and related systems (see, e.g., Bretz et
al.,1992; Benza et al., 1993; Jaeger and Nagel, 1992; Liu and Nagel,
1993; Bideau and Hansen, 1993; Satake and Jenkins, 1988). How these
ingredients mix together to produce booming is still an open problem.
The actual mechanism has been the subject of speculation for
centuries and still remains unresolved.
Many avenues of investigation remain open. Determination of the
mineral composition of the grains, particularly at the granular
surface, has only recently been attempted1.
Shearometer analysis and X-ray studies of mass density fluctuations
during shear would also be of interest. Examination of the possible
piezo-electric properties of booming sands, alluded to earlier, has
also been minimal, as have been attempts to create synthetic booming
grains. A good understanding of sound-production in loose aggregates
might help our understanding of grain flow in general.
In 1906, Einstein (1906) considered the effect of solid grains,
immersed in a fluid, on the shear resistance of that fluid. Half a
century later, Bagnold (1954a, 1966) considered a regime that was
apparently neglected at the border between hydrodynamics and
rheology, namely the case of a Newtonian fluid with a high
concentration of large solid spheres. In this appendix, we summarize
Bagnold's derivation of the equation for the frequency of vibration
that occurs during a booming event.
First consider the slumping, under the force of gravity, of dry
sand that has been piled up to its critical angle of repose. Grain
flow occurs along a well-defined front and proceeds with a constant
velocity. One would like to be able to express this terminal velocity
as a simple function of such variables as the mean grain size and the
depth of flow. Bagnold's equation for the frequency of vibration that
occurs during a booming event is derived from the analysis of the
terminal velocity of the flowing front.
Two different experiments were conducted by Bagnold to arrive at
his result. In the first, a collection of uniformly-sized spheres was
placed into an immersing fluid. The spatial distribution of spheres
was kept uniform by experimenting under "gravity-free" conditions,
where the density of the grains, , was
balanced against the density of the immersing fluid, . In order for uniform shear strain
to be applied to all the spheres, the spheres were sheared inside the
annular space between two concentric drums. This allowed for the
accurate measurement of the intergranular stresses and strains that
occur during shearing. In the second set of experiments, these
results were extrapolated to model the dynamics of dry sand
avalanches, where the grains are not uniform in size, nor is the
interstitial fluid (air) of the same density as the grains.
We now turn out attention to the theoretical considerations
involved in the first experiment (Bagnold, 1954a). Suppose that we
have a collection of uniformly-sized spheres, with diameter . Clearly, in the closed-packed
limit, the distance that separates the centers of two adjacent
spheres is . Recall that
the linear concentration, , is defined to be the ratio of the mean grain
diameter to the mean intergranular
separation . For the case
of equal spheres, l approaches infinity
since is very small. Also, the volume
fraction of space
occupied by the grains becomes for
the case of close-packed equal-sized spheres.
Next consider what happens when the spheres are uniformly
dispersed, so that the mean distance separating two adjacent centers
becomes , where (see Figure 3a). In this case, the mean intergranular
separation , as measured
from the surface of the grains, is larger than the of the close-packed equal sphere
This result can be expressed in terms of the volume fraction
The smaller is, the easier
it is to induce shearing. In general, shearing becomes possible for
values of . Moreover,
Bagnold made the following assumptions about the flow of spheres and
the shear that occurs:
(i) The spheres are in a uniform state of shear strain, ; and the mean relative velocity
between the spheres and the interstitial fluid is zero everywhere
(see Figure 3a). is the
velocity field of the granular material whose average motion is along
the "downhill" -direction.
(ii) Frictional losses maintain a constant kinetic energy per unit
volume of the system.
(iii) In addition to the general drift in the -direction, the spheres also make
small oscillations in all three directions.
Now, consider oscillations produced by a sequence of small jumps
or saltations as the spheres of one layer jump over the neighboring
sphere layer. Since the "gravity-free" conditions in principle
provide a uniform spatial distribution of spheres, the spheres can be
treated as being arranged into sheets lying parallel to the plane, with one plane shearing
across the top of the layer immediately below it (see Figure 3b). Let
us designate the top layer as B, and the bottom layer as A. Then the
mean relative velocity of a sphere in layer B with respect to plane A
is , where is a constant that varies from to depending on
geometry (Bagnold, 1966). In this model, the spheres in plane A can
be thought of as constituting a rigid plane, across which the spheres
in layer B saltate at the differential velocity .
Suppose that on average, any given sphere in layer B makes collisions per unit time with
layer A. The number of spheres in a unit area of the B plane is , and with each collision a sphere
in layer B will experience a change in momentum in the -direction (the angle is
determined by collision conditions, and will be discussed in more
detail shortly). This implies that a net repulsive pressure should exist between the two
layers, with magnitude
Also, there is a corresponding tangential shear stress given by
Indeed, Bagnold's experiment found that, at sufficiently high
speeds, both and become proportional to . The only unknowns in the above
equations are and . Bagnold's experiment (Bagnold,
1966) with uniform spheres sheared inside a rotating double-cylinder
found that in the case where the spheres are sufficiently far apart
for the material as a whole to take on the properties of a Newtonian
liquid, , ,
and . Thus for , we can
At this point, there is no a priori reason to suppose that
these results should be applicable to dry sand avalanches, since
silicon dioxide does not have the same density as air, nor are grains
of sand uniform in size or shape. Nevertheless, let us now make the
appropriate substitutions and try to derive a general expression for
the terminal velocity of flow using known grain parameters. On any
constant plane below the upper,
free surface of the sand, the applied shear stress is:
with being the angle of incline and
the volume fraction (Eq. (2)).
Since air has a very low viscosity, we can apply the above results,
and equate expressions (7) and (8). Thus
which implies that
Making the reasonable assumption that is roughly uniform through the depth of the flow, and
that it has a value of (obtained
empirically by Bagnold), the integral in Equation (10) reduces to
Finally, taking at the shear
plane, and , equation (10)
then reduces to:
where is the
overall depth of the flow and refers to the top surface.
We can further simplify the above expression by making the
following assumptions about the flow: for the relatively high
concentrations that we are working with here, the relative
interfacial velocity at any shear
surface can be substituted for the expression . This simplifies the expression for the pressure:
Also, for slow continuing shear, we can take for the local linear
Having performed the 1954 experiments, Bagnold next needed to
verify that these results for and
apply to actual sand avalanches,
and not just in the ideal case of uniform spheres in a rotating drum
(1966). In order to verify these results, he performed a bull-dozing
experiment in which a heap of sand was pushed, at a constant depth
below the surface, by a push
plate. The results he obtained indicated that the above equation for
terminal velocity of flow also holds for dry sand shearing.
In order to explain the frequency of emission that occurs in a
booming dune, first consider the force , which is the normal compressive stress on the shear
surface due to the weight of the sand above it. In equilibrium, any
increase in in excess of
requires a dilatation increase at
the shear surface, so that falls to some
value smaller than 17.
If the velocity of flow does momentarily exceed the terminal velocity, then will briefly exceed , and the sand mass is accelerated slightly upward. However, the upward stress decreases rapidly as the dilatation increases. The sand would then collapse back under the weight of gravity, decreasing the dilatation, and again causing to temporarily exceed .
If the mass of flowing sand is m, then it would be subject to the
oscillating force , which would
create oscillations in the normal direction. Also the minimum mean
local dilatation at the shear surface, at which oscillations could
still occur, is .
The stress is only effective when the dilatation, , is near its minimum, because the contact faces (of the sliding planes) just clear one another and since very rapidly varies with . Thus, the rise and fall of the overburden through the distance will be an almost free-fall. Hence, the minimum frequency of oscillation is given by
This expression provides an estimate of the vibration frequency
that occurs during a booming event. Using and mm, this expression then predicts that 240 Hertz, which is in the
observed range of values for booming acoustic emissions during an
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Fig. 1. A composite diagram (from top left to right) of
normal beach (top left), squeaking beach (top right) and booming
desert (bottom) sand grains using low-magnification electron
microscopy. Samples were collected from Lake Huron at Bay City, MI
(top left), Lake Michigan at Ludington, MI (top right) and Sand
Mountain, NV (bottom). The sample in the bottom right panel was
sieved and consists of grains smaller than ~200 mm. All micrographs were made on the 100 mm length scale. These photos suggest that the
normal beach sand is poorly polished and irregular in shape, while
the squeaking sand is more polished. Occasional scour marks appear on
both types of beach sands, but not on booming sand. While squeaking
grains are by and large rounded, booming sand contains a variety of
erosional grain states as shown in the bottom left panel, including
many smaller, well-polished, well-rounded grains, as seen in the
bottom right micrograph. The top-right grain in the bottom left panel
is highly unusual in booming sand.
Fig. 2. Sand Mountain, near Fallon, Nevada. Other booming
dunes in the western United States include: Big Dune, near Beatty,
Nevada; the Kelso Dunes, near Kelso, California; and Eureka Dune,
located at the western edge of Last Chance Ridge in California.
Fig. 3. (a) Schematic diagram of two layers, A and B, of grains moving along the "downhill" direction. The average intergranular separation , and the average distance between two adjacent centers, , are indicated. (b) Schematic illustration of a granular flow and its velocity profile.