Sound-Producing Sand Avalanches

ContemporaryPhysics, 1997,volume 38, number 5, pages 329 ± 342

Sound-producingsand avalanch es

PAUL SHOLTZ, MICHAEL BRETZ and FRANCO NORI*

Sound-producingsand grains constitute one of nature’ smore puzzlingand least understood physicalphenomena. They occur naturally in two distinct types: booming and squeaking sands.Although both varieties ofsand produce unexpectedly pure acoustic emissions when sheared,they diŒ er intheir frequency range and duration of emission, as well as the environmentin which they tend to be found. Large-scale slumping events ondry booming dunescan produce acoustic emissions that can be heard up to 10 km awayand which resemble hums,moans, drums, thunder, foghorns or the drone of low-¯ ying propeller aircraft.These analogies emphasize the uniqueness of the phenomenon and the clarity of the producedsound. Although reports of these sandshave existed inthe literature for overone thousandyears, a satisfactoryexplanation for either typeof acoustic emission is still unavailable.

1.Introduction sands haveled to aconsensus that although both types of There existtwo distinct types of sand that areknown to sand produce manifest acoustic emissions, their respective produce manifest acoustic emissions when sheared. The sounding mechanisms must be substantially diŒerent. More more common of the two, known colloquiallyas ` squeak- recent laboratory production of `squeaks’ in booming sand ing’or `whistling’sand, produces ashort ( <1/4 s), high- (HaŒ1979) has nonetheless suggested acloser connection frequency (500± 2500Hz) `squeak’ when sheared or com- between the two mechanisms. Asatisfactory explanation pressed. It is fairlycommon in occurrence, and canbe for either type of acoustic activityis still unavailable. found atnumerous beaches, lakeshores and riverbeds This brief reviewis not aclosed chapter in awell- around the world. The other, rarer type of sound- understood research area,but is asummary of the producing sand occurs principallyin large,isolated dunes incomplete and unsatisfactoryproposals put forward to deep in the desert (Nori et al.1997,Criswell et al. 1975). explainsound production in squeaking and, especially, The loud, low-frequency(typically50 ± 300Hz) acoustic booming sand. Our reviewpoints out the serious problems, output of this `booming’ sand, resultant upon avalanching, incomplete theories, and the severelimitations of diŒerent has been the subject of desert folklore and legend for approaches published so far. centuries. Marco Polo (1295)wrote of evildesert spirits Booming and squeaking sands eachshow amarkedly which `attimes ®llthe airwith the sounds of allkinds of diŒerent response to waterexposure. Booming occurs best musical instruments,and alsoof drums and the clashof when the grainsare very dry, preferably severalweeks after arms’.References canbe found datingas far back asthe the lastrain. Small amounts of atmospheric humidity, ArabianNights (Carus-Wilson 1915),and asrecently asthe which creates a¯uid surface coatingon the grains, science ®ction classic Dune (Herbert 1984).Charles Darwin eŒectively preclude booming emissions in these desert (1889)also makes mention of it in his classic Voyagesof the sands (Lewis1936). Even mixing as little as® vedrops of Beagle.At least31 desert and back-beach booming dunes waterinto aone litre bagfull of booming sand cansilence havebeen located in North and South America, Africa, the acoustic emissions (HaŒ1979). Similarly, squeaking Asia,the Arabian Peninsula and the HawaiianIslands sand that is visiblymoist is not acousticallyactive either. (Lindsay et al.1976,Miwa and Okazaki1995). Sharply However,sound is most easilyproduced from squeaking contrasting diŒerences between squeaking and booming sand immediatelyafter the grainshave been `washed’in waterand subsequently dried. It is not clearwhether this is due to the washingaway of ®ne impurities in the sample *Authorto whom correspondence should be addressed (Brown et al.1961)or to the creation of alooser, more Authors’address: Departmentof Physics, The University of Michigan, AnnArbor, MI 48109-1120,USA natural grainpacking (Clarke 1973), although it may http://www-personal.engin.umich.edu /~nori/ explainwhy squeaking sand typicallydoes not extend 0010-7514/97 $12.00 Ó1997 Taylor &FrancisLtd 330 P. Sholtz et al. inland more than 30m from the shore (Richardson 1919). located between the grains(Bolton 1889a).Also, a This process of cleaningcan also ` revive’squeaking sand preliminary explanation of booming relied on observed that has lost its abilityto squeak, acondition that often electricalcharging of the grains(Lewis 1936). Such air- occurs after repeated compression (Hashimoto 1951). cushion and electricalcharge models were,however, based Finally,squeaking sand canemit sound evenwhen on misguided evidence and havebeen discredited bymore completely submerged in water(Lindsay et al.1976,Brown recent experiments. Most modern theories stress the et al.1961),suggesting that intergranular cohesion in moist importance of intergranular friction. It has been suggested sand precludes acoustic output. that the unusually smooth and polished surfaces of The mean grainsize (diameter) of most sand, whether or booming grainsmay allow for exaggeratedvibration at not it is acousticallyactive, is roughly 300 l m. The the natural resonant frequency of sand (Criswell et al. 1975, frequency of emission generated bysqueaking sand is Lindsay et al.1976).DiŒ erences in grainpacking have also thought to varyas the inverse square root of the mean grain been considered (Humphries 1966).The most complete size (Bagnold1954a, 1966), although mean grainsize does development is givenby Bagnold (1954a, 1966), who argues not byitself determine the abilityof sand to sound (Lindsay that both types of emission result from nonlinear oscilla- et al.1976).It is unlikely that booming frequencies depend tions of dispersive stress in the layerof movinggrains, or similarlyon grainsize alone, asa fairlywide range of shear layer.Even this analysis,however, fails to address key fundamental frequencies is often generated in large-scale aspects of the booming mechanism. slumping events. Particle-size distributions (Folk and Ward 1957)of sound-producing grainsusually only extend over a narrow range,a condition that is called` being well-sorted’. 2.Acoustic and seismic outputof a boomingevent Also, booming and squeaking grainstend to be both Booming emissions produce awidevariety of sounds that spherical and lackabrupt surface asperities. The latter havebeen compared to moans, hums, roars, drums, condition is called` being well-rounded’(Powers 1953),a tambourines,thunder, cannon ®re, the rumble of distant term not to be confused here byour use of the word carts, foghorns, the buzzing of telegraph wiresand the `polished’,which wetake to mean granularsurfaces which drone of low-¯ying propeller aircraft(Lindsay et al. 1976, aresmooth on the 1 l mlength scale.Both types of sand Curzon 1923).These analogiesall emphasize the distinct further exhibit anunusually high shear strength, and in the qualityof the booming sand phenomenonand the clarity caseof squeaking sand, adecrease in shear strength has with which these sounds areproduced. The comparisons to been shown to correspond to adecrease in sounding ability drums and tambourines moreover illustrate how the (Hashimoto 1951,Humphries 1966).Experiments in which booming mechanism canproduce characteristic beat spherical glassbeads produce acoustic emissions similarto frequencies (1to 10Hz), though to result from periodic those of genuine squeaking sand when compressed, albeit amplitude modulation of the acoustic emissions which are under somewhat contrived experimental conditions (Brown often observed in prolonged, large-scale¯ ows.Perhaps et al.1965),provide further support for the notion that most illuminating,however, are those analogieswhich squeaking is caused primarilyby frictional eŒects. The compare booming emissions to musical instruments, such combination of these four grainproperties (ahigh degree astrumpets, bells or low-stringed instruments.Such clear of: size-sorting, sphericity, roundedness and resistance to emissions usuallyoccur only in small-scale¯ ows,whereby shear) is thought to be criticalfor the onset of squeaking. only one fundamental frequency of vibration is produced. Booming, on the other hand, is substantially less sensitive Criswell et al.(1975)point out the similarities in the to diŒerences in grainshape and sorting, and is likely acoustic amplitude trace of asmall-scale, in situ booming governed bythe unusually smooth and polished surface emission and that of apipe organ.It is remarkable that an texture present in allbooming grains(Lindsay et al. 1976). avalancheof granularmaterial could produce anacoustic Exactlywhat governs either sounding mechanism is still oscillation comparable in purity to a®nely-crafted musical anopen question. Research has been hindered both bythe instrument. Our 1994observations of small,induced rarityof the phenomena and the diculty in reproducing avalanchesat Sand Moutain, Nevada,during the second the sounds in alaboratory environment. Moreover, the driest summer on record, revealemissions similarto a increasingtra c of vehicleson dunes appears to suppress didjeridoo (an aboriginalinstrument from Australia) with the natural sounds of sands. Furthermore,manyresearch- its low,droning cadence. ers had problems diŒerentiating between booming and In general,several fundamental frequencies of vibration squeaking sands, and the earlyliterature on the topic is areoften present, especiallywhen largevolumes of shearing often marred byinconsistencie saswell as vague and sand areinvolved. Exactly what precipitates the transition imprecise terms such as` musical’or `barking’sands. One of from one to severalmodes of fundamental vibration is not the ®rst scienti® ctreatments of the subject attributed the completely understood.Eachfundamental frequency seems squeaking sound to periodic oscillations of airpockets to exhibit its own rise and falltime, independent of the Sound-producing sand avalanches 331 others, and they arethought to result from the collective originate on the slopes of Cone Crater,that havebeen vibration of sand grainswhich canoccur alongall recorded atthe Apollo 11,14, 15 and 17landing sites directions (Criswell et al.1975).Taken together, these (Criswell et al.1975,Dunnebier and Sutton 1974).These frequencies cancover afairlybroad range,the width of moonquakes begin abruptly two earth daysafter the lunar which is determined bya varietyof factors which diŒer sunrise, continue nearlyuninterrupt ed throughout the from dune to dune: 50± 80Hz atSand Mountain, Nevada lunar day(lunation), and ceasepromptly atsunset. It is (Criswell et al.1975);50 ± 100Hz atKorizo, Libya likelythat these seismic events aretriggered byheat- (Humphries,1966);130 ± 300Hz in booming sand from induced slumping of lunar soil. However,if this seismic the Kalaharidesert in South Africa (Lewis1936); and 300± activitywere due to conventional conversion of shearing 770Hz atDunhuang, China(Jianjun et al.1995).Such energyinto seismic energy,soil would haveto shear in such broad-band output tends to be muddy in quality,but by largevolumes that the leadingangle of Cone Craterwould virtue of the largervolumes of shearing sand, alsoloud. fallbelow the static angleof repose in less than 100years Comparisons to thunder or the drone of low-¯ying (Criswell et al.1975).This is clearlyat odds with the fact propeller aircraftare common. Also, the terms `roaring’ that Cone Crateris atleast 30 million yearsold. and `humming’ seem to have® rst been introduced rather loosely byLewis (1936), and their subsequent use bylater authors has attimes been somewhat confusing. It seems 3.Grain size andmorphology of booming sand that `humming’ wasthe term used byLewis when only one Much attention has been givento examiningthe morphol- fundamental frequency of emission washeard, whilein ogyof sound-producing grainsand, in particular, to `roaring’,two or more werepresent. addressing the role that exceptional granularpolishing Fullydeveloped avalanches,in which sliding plates of might playin both types of sounding mechanisms. Figure1 sand remain intact for most of their motion, exploit the (® gure 2)presents acomparison of the morphological shear (and hence acoustic) potential of booming dunes to features of silent beach, squeaking beach and desert the utmost extent. Soundings canquickly growto near- booming sand grainsusing electron (optical) microscopy. deafening volumes, comparable in intensity to rumbling thunder (Curzon, 1923).Under the right conditions, emissions canbe heard up to 10km awayand lastas long as15 min. Perhaps more surprisingly, the booming mechanism produces seismic gound vibrations roughly 200to 400 times more eciently than the coincident oscillations in air pressure (Criswell et al.1975).These ground vibrations, thought to occur alongall directions, haveon occasion been reported asbeing so intense asto make standing in the midst of afullydeveloped ¯ownearly impossible (Curzon 1923). Of course, shearing that takes placeover amuch smaller area,like that caused byrunning one’shand through the sand, alsocreates acoustic emissions. In anycase, a critical amount of booming sand must shear before acoustic emissions of anysort areproduced. Bagnold(1954b) cites that running one’shand through the sand provides just Figure 1. Acomposite diagram (from top left to right) of about the minimum amount of displacement needed in normal beach (top left), squeaking beach (top right) and booming order to produce soundings. Although decidedly lower in desert (bottom) sand grains using low-magni® cation electron intensity, these small-scaleslumpings, or sudden small microscopy. Samples werecollected from Lake Huron at Bay slides, produce soundings that areoften likened to the low City, MI (top left), Lake Michigan at Ludington,MI (top right) notes of acello or bass violin.The relationship between and Sand Mountain,NV(bottom). The sample in the bottom right panel was sieved and consists of grains smaller than intergranular frictional eŒects to sound production in ~200 l m. All micrographs weremade on the 100 l m length booming sand becomes clearupon actuallysensing the scale. These photos suggest that the normal beach sand is poorly tactile vibrations caused bythe grainsresonating in a polished and irregular in shape, while the squeaking sand is more coherent manner during abooming event. Tactilevibrations polished. Occasional scour marks appear on both types of beach created bysmall-scale slumpings areoften compared with a sands, but not on booming sand. While squeaking grains are by minor electric shock (Criswell et al. 1975). and large rounded, booming sand contains avariety of erosional This eciency in converting mechanicalshearing energy grain states as shown in the bottom left panel, including many smaller, well-polished, well-rounded grains, as seen in the bottom into seismic vibrations suggests that booming mayalso be right micrograph. The top-right grain in the bottom left panel is responsible for the curious `moonquakes’,thought to highly unusual in booming sand. (Adapted from Non et al, 1997). 332 P. Sholtz et al.

Criswell et al.(1975)and Lindsay et al.(1976)suggested that ahigh degree of polishing alone is responsible for booming, and that allthe other physicalparameters typicallyassociated with booming sand simply serve to enhance granularsmoothing in desert environments. They arguethat verysmooth granularsurfaces willdecrease the amount of energylost during shearing asa result of low mechanicalcoupling between the grains.Su ciently smooth surfaces would allowalmost elasticcollisions, thus narrowingthe resonant frequency of whatis otherwise ordinary dry sand (which is ~80Hz (Ho and Burwash 1969),well within the rangeof most booming dunes). Although straightforward,this analysisdoes not take into account the largeamplitude of oscillation that grainsin the shear plane experience during abooming event. These vibrations arenearly as large as the grainsthemselves (Criswell et al.1975)and liewell outside the bounds of the linearanalysis (Wu 1971)used to model the elastic deformation of grains.Moreover, it appears that this amplitude represents only the lower bound of magnitudes attainableduring abooming event (Criswell et al. 1975). Any realisticmodel of booming must be based on nonlinear pressure vibrations. Booming dunes (shown in ®gures 3and 4)are often found atthe downwind end of largesand sources, which is alikely consequence of the need to optimize granularpolishing. Speci® cally,wind-driven saltations of sand (the bouncing of grainsupon agranularsurface) across the desert ¯oor Figure 2. Optical micrographsof sand grains from: anormal increasinglyrounds oŒgranular asperities the further the beach in Bay City, Michigan, (top left photo); the booming dune sand is blown (Sharp, 1966).Since some preliminary degree Sand Mountain,near Fallon, Nevada (top right); and a squeaking beach in Luddington, Michigan (bottom). Tungsten of rounding is essential before agraincan be polished, one illumination was used to produce the photos. The lengths of the would expect the probability of ®nding largeaccumulations bottom edges of the photos are: 2mm(top left and bottom right of highlypolished desert grainsto increase with distance panels), and 3mm(top right and bottom left panels). downwind from largesources of sand. Granular polishing

Figure 3. 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. (Photo by Rudy Bretz) Sound-producing sand avalanches 333

Figure 4. Views of Kelso Dune, California (Photos by Terrence Moore).

cantake placeeither during the long-range saltation variablesand source distance to Sand Mountain exists, as transport of desert sands, e.g.35 miles in the caseof the weobserved that particular booming sand to possess a booming Kelso dunes of California(Sharp 1966),or bya spectrum of polishing histories. suciently long residence time within the dune itself Booming dunes usuallyform far enough downwind from (Lindsay et al.1976).Examples of the latter eŒect arethe largesand sources to permit development of reasonably collections of back-beach booming dunes of the Hawaiian well-sorted grain-sizedistribution s(Lindsay et al. 1976, islands of Kauaiand Niihau, located no more than 300m Sharp, 1966).This fact has prompted much speculation as inland from their source beach sand (Lindsay et al. 1976, to how important certain grain-sizeparameters areto the Bolton 1889b).In this case,it is likelythat preliminary sounding mechanism. Highlysorted grain-sizedistributions rounding is brought about bythe relativelylong residence arein fact common, although byitself, this is not likelyto time of the grainson the beach itself, the result of limited determine the abilityof sand to boom. On the contrary, the sediment supply and slowoŒ -shore currents (Lindsay et al. booming sands of Korizo and Gelf Kebib, both in Libya, 1976).Substantial rounding is likelyto take placebefore the havebeen noted for their uncharacteristicallybroad range grainsare blown into the dunes behind the beach, where of particle sizes (Humphries 1966,Bagnold 1954b). More- subsequent polishing presumably occurs. Some combina- over,silent dune sand is often aswell-sorted asnearby tion of the abovearguments probably accounts for the booming sand (Lindsay et al.1976),and monodisperse (i.e. existence of the Jebel Nakus and BedawinRamadan identical-size) glassbeads havenever been observed to booming dunes of Egypt,located on the Sinai Peninsular boom. Booming is alsolargely independent of grainshape. 3km inland from the Gulf of Suez. We alsosuspect that a Close inspection of Sand Mountain (see ®gure 1(bottom, 334 P. Sholtz et al. left)) and Kalaharibooming sand (see ®gure 7in Lewis or steeply inclined asleeward sand, and hence does not (1936))reveals that not allgrains are highly spherical or shear spontaneouslyas easily, but when properly loosened rounded. Furthermore,quartz grainsin Dunhuang, China, up, it frequently emits sound just asreadily as leeward sand haveobtuse edges and irregularlyshaped pits distributed (HaŒ1979). For completeness,wenote that during the on their surfaces (Jianjun et al.1995).Also, Lewis(1936) unusually dry conditions of summer 1994,our tests atSand claimsto haveproduced booming in ordinary table salt, Mountain revealedbooming over much of the dune’s which has cubical grains. leewardsides, with the most robust emissions occurring The role that other grain-sizeparameters might playin the near its base. booming mechanism has alsobeen the subject of consider- Wind carryingairborne sand grainsacross the dune crest ableresearch. For instance, booming sand usuallycontains has agreatertendency to deposit the grainscloser to the anexcess of slightly® ner-sized grainsthan its averagegrain top of the leewardthan near its bottom. Consequently, size (Lindsay et al.1976).The asymmetrythat this condition sand accumulates faster in the upper portions of the creates in the particle-size distribution of the sand is called leewardslope than in the lowerportions, slowlyincreasing `®ne-skewness’.Humphries (1966),in particular, gave the anglethat the dune’sleadingedge makes with the substantial consideration to the role that ®ne-skewness horizontal. General slumping occurs when this angle might playin the sounding phenomena. However,since reaches ~348 ,the angleof dynamicrepose for dry desert nearlyany size-fraction of booming sand exhibits pro- sand (Bagnold1954d). Typically, large plate-like slabs of nounced acoustical activity(HaŒ 1986, Leach and Rubin sand break oŒalong clearly de® ned cracks near the crest. 1990, Miwa et al.1995),it is unlikely that the presence of a Especiallyin booming sand, it is possible that these cracks ®nely-skewed particle-size distribution alone directly aŒects form in regions of more symmetrical skewness, which are the sounding abilityof the sand. Booming is alsovery typicallyless resistant to shear (Humphries 1966).The sensitive to the addition of very® ne-sized fragments and plates themselves usuallyremain more or less intact until grains(on the order of 1 l min diameter), which seem to they getnear the base of the dune, where the changeto a disrupt the collectivegrain behaviour (HaŒ1986). gentler slope slows their slide (HaŒ,1979). Booming sand is often observed to boom best ator near An unusual, and asyet not fullyunderstood aspect of the leewarddune crest. Severalfactors maybe responsible booming sand is the manner in which these plates for this, including the fact that crest sand tends to be better subsequentlybreak apart.Rather than simply disintegrat- sorted and the grainsare more rounded and polished inginto loose ¯owupon hitting the gentler basalslopes, the (Lindsay et al.1976).Another important factor is that sand upper (trailing) portions of booming sand plates areat around the crest tends to dry most quickly. Although times seen collapsing,or telescoping, into the lower precipitation is rarein desert environments,when it does (leading) portions. HaŒ(1979) compares the appearance occur, sand dunes retain the waterthey absorb with of this eŒect to that of asheared deck of cards, and suggests remarkable eciency. Sand near the dune surface that it mayresult from distinct boundaries formed between (<20cm deep) dries oŒfairly quickly, since water shearing and stationary grainswithin the plate itself. evaporates oŒthe grainsinto the interstitial air,which, Furthermore,the free-¯ow of sand that ®nallydoes result when heated during the day,expands and carries the water from the `break-up’ is unusually turbulent, resembling à vapour out of the dune. At night, cold, dry air® llsthe rush of waterseen in slowmotion’ (Bagnold1954b). The granularinterstices and repeats the process until the surface connection between this ¯owphenomeno nand the boom- sand is completely dry.However, sand is apoor conductor ingmechanism is asyet not fullyunderstood .It is not clear of heat, and this temperature gradient rarelyreaches more from the literature whether this `stacking’eŒ ect and the than 20cm into the dune. Beyond that, with no mechanism subsequent turbulent motion occurs only when the sand is to driveit anywhere,interstitial aircan become completely booming, and hence is the result of sound propagation saturated with watervapour and remain trapped for years through the sand, or whether it alwaysoccurs, and is the (Bagnold1954c). Since booming sand tends to shear oŒin result of some unique grainpacking. On the other hand, no plates that areroughly 4inches deep (Humphries 1966), such rippling motion has been reported byCriswell et al. sounds occur in those parts of the dune which dry oŒthe (1975)and Lindsay et al.(1976),suggesting that this eŒect fastest. Nearthe leewarddune crest, the combination of maybe subtle or completely absent in some booming smooth, well-sorted grainsand the constant recirculation of dunes. More detailed ®eld observations areneeded. interstitial airresulting from the ¯owof wind over the crest helps promote complete granulardrying deep into the dune. Note alsothat although spontaneous acoustical 4.Piezoelectric properties of quartz and the booming activitynormally only occurs on leewardslopes, sand from mechanism the windwardside usuallypossesses comparable acoustic The piezoelectric properties of quartz crystalswere at one potential. Windward sand is usuallynot asloosely packed time thought to playa signi®cant role in the booming Sound-producing sand avalanches 335 mechanism, although asyet there has been no evidence to out, most of the `musical’sand which Bolton sampled was suggest that this maybe the casewith squeaking sand. It is actuallysqueaking beach sand (although he did use this wellknown that electricalpolarization arises when pressure model to explainbooming sands aswell, and did not seem is applied to both ends of certain axeswithin aquartz to attach much importance to the diŒerences between the crystal(Cady 1946). It has been proposed that the wayin two). However,concrete empirical evidence in support of which stress is applied to booming grainswhen sheared this theory has never been produced, and the fact that maycause anaccumulation of these tiny piezoelectric tactile vibrations observed in booming sand from Sand dipoles, which would then somehow be responsible for the Mountain, Nevada(shown on ®gure 3),at atmospheric pronounced acoustic output of the sand (Bagnold1954b). pressure areno diŒerent from those observed at Such speculation beganafter Lewis(1936) observed that 1.5mm Hgair pressure (Criswell et al.1975)eŒ ectively upon slowlypouring Kalaharibooming sand, grainswould discredits anair-cushion mechanism asthe cause of occasionallyadhere to one another so asto form ®laments booming emissions. To the authors’ knowlege,the last aslong asa half inch. An electroscope veri®ed that these paper published in support of Bolton’sair-cushion model ®laments did indeed exhibit electricalcharge. Furthermore , wasby Takahara (1965). It is,however, merely a if booming sand wasshaken inside aglassjar, a signi®cant rearmation that squeaking sand produces better acoustic number of grainswere observed to adhere to the sides of emissions immediatelyafter washing. the glass,where they remained for severaldays. The grains As wasstated atthe outset of this paper, it is likelythat werealso noted to clingmost densely atplaces where the the cause of the acoustic emissions is closelyrelated to the temporary surface of the sand had rested againstthe glass. frictional behaviour between the grainsduring shearing. However,this has alsobeen observed in normal sand. Carus-Wilson (1891),a contemporaryof Bolton, wasthe One could evenextend this reasoning to explainwhy ®rst to propose that intergranular frictional eŒects may booming is precluded bythe adsorption of verysmall create sound in certain types of `musical’(in fact squeaking) amounts of water,which could eŒectively interfere with the sands. He was,it seems, the ®rst to correctly conclude that necessary polarization of grains.Nonetheless, Lewis(1936) the grainsare in generalhighly spherical, well-rounded, wasable to demonstrate that grinding the sand had no well-sorted and unusually smooth, and postulated that eŒect on its acoustic output. Moreover, since booming acoustic emissions must result from collectively` rubbing’ occurs naturallyin the calciumcarbonate sand dunes of grainswhich exhibit these four properties. He, nonetheless, Hawaiiand Lewisclaims to haveproduced booming in drew criticism for not elaboratingon precisely how a sodium chloride crystals,this èlectricalconnection’ should frictional sounding mechanism might work. Most notable be considered tenuous atbest. amonghis critics wasthe physicist Poynting (1908),who showed that if the sound should result only from the natural vibrations of the grainsthemselves, frequencies of 5.Development of modern friction theories no less than 1megahertz could be produced. Working The earliest reports of booming sands weremade bydesert together, Poynting and J.J.Thomson (1922)made their nomads, who interpreted the noises assupernatural ghosts own attempt to extend Carus-Wilson’sreasoning by and demons (Curzon 1923).Similarly unorthodo xnotions, coupling it with the principles of granulardilatancy, as such asthat the soundings result from the eruption of put forth byReynolds (1885).Dilation is simply the subterranean volcanoes, persisted wellinto the late-nine- expansion in volume that agranularsubstance undergoes teenth century (Carus-Wilson 1888).At this time more when it deforms under applied shear. This principle canbe serious systematic investigations into the phenomena ®rst rephrased bystating that ®xingthe volume of agranular began.In 1889,Bolton (1889a),one of the ®rst to mass precludes its deformation.Afamilarexample of this extensivelystudy the phenomena, published his model of volume expansion is the dryingout, when stepped on, of àir-cushion’ theories. He proposed that the sounds result wetsand around one’sfoot. The pressure applied bythe from thin ®lms of adsorbed gasesdeposited on the grains foot creates adeformation in the sand, causingexpansion bythe gradualevaporation of water.The acoustic in the region immediatelysurroundin gthe placeof emissions would arisefrom the vibration of elasticair compression.This expansion, or dilation, is largeenough cushions, and the volume and pitch of the emissions would to cause temporary drainagefrom the compressed to the be modi® ed bythe surface structure of the grains expanded regions. themselves and extinguished bysmaller fragments and Poynting and Thomson (1922)reasoned that if aclose- debris in the sample. Given the importance assigned to packed granularsubstance consists of monodisperse,sphe- waterin this model, the reader might assume that Bolton ricalparticles, then the dilation caused byshearing should be wasconcerned exclusivelywith squeaking beach sands. (roughly) periodic in time. Such uniform variationsin Booming dunes existin extremelyarid desert regions which dilation, they conclude, arelikely related to the uniform receiveessentially no rainfallfor yearsat a time. As it turns oscillations which produce booming and squeaking. Speci- 336 P. Sholtz et al.

®cally,suppose ashearing stress is applied to one layerof Bagnoldintroduced the concept of `dilatation’,1 /k , of a such grains,causing it to slide over another layerof identical granularsubstance, and de® ned it to be the ratio of the grains.Following Reynolds (1885),they reasoned that this mean intergranular surface-to-surface separation, s, to motion requires some grainsto dilate,or rise out of their mean graindiameter, D.This ratio is qualitativelyidentical interstices, move over neighbouring grains,and then fall to the `dilation’ of Reynolds, but is easierto work with down into adjacentinterstices. So long asthe shearing stress mathematically.Since for most natural packingdensities, remains constant, successive expansions and contractions dilatation, asde® ned above,is far smallerthan unity, its should occur periodically.Needless to say,real grain ¯ owis inverse, the linearconcentration k = D/s,is generallyused. far more complex than speci® ed in this simple model. It is In the limit of closest possible packingdensities, the linear unrealistic to expect that the uniform size and spherical concentrationapproaches in® nity.Bagnold showed that shape of the grainsis enough to ensure such highly-ordered most granularmaterials remain fairlyrigid for linear behaviour in loose aggregates.In fact, sound-producing concentrations down to k = k 2»17,the point atwhich grainsare not, in general,perfectly monodisperse.Their dilatation becomes just largeenough to allowgeneral model alsoimplies that the frequency of emissions should slumping. Granular materialswith k <17begin to take on depend on grainsize and the rate of shear alone. This the properties of a¯uidized bed of particles. The system contradicts the fact that squeaking emissions areusually 5 to behavesas a non-Newtonian¯ uid for smallmean inter- 10times higher in frequency than booming emissions, even granularseparation, that is resistance to shear exists atzero though both types of grainsare usually ~300 l m in shear rate.Below some linearconcentratio n, k 3»14, the diameter. Also, if sand is alreadyshearing in athin layer grainsare too disperse to eŒectively transmit intergranular (as in most squeaking sand), the addition of shear stress at stress. The system then becomes aNewtonian ¯uid, losing the surface is more likelyto create newshear planes parallel allresistance to shear. to the existingone rather than increasingthe rate of shear at Pilingup sand to aboveits dynamicfriction angleresults the existingplane (and hence, bythe Poynting and Thomson in the shearing of athin layerof grainsnear the surface of model, increasingthe frequency of emission) (Ridgwayand the sand. Arepulsive stress in the plane of shear ( k »17) Scotton1973).Of course, not allof the added shearstress can results from the successive collisions of ¯owinggrains upon be accommodatedatnew shear planes, and some increase in stationary grains.Of particular interest is the stress frequency is bound to occur. But in neither booming nor component normal to the plane of shear, which is squeaking sanddoes frequency of emission varylinearly with responsible for sustaining the dilation in the ®rst place.In rate of shear (Lewis1936, Bagnold 1954b). This argument steady-state equilibrium ¯ow,this component is equal and would not applyto booming sand, which shears in rather opposite to the normal component of the compressive stress thick (~4inches) layers.Lewis (1936) nonetheless found due to the weightof the shearing grainsunder gravity. that quadrupling the rate of shear roughly doubled the However,Bagnold argues, should the shearing grainsattain frequency of emission and resulted in apronounced increase arelativeinterfacial velocity in excess of the system’s in volume. preferred velocity,without internal distortions occurring in the shearing layer,the shearing grainscould begin vibrating collectively.He reasons that alarge,sudden increase in 6.The Bagnold model of booming and squeaking emissions dispersive stress must be created to compensate for the In 1966the British engineer and ®eld commander R. A. increase in velocity.This creates dilation throughout the Bagnold,having already done alargeamount of work on entire sliding layer,raising it slightlyup oŒthe plane of granularmechanics in general,put forth the most complete shear. The dispersive stress itself in turn quickly decreases attempt atexplaining the booming mechanism to date. in magnitude asdilation increases, and the sliding grains According to Bagnold,both types of acoustic emissions soon collapse under their own weightback into the bulk result from nonlinear oscillations of dispersive stress along sand. This compacts the original k »17slip faceinto a the shear plane.The ideathat the cause of the sound may denser (k >17)packing, creating a new k »17slip plane result from disturbances in the shear plane itself stems from slightlycloser to the surface of the sand. This process experiments performed byBagnold (1954b) in which repeats so long asthe grainsare shearing. The expression booming wasproduced byshaking sand in ajar.He found 1 /2 ¸3g that acriticalamount of downward force must be applied f 5 8D to eachstroke in order to keep the sand sounding and estimated the magnitude of this force to be approximately wasderived for the frequency atwhich this saltation should equal to the weightof 8±10cm of sand; roughly the depth occur, where g is the localgravitational acceleration. The atwhich shearing planes form in booming sand. The weight Appendix summarizes the derivation of this relation for f. of the sliding grainsalone, it seems, exerts just enough force Although elegant,this analysisdoes not completely on the shear plane to sustain the oscillations. describe abooming event. In the ®rst place,a booming Sound-producing sand avalanches 337 sample with amean grainsize of 300 l mshould, according We too wereunsuccessful in gettingbooming sand to to Bagnold,boom ata frequency of ~240Hz; welloutside squeak. Acloser look atthe older literature maynone- the rangeof Korizo, Libya(50 ± 100Hz) and Sand theless suggest otherwise. For instance, in 1889Bolton Mountain (50± 80Hz). Equallyproblematic is that only (1889b)writes: one frequency is predicted. It is not clearhow four or ®ve separate modes of ground vibration, allwith diŒerent axes `The sand of the Hawaiianislands possesses the acoustic of vibration, could be created simultaneouslyfrom asingle, properties of both classes of places [beaches and deserts]; saltatinglayer of grains.Consideration of the low- it givesout the samenotes asthat of Jebel Nagous [an frequency (1± 10Hz) beats that typicallyaccompany Egyptianbooming dune] when rolling down the slope, prolonged ¯ows,and of the frequency modulation that and it yieldsa peculiar hoot-like sound when struck occurs byvarying the rate of shear, is alsonotably absent together in abag,like the sands of Eigg,Manchester, from Bagnold’sanalysis.Grain size maybe one component Mass., and other sea-beaches [squeaking sand].’ that ®xesthe frequency rangefor agivensample of booming sand, but there must be other factors that More recently, HaŒ(1979) has alsobeen ableto produce Bagnoldfails to take into account. Furthermore, other similarhigh-frequen cysqueaking using booming sand from experimental results (Leachand Rubin 1990,Leach and the Kelso dunes, both in situ on the dune aswell as in a Chartrand 1994)indicate that the frequency of emission for laboratory. The fundamental frequencies of these vibra- agivensize fraction of booming sand seems to decrease tions areclose to 1200Hz, implyingthat compression of linearlywith increasinggrain size, rather than decrease with booming sand does not just amplifyhigh-order harmonic the square root of grainsize asproposed byBagnold. frequencies of low-frequency fundamental tones (e.g.50 ± Bagnoldapplies asimilarline of reasoning in explaining 300Hz). This provides some support for Bagnold’stheory, squeaking, the primarydiŒ erence being that here a which implies that the only diŒerence between the two compressive stress givesrise to the sound, rather than a modes of emission arecompression alversus shear-induced generalslumping shear. This makes sense, since acoustic slumping. Subtle diŒerences do exist between booming emissions in squeaking sand aremost naturallycreated bya sand that squeaks and genuine squeaking sand, however. quick, sharp compression,like when walkingon it, rather For instance, frequency analysisby HaŒ clearly shows that than bygeneral shearing like avalanchingin dune environ- multiple fundamental frequencies arestill present in ments. Bagnoldproposes that dispersive stress should again squeaking emissions from booming sand. It is further be created in the shear planes in asimilarway as in interesting to note that all desert sands HaŒsampled were booming sand. However,since compression subjects the ableto produce some type of squeaking emission when grainsin the shear plane to K times more acceleration than compressed verticallyin acontainer. Bolton (1889a),on the they would experience during slumping, the emission other hand, noted that the sand of Jebel Nakus, an frequency should be K1/2 times higher. The constant K Egyptianbooming dune, could not produce such `squeaks’ variesamong diŒ erent types of granularsystems and when compressed. It is possible that since Bolton was depends on, amongother things, the areaof sand working in situ,adequate pressure might not havebeen compressed and the maximalangle of respose of the sand applied. Moreover, HaŒfound that silent beach sand does (Terzaghi1943). The frequencies emitted bysqueaking not necessarilysqueak when compressed.Most likely,the sand seem to ®tBagnold’smodel better than those of occurrence of squeaking in sand correspondsverydirectly booming sand (Lindsay et al. 1976). with grainshape and morphology. Most silent desert sand grainstend to be more spherical, rounded and polished than silent beach sands (Lindsay et al.1976).It would be 7.Comparisons between booming and squeaking sands interesting to see if table salt,which Lewis(1936) claims can There aremany qualitative diŒ erences between booming be made to boom, could alsobe made to squeak. We have and squeaking emissions. For instance, squeaking emis- observed some booming after sievingsilent sand. sions almost alwaysproduce only asinglefundamental The back-beach booming dunes of Hawaii,which diŒer frequency of vibration. The multiple-band phenomena from most other booming dunes in anumber of funda- observed in largebooming events almost never occurs. mental ways,provide aninteresting casestudy. In the ®rst Conversely,squeaking sand often produces four or ®ve place,they arecomposed primarilyof calciumcarbonate harmonic overtones (Takahara1973), while at most only grainsfrom seashells, and arethought to be the only one harmonic of the fundamental tone has been observed in naturally-occurring booming sands not made out of quartz. booming sand (Criswell et al.1975).Understandab ly,a The grainsthemselves areunusually large( ~460 l m) and consensus that squeaking sand never booms and that the frequency rangeof their emissions appears to be wider booming sand never squeaks has arisen in the literature than that of most other booming sands, although the latter (Criswell et al.1975,Lindsay et al.1976,Bagnold 1954b). point has yetto be precisely determined (Lindsay et al. 338 P. Sholtz et al.

1976).Also, they arethe only non-desert booming dunes grainsin the sample. Booming occurs best atvery low and seem to exhibit aslightlyhigher tolerance for water humidity, since humidity creates a¯uid surface coatingthat exposure than do other booming dunes (HaŒ1986). acts like alubricant and lowers the shear resistance. Nonetheless,the loose packingand the rough surface Humidity alsoincreases the cohesion between grains. pro® leof back beach dunes maybe whatkeeps them from Booming is enhanced when the grainsare loosely packed, squeaking. More comprehensiveinvestigations of the since averytight packingwill preclude shearing.EŒ ects Hawaiiandunes areneeded. due to collectiveor resonant motion arealso considered to The speculative ideathat booming and squeaking are be important. However,these collectiveeŒ ects arestill not produced bythe samemechanism, the output of which is wellunderstood, in spite of research work on collective modulated either bycertain intrinsic grainparameters or by motion in granularmaterials and related systems (see, e.g. the method of shearing used to produce the sounds, is Bretz et al.1992,Benza et al.1993,Jaeger and Nagel1992, appealingand warrantsfurther investigation.We choose Liuand Nagel1993, Bideau and Hansen 1993,Satake and here to remain with convention and treat the two types of Jenkins 1988).How these ingredients mixtogether to emission asif they wereproduced bytwo distinct sounding produce booming is still anopen problem. The actual mechanisms. mechanism has been the subject of speculation for centuries and still remains unsolved. Manyavenues of investigation remain open. Determina- 8.The importance of shearing tion of the mineral composition of the grains,particularly It seems unlikely that booming and squeaking sand grains, atthe granularsurface, has only recently been attempted ² . which both exhibit mean graindiameters of ~300 l m and Shearometer analysisand X-raystudies of mass density appearsomewhat similarin electron micrographs, could ¯uctuations during shear would alsobe of interest. produce such substantially diŒerent modes of acoustic Examinationof the possible piezoelectric properties of emission simply on the basis of some intrinsic grain booming sands, alluded to earlier,has alsobeen minimal, parameter. Instead, the waythe sand is sheared probably ashave been attempts to create synthetic booming grains. determines the frequency of emission, especiallyin boom- Agood understandingof sound-production in loose ingsands. Increasing the rate of shear seems to increase its aggregatesmight help our understandingof grain¯ owin frequency of emission. Quickly compressing the booming general. sand vertically,thus creatinga veryhigh rate of shear, reportedly produces emissions which resemble genuine squeaking. Appendix.Derivation of the equation for thefrequency of While the two modes of acoustic output from sand may vibrationof booming sand be inherently connected and mostly governed bythe In 1906,Einstein (1906)considered the eŒect of solid method of shearing,the actualonset of acoustical activity grains,immersed in a¯uid, on the shear resistance of that in asand sample probably is controlled bycertain intrinsic ¯uid. Halfa century later,Bagnold (1954a, 1966) con- grainparameters. The most criticalparameter that governs sidered aregimethat wasapparently neglected atthe the abilityof sand to boom seems to be ahigh shear border between hydrodynamics and rheology, namelythe resistance. Both varietiesof sound-producing sand possess caseof aNewtonian ¯uid with ahigh concentrationof large alargefraction of smooth grains,and booming sand grains solid spheres. In this appendix, wesummarize Bagnold’s smallerthan ~200 l mtend to be verysmooth. In addition, derivation of the equation for the frequency of vibration squeaking sand is typicallywell-sorted. Monodisperse glass that occurs during abooming event. spheres, which exhibit alower shear resistance than any First consider the slumping, under the force of gravity,of type of sand, cannot boom, and squeak only under dry sand that has been piled up to its criticalangle of somewhat contrived experimental conditions. Ahighly- repose. Grain ¯owoccurs alonga well-de® ned front and sorted collection of smooth, well-rounded spherical grains proceeds with aconstant velocity.One would like to be able is thus not asucient condition for sound-production in to express this terminal velocityas a simple function of such sand. variablesas the mean grainsize and the depth of ¯ow. Bagnold’sequation for the frequency of vibration that occurs during abooming event is derived from the analysis 9.Conclusions of the terminal velocityof the ¯owingfront. In summary, important physicalproperties that distinguish booming sands from their silent counterparts are:high surface smoothness and high-resistanceto shear. Other ² See,for instance,Leach and Chartrand (1994). Takahara (1973) does factors that contribute arethe roundedness, the sphericity presenta smalltable of per cent weight by mass analysis of the chemical of the grainsand diŒering roughness between the larger compositionof squeaking grains. Sound-producing sand avalanches 339

TwodiŒ erent experiments wereconducted byBagnold (ii) Frictional losses maintain aconstant kinetic to arriveat his result. In the ®rst, acollection of energy per unit volume ofthe system. uniformly-sized spheres wasplaced into animmersing (iii) In addition tothe general drift in the x-direction, ¯uid. The spatialdistribution of spheres waskept uniform the spheres also make small oscillations in all three byexperimentin gunder `gravity-free’conditions, where directions. the density of the grains, r ,wasbalanced againstthe density of the immersing ¯uid, q .In order for uniform Now,consider oscillations produced bya sequence of shear strain to be applied to allthe spheres, the spheres smalljumps or saltations asthe spheres of one layerjump weresheared inside the annular space between two over the neighbouring sphere layer.Since the `gravity-free’ concentric drums. This allowedfor the accuratemeasure- conditions in principle provide auniform spatialdistribu- ment of the intergranular stresses and strains that occur tion of spheres, the spheres canbe treated asbeing arranged during shearing.In the second set of experiments, these into sheets lyingparallel to the x ± z plane,with one plane results wereextrapolated to model the dynamics of dry shearing across the top of the layerimmediately below it sand avalanches,where the grainsare not uniform in size, (see ®gure 5( b)). Let us designate the top layeras B, and nor is the interstitial ¯uid (air) of the samedensity asthe the bottom layeras A. Then the mean relativevelocity of a grains. sphere in layerB with respect to plane Ais d U = kbD(dU/ We now turn our attention to the theoretical considera- dy), where k is aconstant that variesfrom 1 /21/2 to (2/3)1/2 tions involvedin the ®rst experiment (Bagnold,1954a). depending on geometry (Bagnold1966). In this model, the Suppose that wehave a collection of uniformly-sized spheres in plane Acanbe thought of asconstituting arigid spheres, with diameter D.Clearly,in the closed-packed plane, across which the spheres in layerB saltateat the limit, the distance that separates the centres of two adjacent diŒerential velocity d U. spheres is D.Recallthat the linearconcentratio n, k = D/s, is de® ned to be the ratio of the mean graindiameter D to the mean intergranular separation s.For the caseof equal spheres, k approaches in® nity since s is verysmall. Also, the volume fraction C of space occupied bythe grainsbecomes 1/2 C0 = p /3(2) »0×74for the caseof close-packed equal- sized spheres. Next consider whathappens when the spheres are uniformly dispersed, so that the mean distance separating two adjacentcentres becomes bD, where b>1 (see ® gure 5 (a).In this case,the mean intergranular separation s, as measured from the surface of the grains,is largerthan the s of the close-packed equal sphere arrangement s b 1 1. (A 1) 5 D This result canbe expressed in terms of the volume fraction C C C C 0 0 A 2 5 3 5 3 . ( ) b (¸2 1 1 1) The smaller k is, the easierit is to induce shearing.In general,shearing becomes possible for valuesof k < 17. Moreover, Bagnoldmade the following assumptions about the ¯owof spheres and the shear that occurs:

(i)The spheres are in auniform state ofshear strain, = (dU/dy) »0; and the mean relative velocity between the spheres and the interstitial ¯uid is Figure 5. (a)Schematic diagram of two layers, Aand B, of zero everywhere (see ®gure 5( a)). U(y) is the grains moving along the `downhill’ x-direction. The average velocity ®eld ofthe granular material whose intergranular separation s,and the average distance between two average motion is along the `downhill’ x-direc- adjacent centres, bD,are indicated. ( b)Schematic illustration of tion. agranular ¯ow and its velocity pro® le. 340 P. Sholtz et al.

Suppose that on average,any given sphere in layerB which implies that makes f(k )d U/s collisions per unit time with layerA. The 1/2 Ð 2 0 number of spheres in aunit areaof the Bplane is ( bD) , 1 2 C(y ) dy dU g sin b / y ¢ ¢ and with eachcollision asphere in layerB willexperience a . (A 10) dy 5 r sin ¸D changein momentum 2 md U cos a in the y-direction (the a angle a is determined bycollision conditions, and willbe Makingthe reasonable assumption that C is roughly discussed in more detail shortly). This implies that anet uniform through the depth of the ¯ow,and that it has a repulsive pressure Py should existbetween the two layers, value of C = 0×6(obtained empiricallyby Bagnold), the with magnitude integralin equation (A 10)reduces to 2 2 f(¸) d U 0 Py (bD) 2m d U cos (A 3) 5 s a C dy¢ 5 0.6 y . (A 11) y | | 2 2 dU Finally,taking k =17at the shear plane, and r sin a Py 5 rr ¸ f(¸)D cos a. (A 4) dy = 0×076,equation (A 10)then reduces to: with dU y1 /2 . b 1/2 2mk2 5 0 165(g sin ) . (A 12) r (A 5) dy D 5 r D3 Thus, Also, there is acorrespondingtangentialshear stress Tx y 3 /2 given by 2 1 2 y U (0.165)(g sin b ) / 0 , (A 13) 5 3 D Txy 5 Py tan a. (A 6) where y0 is the overalldepth of the ¯owand y = y0 refers to Indeed, Bagnold’sexperiment found that, atsu ciently the top surface. high speeds, both Tx y and Py become proportionalto (d U/ We canfurther simplify the aboveexpression bymaking dy)2.The only unknowns in the aboveequation are f(k ) and the following assumptions about the ¯ow:for the relatively a .Bagnold’sexperiment (Bagnold1966) with uniform high concentrations that weare working with here, the spheres sheared inside arotating double-cylinder found relativeinterfacial velocity U atany shear surface canbe that in the casewhere the spheres aresu ciently far apart substituted for the expression D(dU/dy). This simpli® es the for the materialas a whole to take on the properties of a expression for the pressure: Newtonian liquid, k <12, f(k ) »k , tan a »0×32 and P 5 rar ¸2U2 cos a. (A 14) r = 0 042.Thus for k 12,we can write × < Also, for slowcontinuing shear, wecan take k 17 for the dU 2 » P 0.042 r ¸2D2 cos . (A 7) locallinear concentration ,giving: 5 dy a 1 P 1/2 At this point, there is no a priori reason to suppose that Uc . (A 15) 5 17 rr cos these results should be applicableto dry sand avalanches, a since silicon dioxide does not havethe samedensity asair, Havingperformed the 1954experiments, Bagnoldnext nor aregrains of sand uniform in size or shape. Never- needed to verifythat these results for U and P apply to theless, let us now make the appropriate substitutionsand actualsand avalanches,and not just in the idealcase of try to derivea generalexpression for the terminal velocity uniform spheres in arotating drum (1966).In order to of ¯owusing known grainparameters. On any y = con- verifythese results, he performed abulldozing experiment stant plane below the upper, free surface of the sand, the in which aheap of sand waspushed, ata constant depth h0 applied shear stress is: below the surface, bya push plate.The results he obtained 0 indicated that the aboveequation for terminal velocityof Txy 5 r g sin b C(y ¢) dy ¢ (A 8) ¯owalso holds for dry sand shearing. y In order to explainthe frequency of emission that occurs with b being the angleof incline and C the volume fraction in abooming dune, ®rst consider the force Q,which is the (equation (A 2)). Since airhas averylow viscosity, we can normal compressive stress on the shear surface due to the applythe aboveresults, and equate expressions (A 7)and weightof the sand aboveit. In equilibrium, anyincrease in (A 8). Thus U in excess of Uc requires adilatation increase atthe shear dU 2 0 surface, so that k fallsto some valuesmaller than 17. rr ¸ f(¸)D2 sin r g sin b C(y ) dy , If the velocityof ¯owdoes momentarily exceed the dy a5 ¢ ¢ y terminal velocity,then P willbrie¯ y exceed Q,and the sand (A 9) mass is acceleratedslightly upward. However,the upward Sound-producing sand avalanches 341 stress P decreases rapidlyas the dilatation increases. The Clarke, J. A. R. C., 1973, `Singing Sands’, New Sci., 59, 222. sand would then collapse back under the weightof gravity, Criswell, D.R., Lindsay, J. F., and Reasoner, D.L., 1975, `Seismic and Acoustic Emissionsof aBooming Dune’, J.Geophys.Res. , 80, 4963. decreasing the dilatation, and againcausing P to tempora- Curzon, G.N., Marquessof Kedleston, 1923, Talesof Travel (New York: rily exceed Q. GeorgeH. Doran), pp. 315± 398. If the mass of ¯owingsand is m,then it would be subject Darwin, C., 1889, Voyageof H.M.S. BeagleAround the World (London: to the oscillatingforce mgÐ P,which would create Murray). oscillations in the normal direction. Also the minimum Dunnebier, F.K., and Sutton, G.H., 1974, `ThermalMoonquakes’, J. Geophys.Res. , 79, 4351. mean localdilatation atthe shear surface, atwhich Einstein, A., 1906, `Zur Theorie der Brownschen Bewegung’ Ann. Phys. oscillations could still occur, is D/k min. Leipzig, 19 (4), 371. The stress P is only eŒective when the dilatation, k , is Folk, R. L., and Ward, W. C., 1957, `BrazosRiver Bar:A Study in the near its minimum, because the contact faces (of the sliding Signi® cance of Grain Size Parameters’, J.Sediment. Petrol. , 27, (1), 3. planes) just clearone another and since P veryrapidly HaŒ,P. K., 1979, `Booming Dunes of the Mojave Desertand the Basin and Range Province’, Internal Report: PHY76-83685 (California Institute varies with k .Thus, the rise and fallof the overburden of Technology ,CA: Pasadena), unpublished. through the distance D/k min willbe analmost free-fall. HaŒ,P. K., 1986, `Booming Dunes’, Amer. Sci., 74, 376. Hence, the mimimum frequency of oscillation is givenby Hashimoto M., 1951, `Some Propertiesof Singing Sand’, Proceedingsof the FirstJapan National Congressfor Applied Mechanics , 261. g¸ 1 /2 f min . Herbert,F., 1984, Dune (NewYork: Putnam). 5 8D Ho, M. M. K., and Burwash,W. J., 1969, `Vertical Vibration of aRigid Foundation Resting on Sand’, Vibration EŒects of Earthquakeson This expression provides anestimate of the vibration Soilsand Foundations ,ASTM STP450 (Philadelphia, PA: American frequency that occurs during abooming event. Using Society for Testing and Materials), p. 197. Humphries, D. 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Ridgway, K., and Scotton, J.B., 1973, `Whistling Sand Beachesin the the adsorption research community. He ispre- British Isles’, Sedimentology , 20, 263. sentlyProfessor ofPhysics atthe Universityof Satake, M., and Jenkins, J. T., (editors), 1988, Micromechanics of Granular Michigan, Ann Arbor.In addition toongoing Materials (Amsterdam: Elsevier). research on criticalphenomenon, he performs Sharp, R. P., 1966, `Kelso Dunes, Mojave Desert,California’, Geol. Soc. granular materialsand earthquake studies,and Amer. Bull., 77, 1045. Takahara, H., 1965, `Frequency Analysis of Squeaking Sand’, J. Acoust. pursues carbon nanotubule development. Soc. Amer., 39 (2), 402. Takahara, H., 1973, `Sounding Mechanism of Singing Sand’, J. Acoust. ProfessorFranco Nori obtained hisPhD in 1987 Soc. Amer., 53 (2), 634. fromthe Universityof Illinois at Urbana- Terzaghi, K., 1943, Theoretical Soil Mechanics (NewYork: Wiley), p. 118. Champaign.Afterwards, he became apostdoc- Wu, T.H., 1971, Soil Mechanics (Boston: Allyn and Bacon), p. 272. toralresearch fellowat the Institutefor Theore- ticalPhysics ofthe Universityof California, Paul J.Sholtz obtained BS degrees (Honors)in Santa Barbara. In 1990,he joined the facultyof Physics and Mathematicsin 1996 fromthe the Physics Departmentat the Universityof Universityof Michigan, Ann Arbor.As an Michigan, wherehe isan AssociateProfessor. He undergraduate, he did research on sound-produ- has done research in severalareas of condensed cing sand. He ispresently developing softwareat matterphysics, including: collectivetransport in Omix,a software® rmin Silicon Valley,Califor- disordered systems,dynamical instabilities,ava- nia. lanches, vortexdynamics in superconductors, hole dynamics in antiferromagnets,quantum ProfessorMichael Bretz obtained hisBS degree in interferencedue toelectron motion in magnetic 1961from UCLA and hisPhD in 1971from the ®elds,superconducting networksand Josephson Universityof Washington. Hislow temperature junction arrays,quasicrystals, acoustic interfer- physics thesis,entitled `Submonolayer Helium ence, squeezed phonons and other formsof Filmson Graphite’,alsointroduced `Grafoil’to quantum noise manipulation in solids.