Illustrated Physics
physics, drawn
Fluids

The sound sand carries faster the deeper it goes

A solid carries sound at a speed set by what it is made of. A heap of quartz grains is made of quartz, and near its surface it carries sound more slowly than the air above it. The grains are not the solid; their contacts are, and a contact between two spheres grows stiffer the harder it is pressed. So the speed of sound in dry sand rises as the sixth root of the load on it — from about air's speed a few centimetres down to twice that at six metres — the paths of sound in a beach curve back up to the surface, and a water table is a mirror to one kind of wave and almost invisible to the other.
15 min read 5 figures The shape decidesWhat happens at the edge

Assumes: The heap that becomes a solid · The silo that does not weigh what it holds

The heap that becomes a solid found that sand changes from something that pours to something that holds a load when the number of contacts per grain crosses a threshold: below it the packing can rearrange without deforming any grain, above it every rearrangement must squeeze a contact. The silo that does not weigh what it holds found that the load in a column of grains does not grow with depth the way a liquid’s pressure does, because friction hands it to the walls. Both arguments asked whether grains carry load and where it goes. Neither asked how stiff the resulting solid is — how hard it pushes back when squeezed a little more, which is what sets how fast a disturbance travels through it.

The answer turns out to depend on the load itself, so strongly that the speed of sound in a sand pile is a measure of how much weight is resting on it. Near the surface of a dry beach, sound travels through the sand no faster than through the air above it. Six metres down it travels twice as fast. The reason is in the geometry of two spheres pressed together.

A contact that stiffens

When two elastic spheres are pressed together they do not touch at a point. They flatten against each other over a small circle, and as the force grows the circle grows. Heinrich Hertz worked out the shape of the contact in 1882, while still a student, from a question about glass lenses pressed together: the force rises as the three-halves power of how far the spheres’ centres have approached, F∝δ3/2F \propto \delta^{3/2}. The stiffness of the contact — the extra force needed per extra approach — is therefore not a constant like a spring’s. It is dF/dδ∝δ1/2∝F1/3dF/d\delta \propto \delta^{1/2} \propto F^{1/3}.

A contact that stiffens as it is pressed. The stiffness of the contact between two quartz spheres half a millimetre across — the force per unit of further squeezing — against the force pressing them together, on logarithmic scales, from Hertz's theory of elastic contact. The spheres touch over a circle that grows as they are pressed, so the contact stiffens: as the cube root of the load, slope 0.333 on the drawing. A micronewton of load gives 1.2·10⁴ N/m; a millinewton, 1.2·10⁵; a newton, 1.2·10⁶. Dashed, a contact between rough grains that touch first at a conical asperity, whose stiffness rises as the square root of the load. A pile of grains is a network of such contacts, and its stiffness — and so its speed of sound — inherits the exponent.
Fig. 1 The stiffness of the contact between two quartz spheres half a millimetre across against the force on it, from Hertz’s theory: 1.2 × 10⁴ N/m at a micronewton, 1.2 × 10⁵ at a millinewton, 1.2 × 10⁶ at a newton — the cube root of the load. Dashed: grains touching at a conical roughness, stiffening as the square root.

A thousandfold increase in load stiffens a contact tenfold. Rough grains, which touch first at the tips of small asperities, stiffen faster: a cone pressed onto a flat has a contact that grows linearly with the approach, and its stiffness goes as the square root of the load. Either way the contact is soft when lightly loaded and hard when heavily loaded, and that is the property a heap inherits. The same law, applied to a row of steel balls in a Newton’s cradle, is what decides how the impact passes from ball to ball.

From contacts to a medium

A packing of grains under pressure PP carries the pressure as forces through its contacts. In a random packing with CC contacts per grain and a fraction 1−ϕ1-\phi of space filled, the average contact force is proportional to PP times the grain’s cross-section, divided by the number of contacts sharing it. Each contact’s stiffness goes as the cube root of its force, and adding up contacts in a box, as the effective-medium theory of Hertz–Mindlin does, gives bulk and shear moduli that go as the cube root of the pressure:

K=[C2(1−ϕ)2μ2P18π2(1−ν)2]1/3,K = \left[\frac{C^2(1-\phi)^2\mu^2 P}{18\pi^2(1-\nu)^2}\right]^{1/3},

with μ\mu and ν\nu the grains’ own shear modulus and Poisson ratio, and a similar expression for the shear modulus with a slightly larger numerical factor. A wave’s speed is the square root of a modulus over a density, and the density does not change. So the speed goes as the sixth root of the pressure.

The contacts doing this are tiny. Six centimetres down in a dry sand of half-millimetre grains, the average contact carries about a sixth of a millinewton, and Hertz’s formula puts the radius of the flattened circle at about seven-tenths of a micrometre — a thousandth of the grain. Six metres down the force is a hundred times larger and the circle about three micrometres across. Everything the packing does elastically happens in these specks, and the grains between them are, for the purpose of a passing wave, rigid lumps of mass connecting one contact to the next. That is why a medium made of one of the stiffest common minerals can be as soft as a sponge: almost none of the mineral is being squeezed.

The speed of sound in dry sand rises with the load on it. The speeds of compressional and shear waves in a dry random packing of quartz grains, porosity 0.36, against the pressure on it, from the Hertz–Mindlin effective-medium theory, for coordination numbers from 6 to 9 (the bands). The pressure axis is logarithmic. Both speeds rise as the sixth root of the pressure. At 1 kPa — the load about six centimetres down in dry sand — compressional waves travel at 316–362 m/s; at 100 kPa, about six metres down, 681–780 m/s; at 10 MPa, 1468–1680. The dashed lines are the speed of sound in air, 343 m/s, and in water, 1,480 m/s. Near the surface dry sand carries sound no faster than air; measured shallow speeds are lower still, by up to a factor of two, because real contacts are fewer and weaker than the theory assumes.
Fig. 2 Compressional and shear wave speeds in a dry quartz packing of porosity 0.36 against pressure, for coordination numbers 6 to 9: 316–362 m/s at 1 kPa, about 6 cm deep; 681–780 at 100 kPa, about 6 m; 1,468–1,680 at 10 MPa. Dashed: air at 343 m/s and water at 1,480.

Quartz itself carries compressional waves at about six thousand metres a second. A packing of quartz carries them at a few hundred. At the load six centimetres below the surface of dry sand, about a kilopascal, the model gives three hundred and something metres a second, the speed of sound in air. Six metres down, under a hundred kilopascals, the speed has doubled. Measurements in dry beach sand made with hammer blows and buried geophones around the year 2000 found speeds in the top metre even lower than this, between about a hundred and three hundred metres a second, rising roughly as a power of depth. The theory gets the exponent right and the magnitude high, because real contacts are fewer, smaller and more irregular than the average packing it assumes.

Sand scorpions in the Mojave Desert hunt by feeling the surface waves a burrowing insect sends through the sand, and the waves they use travel at a few tens of metres a second — slower than a car on a motorway. A shear-dominated wave in the loosest few centimetres of a sand surface is about as slow as anything elastic gets.

The medium with no sound at all

The sixth-root law has a startling limit. As the pressure falls to zero, the stiffness of every contact falls to zero and so does the speed. A packing of grains under no load has no linear speed of sound whatever: an infinitesimal disturbance does not propagate at any fixed speed, because there is nothing to push back against it until it has pressed the contacts.

Vitali Nesterenko called this a sonic vacuum. In 1983 he showed that a chain of touching but unloaded elastic beads carries a disturbance as a single compact pulse, a solitary wave a few beads long, whose speed depends on its own strength: the harder the push, the stiffer the contacts it creates and the faster it goes, as the sixth root of the force it carries. The pulse is entirely nonlinear, there is no wave of small amplitude for it to be a large version of, and it was soon seen in chains of steel and glass beads struck at one end. A pile of sand near its free surface is a three-dimensional relative of that chain, and the shallow, soft layer where the load approaches zero is why the speeds measured there are so low and so variable.

A water table that sound sees

Below the water table the pores are full of water, and the two kinds of wave respond to that in opposite ways.

A water table that sound sees and shear does not. Compressional and shear wave speeds against depth in quartz sand with a water table 2 m down, from the Hertz–Mindlin moduli for coordination 7.5 and, below the table, Gassmann's relation for pores filled with water. Above the table both speeds grow as the sixth root of depth. At the table the compressional speed jumps from 611 to 1737 m/s, because water, nearly incompressible, now resists squeezing the pores; the shear speed falls slightly, from 430 to 391 m/s, because water adds weight and no resistance to shearing. So the ratio of the two speeds leaps from 1.42 to 4.45 at the table: a compressional wave reflects strongly there and a shear wave passes almost unaware of it. Below the table the grains carry less load, being buoyed by the water, and the shear speed grows more slowly with depth.
Fig. 3 Compressional and shear wave speeds against depth in quartz sand with a water table 2 m down. At the table the compressional speed jumps from 611 to 1,737 m/s; the shear speed falls from 430 to 391; their ratio leaps from 1.42 to 4.45. Dashed vertical: sound in air.

A compressional wave squeezes the pores as it passes. In dry sand the air in them offers no resistance, and only the contacts push back. With water in them, the water must be squeezed too, and water is nearly incompressible. Fritz Gassmann worked out the combined bulk modulus in 1951, and it jumps at the table from tens of megapascals to gigapascals. The compressional speed in the model rises from about six hundred to over seventeen hundred metres a second, faster than sound in water itself, since the grain framework adds its stiffness to the water’s.

A shear wave does not change the volume of the pores, and water, having no resistance to shearing, adds nothing to the shear stiffness. All it adds is weight, and the shear speed falls slightly. Below the table the water also buoys the grains, taking part of the load off their contacts, so the shear speed grows more slowly with depth there.

The result is one of the plainest signatures in shallow geophysics. The ratio of the two speeds is about 1.4 in dry sand and leaps past four at the table. A compressional wave meeting the table sees a sharp jump in the product of speed and density that decides reflection and is strongly reflected; it also travels along the top of the faster saturated layer, and arrives first by going the long way at a distance from the source, which is how refraction surveys map the depth of the water table from the surface. A shear wave passes the table with hardly a reflection. Engineers who need to know the stiffness of the sand — whether it will settle, or liquefy in an earthquake — therefore measure shear waves, which see the grains, rather than compressional ones, which below the table mostly see the water.

When the load is taken away

Below the water table the load on the grains is not the weight of everything above them. The water pushes back on every grain, carrying part of the overburden itself, and the grains carry only the remainder — the effective stress, in the language Karl Terzaghi introduced to soil mechanics in the 1920s. It is the effective stress that presses the contacts, so it is the effective stress that sets the shear speed, and a sand whose pore water is pushed up to a higher pressure carries less load through its contacts and becomes slower, softer and weaker at once.

Earthquakes do exactly that to loose, saturated sand. Shaking makes the grains try to settle into a denser packing; the water in the pores cannot escape fast enough to let them; the pore pressure rises, and the effective stress falls. When it reaches zero the contacts carry nothing, their stiffness vanishes, the shear speed falls to zero, and the ground that was a solid flows like a liquid. Buildings tilt into it, buried tanks float up through it, and sand boils erupt at the surface. Liquefaction is the sonic vacuum reached from below: the sixth-root law followed down to a load of nothing.

The same dependence makes shear waves the engineer’s instrument for the opposite question. Small piezoelectric benders pushed into a soil sample, or seismic sources at the surface with geophones down a borehole, measure the shear speed, and from it the effective stress and the stiffness that a foundation will meet. Empirical laws used in geotechnical practice for decades put a sand’s small-strain shear modulus at about the square root of the effective stress — a shear speed rising as the quarter power — the same rough-contact exponent the laboratory finds, arrived at independently by engineers who needed a number.

Rays that come back

A speed that grows with depth bends sound back towards the surface, as surely as a temperature gradient over hot ground bends light into a mirage.

Sound that curves back up out of the sand. Ray paths of sound from a source at the surface of dry sand, in a medium whose wave speed grows from zero as the sixth root of depth, for rays that turn at 0.5, 1, 1.5 and 2 m; depth runs downward and the two scales are equal. By Snell's law a ray's sine of angle over speed is fixed along it, so each ray bends away from the faster sand below, turns where the speed has risen enough, and returns to the surface. Because the speed is a power of depth the pattern has no length of its own: every ray is a scaled copy of every other, and returns 5.89 times its turning depth from the source, whatever the sand's moduli. The geometry needs only the exponent.
Fig. 4 Ray paths from a source at the surface of dry sand whose wave speed grows as the sixth root of depth, for rays turning at 0.5, 1, 1.5 and 2 m, on equal scales. Every ray is a scaled copy of every other and returns to the surface 5.89 times its turning depth from the source.

By Snell’s law the sine of a ray’s angle divided by the local speed stays fixed along the ray, so a ray heading down into faster sand tilts steadily towards the horizontal, levels off where the speed has grown enough, and curves back up. The ray that bends without a surface found the same arithmetic in air over a hot road; here the medium does it to sound, and it does it everywhere at once.

The figure carries a less obvious point. When the speed grows as a power of depth, there is no length in the problem: double every distance and the speed at each point is multiplied by the same factor, which Snell’s law does not notice. So every ray is a scaled copy of every other, and each returns to the surface a fixed multiple of its deepest point away from the source — 5.89 for the sixth-root law. A survey that measures how the arrival time of sound grows with distance along the surface is therefore measuring the exponent, and it can do so without knowing the grains’ moduli, the coordination number, or the prefactor that the theory gets wrong. Shallow seismic surveys of sand fit exactly that.

Sound in the deep ocean does something similar for a different reason, trapped in a channel with no walls by a minimum of speed at a kilometre’s depth. In sand there is no minimum, only a speed that rises from the surface down, so energy is not trapped but returned: a beach throws the sound of a footstep back up to the surface a few metres away.

The exponent is the measurement

How much faster sound gets, by kind of contact. The compressional wave speed relative to its value at 100 Pa, on a logarithmic scale, against pressure, for a packing of smooth spheres (Hertz, speed as the 1/6 power of pressure), of grains touching at rough conical asperities or gaining contacts as they are squeezed (1/4 power, which is closer to what dry sands show at low pressure), and for an uncracked solid, whose speed hardly changes with pressure. Over five decades of pressure the Hertzian packing speeds up 6.8-fold, the rough one 17.8-fold. A solid's stiffness belongs to its material; a packing's belongs to its contacts, and contacts that grow under load make a medium whose sound speed is a gauge of the load it carries.
Fig. 5 The compressional speed relative to its value at 100 Pa against pressure, for smooth Hertzian contacts (1/6 power), rough contacts or contacts multiplying under load (1/4), and an uncracked solid, nearly flat. Over five decades of pressure the first speeds up 6.8-fold, the second 17.8-fold.

Dry sands measured in the laboratory at low pressures usually show speeds rising faster than the sixth root — often close to the quarter power. Two explanations compete, and the experiments do not cleanly separate them. One is roughness: grains touch first at asperities whose stiffness rises as the square root of the load, so the packing’s speed rises as the quarter power until the load is large enough to flatten them. The other is that the number of contacts is not fixed: a lightly loaded packing is close to the threshold at which it became a solid, many of its grains are only barely held, and pressing it brings new contacts into play, each adding stiffness. Near that threshold the theory of jamming predicts that the shear modulus vanishes faster than the bulk modulus, because a barely rigid network can be sheared by rearranging without compressing anything — so shear waves in loose sand are slower relative to compressional ones than the Hertz–Mindlin model allows.

A solid rock without cracks shows almost no change at all: its stiffness belongs to its mineral, not to its contacts. The pressure dependence of sound speed is how a geophysicist tells a granular medium from a solid one at a distance.

What the average packing hides

The effective-medium model treats every contact as average: the same force, the same orientation statistics, the same stiffness. Real packings are not like that. Forces in a pile are carried along chains of strongly loaded contacts, with weakly loaded grains in between, and the distribution of contact forces is broad and roughly exponential. A sound pulse sent through a box of glass beads, in experiments by Chu-heng Liu and Sidney Nagel in 1992, arrived with an amplitude that changed by large factors when a single bead was moved or the box warmed by a fraction of a degree, because the chains carrying the sound were reorganised. The coherent wave the figures draw is the long-wavelength average; at wavelengths comparable to a few grains the transmission is a fluctuating, scattered signal following whichever chains are loaded.

The model also assumes contacts that do not slip, a single grain size and shape, clean dry grains, and a pressure equal in every direction. Real sand has a range of sizes and angular shapes, slides at some contacts under shear, holds films of water at its contacts in all but the driest conditions — which add a cohesive force and a stiffness of their own — and is usually loaded more vertically than horizontally, so that waves travel at different speeds in different directions. In a container, the load stops growing with depth past a few widths, exactly as the silo essay found, and the sound speed stops growing with it. And the water-table figure assumes the water fills the pores completely; a few per cent of trapped gas lowers the compressional speed almost back to its dry value, which is why marine sediments with gas in them are so hard to see through.

The domain of the argument is dry or fully saturated granular packings under a confining load, at wavelengths many grains long, where the contacts stay elastic. Within it the sixth-root law and its quarter-power cousin hold, and the speed is a gauge of the load.

Still open: what carries sound near zero pressure

The low-pressure end of the curve, where real sands depart most from the theory, is where most of the questions are. How the number of load-bearing contacts grows with pressure in a given packing, how much of the measured quarter-power law comes from that and how much from roughness, and whether the vanishing shear stiffness predicted near the jamming threshold can be seen cleanly in a real sand rather than in idealised frictionless simulations, are being tested with packings of photoelastic discs that show their force chains and with ultrasound sent through beads at controlled pressures. The answers matter well beyond physics laboratories, in the interpretation of shallow seismic surveys on Earth and of the regolith of the Moon and Mars, where landers have measured very low seismic speeds in the loose surface layer.

The law that the figures rest on is the contact’s. A Hertzian contact stiffens as the cube root of its load, so a packing of grains carries sound at a speed growing as the sixth root of the pressure on it — about air’s speed a few centimetres down in dry sand and twice that at six metres — rays curve back to the surface as scaled copies of one another, and at a water table the compressional speed jumps from about 600 to 1,700 m/s while the shear speed barely changes. A heap of quartz is not a slab of quartz with holes in it. It is a network of contacts, and its stiffness is the stiffness of the network, which nothing but the load can set.

Part 9 of 9

This essay is one argument about Granular matter. The others:

Shake more gently and the grains choose a side. The steady share of the grains in each of two shaken compartments joined by a slot, against B, the parameter that grows as the shaking gets gentler. Below B = 4 the only steady state is an equal share, and it is stable. At B = 4 it becomes unstable and two clustered states appear, continuously, one with each compartment full: the fuller compartment holds 0.855 of the grains at B = 5, 0.929 at 6 and 0.979 at 8. Which compartment fills is decided by the first fluctuation. Nothing in the rules prefers either side, and the equal state that the rules would suggest is the one the grains avoid. The sand that chooses a side Part 8 — Divide a box of grains into two compartments with a slot in the wall between them and shake it from below. Shaken hard, the grains share themselves equally, as molecules of a gas would. Shaken a little more gently, they gather in one compartment and leave the other almost empty, and the crowded side is cold and the empty side hot. Nothing sorted them. Every collision between grains loses energy, and a compartment that happens to hold a few more grains loses more of it and stops throwing grains over the wall.

Where this goes next

Where a heap stops flowing. The number of motions that cost nothing, against the mean number of contacts each grain has, for a patch of 37 grains. Every point is the rank of an actual rigidity matrix — one row per contact, one column per coordinate — subtracted from the number of coordinates, so it is a measurement of the network rather than a formula about it. The dashed line is Maxwell's count, two coordinates a grain minus half a contact each, which is what the counting alone predicts. The measured curve reaches its floor of three free motions — the rigid-body ones — at a coordination of 4.11, and Maxwell's line reaches zero at exactly four, which is twice the dimension of the plane. That is the isostatic point and the difference between the two numbers is the boundary of a finite patch, whose outer grains have fewer neighbours than they would in an infinite one. Above the threshold the two curves separate: the measured floor stays at three while the count keeps falling, and the gap is the redundancy — contacts whose forces no equilibrium equation can determine. Below it the pack has genuine mechanisms and will rearrange under any load at all. The transition is in the count and not in the material, which is why a fluid and a solid here are made of exactly the same grains. The heap that becomes a solid Sand poured into a jar flows; the same sand shaken down and pressed does not. Nothing about the grains has changed — not their size, their hardness, their friction or their density by more than a per cent — and the thing that changed is a count of contacts, which crosses a threshold set by the dimension of space and by nothing else. 5 balls whose contact force goes as the overlap to the three halves, touching. The velocity of every ball in a line of 5, against time in units of one binary contact, with the first arriving at unit speed. Nothing about collisions is assumed: neighbours push on each other with k times their overlap raised to the power 1.5, and the equations of motion are integrated. With the balls touching there is no such separation — several overlaps are non-zero at once and the disturbance crosses the line as a single compression wave. The far ball leaves at 0.989 of the striking speed and the others keep 0.011 between them, which is why a real cradle's balls do not quite come to rest. Momentum and energy are conserved to 4.4e-16 and 4.1e-8, so the difference between the two cases is the contact law and not the bookkeeping. Five balls, and the law that does not choose The usual account of a Newton's cradle says that momentum and energy conservation force one ball out at the striking speed. For three balls or more they do no such thing: the two laws leave a whole curve of possible outcomes, and what picks one is the shape of the force between two touching spheres. Rays that turn round without meeting anything. Rays leaving an eye 1.5 m above a road, at 0.18°, 0.28°, 0.36°, 0.42°, 0.52° below the horizontal, in air whose refractive index is reduced by 3.0e-5 at the hot surface and recovers over 5 cm. Each is traced by integrating the ray equation, with the conserved quantity n cos θ fixed by where and how the ray set out. The shallow ones come back up without touching anything — the lowest gets to 0.6 cm above the surface and turns — and a ray that returns to eye level from below is seen as sky lying on the road. The steep ones run out of gradient first and hit it. The dividing angle is 0.444°, which is what the whole effect is made of: an unremarkable temperature difference and an angle a fiftieth the width of the Moon. Heights are exaggerated 128× against distances; at true scale every ray here would be indistinguishable from the axis. The conserved n cos θ holds to 4.3e-14 over every trace, which is what says the turns are the physics and not the integrator. The ray that bends without a surface Snell's law is about a boundary, and light bends in air where there is no boundary anywhere. Let the index vary continuously and the law of angles becomes a differential equation — one that carries a conserved quantity, forbids the ray from reaching certain heights, and turns a hot road into a mirror a hundred metres long.

The objects named here

The third axis, after the field and the reading path: the things themselves, and every essay that touches each one.

Effective mediumElastic modulusGranular matterHertz contactJammingRefractionSeismic wavesSpeed of sound