The bounce an elastic plate cannot give back

The bounces that add up to a stop took the coefficient of restitution as given — the fraction of its speed a ball keeps at each bounce — and followed what one fixed number does over infinitely many impacts. It ended by admitting that the number itself is measured, not derived, and by listing the ways a real impact can lose energy: plastic flow in the contact, internal friction in the material, adhesion as the surfaces part, and elastic waves launched into the bodies that do not come back in time to push them apart. The last is the odd one out. It needs no imperfection in the material at all, and it can be computed.

This essay computes it. The bodies are perfectly elastic, the contact is frictionless, and the only physics is Hertz’s law for two elastic surfaces pressed together and Newton’s second law for the waves in a plate and a rod. The rebound comes out below the arrival speed anyway, sometimes far below it. The coefficient of restitution turns out to be as much a property of the shape of the things colliding as of what they are made of.

A pulse that ends too soon #

On a rigid, immovable support an elastic ball’s impact is the classic Hertz problem. The ball flattens against the surface over a small circle whose size grows as it is pressed; the force rises as the compression to the power three halves, because the stiffening contact spreads as it is pushed; the ball stops, springs back, and leaves with exactly the speed it arrived with. The force pulse is symmetric in time, and in units built from the ball’s mass, speed and elastic stiffness it has a universal shape: a peak of 1.14, a duration of 3.21. For a steel ball two centimetres across arriving at a metre a second, the unit of time is about twenty microseconds and the impact lasts about sixty.

Now let the support be a plate, large and thin, rather than a block. When the ball presses on it the plate bends away under the contact, and the bending does not stay local. It runs outward as a flexural wave — the wave that makes a struck sheet of metal ring — and for a plate large enough that nothing reflects back from its edges within the impact, the energy that goes into that wave is gone as far as the ball is concerned. Clarence Zener showed in 1941 that for such a plate the whole response reduces to one line: the plate under a point force moves with a velocity proportional to the force, as if the ball were pressing on a dashpot. The constant of proportionality is the plate’s point impedance, $8\sqrt{D\rho h}$, built from its bending stiffness $D$, its density and its thickness.

With that, the impact has one free number, Zener’s parameter $\lambda$, which measures how soft the plate’s dashpot is compared with the Hertz contact. The drawing solves the impact at four values. As $\lambda$ grows the plate gives way under the ball, the force peaks lower and later, and the pulse ends — the ball leaves the surface — while it is still well short of having returned the ball’s momentum. At $\lambda = 1$ the ball keeps only a third of its speed.

Why a bending plate behaves like a dashpot #

It is not obvious that a plate should respond to a push the way a dashpot does, with a velocity proportional to force and no springiness at all. A stretched string does not: pushed at a point it pushes back with a force proportional to its displacement, like a spring, until the waves it launches have run off. The difference is dispersion. Bending waves in a plate travel at a speed that grows with their frequency, as the square root of it, because a shorter ripple bends the plate more sharply and the plate resists sharp bending more strongly — a dispersion of the kind water waves show, except that on deep water long swell outruns short waves and in a plate it is the other way round. Work out how a plate’s point responds when a force is applied, summing waves of every frequency, and the stiffness and inertia terms cancel exactly for a thin infinite plate, leaving a response that is purely resistive at every frequency. Energy fed into the point flows outward and never comes back; the constant $8\sqrt{D\rho h}$ says how readily it does.

That exact cancellation is why Zener’s problem is so clean, and why its single parameter captures everything. A thick block, whose surface responds through waves that do not disperse in this way, looks more like a stiff spring with a small leak, and returns nearly everything.

Restitution from geometry alone #

The central curve is the coefficient of restitution against $\lambda$. Nothing in either body dissipates energy: no plasticity, no internal friction, no adhesion. Yet the rebound falls steadily with $\lambda$, below a half by about 0.65, and the missing kinetic energy is travelling outward through the plate as a spreading ring of bending waves. In time it will turn into heat, by the plate’s internal friction or by sound radiated into the air, but that happens long after the ball has gone and plays no part in the bounce. The plate itself does no net work on the ball that it does not get back from somewhere — a support that does not move does no work at all — and this one moves, which is exactly how it takes energy.

What sets $\lambda$ is the interesting part. Working through the scales, it grows as the square of the ball’s radius over the plate’s thickness, as the fifth root of the impact speed, and with a combination of the two bodies’ densities and stiffnesses. For steel on steel, a ball two centimetres across at a metre a second has $\lambda$ = 0.06 on a plate two centimetres thick, 0.23 on one centimetre and 0.94 on half a centimetre. The corresponding restitutions run from nearly one to about 0.4. The materials are the same throughout; only the thickness has changed.

A plate is not special in this. Any body that can carry waves away from the contact faster than the contact lasts takes energy from the impact, and the question is always the same comparison: the time the impact lasts against the time it takes the struck body’s response to spread beyond the contact and not return. A thick block is effectively rigid because its response to the contact is compressive and returns within the impact. A thin plate responds by bending, which is slow and spreads far, and the energy is lost to the ball.

The same ball on different plates #

Plotted against the plate’s thickness, the result is a steep transition. A twenty-millimetre ball rebounds almost perfectly from forty millimetres of steel, keeps three quarters of its speed on ten, and barely bounces on four. Doubling the ball’s diameter is equivalent to halving the plate’s thickness, since what enters is their ratio squared.

This is familiar to anyone who has dropped a ball on a tabletop and a floor. A marble on a thin shelf gives a dead thud; the same marble on a thick stone slab rings back up. The dead thud is not the shelf absorbing the energy in the sense of damping it; the shelf is carrying it away as a vibration and then, over the next moments, turning that vibration into a faint note and a little warmth. Pinball machines and billiard cushions are designed around exactly this, and so are the anvils blacksmiths prefer: a heavy anvil returns the hammer’s energy to the hammer, and a thin plate does not.

A number that hardly depends on speed #

The coefficient of restitution is useful at all only because it changes slowly. Zener’s parameter grows as the fifth root of the impact speed, because a faster impact is shorter and stiffer and so excites the plate more, and a fifth root is very flat: over four decades of speed, from a millimetre a second to ten metres a second, the ball on the ten-millimetre plate goes from 0.93 to 0.66. Over a single decade the change is a few per cent. That is why a single number describes a bouncing ball well enough to sum its bounces, as the essay on the stopping ball did, and why that description drifts at the extremes.

The drift has a direction worth noticing. Faster impacts lose a larger fraction, so a ball dropped from higher bounces back a slightly smaller fraction of its speed, and a ball’s last tiny bounces are slightly more elastic than its first. The real series is not quite geometric, and the deviation is in the direction that makes the sum of the flight times a little longer than a constant coefficient predicts.

Rods that keep the energy as a ringing #

The plate loses the energy to waves that leave. A rod keeps it as waves that stay, and the case was solved exactly by Barré de Saint-Venant in the 1860s. A rod of length $L_1$ strikes, end on, a longer rod of the same material and cross-section, $L_2$, at rest. At contact a compression wave starts into each rod from the joint. The striker’s wave reaches its free far end, reflects as a tension wave, and arrives back at the joint after a time $2L_1/c$, where $c$ is the speed of sound in the rod, set by its stiffness and density and nothing else; at that instant the striker is at rest everywhere and simply stops touching. It has handed over all its momentum.

The long rod now carries that momentum, but not uniformly. A compression pulse of length $2L_1$ is travelling along it, and the rod as a whole moves off at $L_1/L_2$ of the striker’s original speed. That ratio is the coefficient of restitution. For a rod three times the striker’s length it is a third, and two thirds of the kinetic energy is left as the pulse, bouncing back and forth between the long rod’s ends for as long as the material lets it. Nothing was inelastic. The energy was put somewhere the collision cannot reach.

Only when the rods are the same length is the exchange complete: the pulse fills the whole struck rod exactly, reflects from its far end as the striker separates, and the struck rod leaves with the striker’s full speed and no vibration. This is why the balls of a Newton’s cradle are identical — and why that essay found the cradle’s clean behaviour to be a fragile thing that depends on how the contacts transmit a pulse.

An instrument that depends on the plate being thick #

The loss to a thin support is not only a nuisance; it is a specification. A common way of measuring the hardness of a metal in the field is to fire a small hard ball or tipped body at its surface and measure the speed it comes back with. A harder surface yields less under the impact and returns more of the energy, and the ratio of the rebound speed to the impact speed, scaled, is read as a hardness number. The instruments that do this carry a requirement that the piece being tested be thick enough and heavy enough — or firmly coupled to a massive base — because a thin sheet would take energy from the impact by bending, exactly as Zener’s plate does, and the rebound would report a hardness far lower than the metal has.

That requirement is the essay’s curve turned into a rule of practice. The test assumes the only losses are in the contact, so the support must be in the regime where $\lambda$ is nearly zero. A plate at $\lambda$ = 0.2 would already read a rebound twenty per cent low, which on the instruments’ scales is the difference between a soft steel and a hard one. The same rule tells anyone measuring a coefficient of restitution by dropping a ball on a slab what the slab must be: much thicker than the ball is wide.

Where the wave energy goes, and what the impact sees #

The two cases are the same statement about how a disturbance crosses a change of medium. At the contact the ball meets the plate, and the plate presents an impedance — a ratio of force to velocity — to the point where it is pushed. A rigid block has infinite impedance: it does not move, and all the energy stays with the ball. A plate has a finite impedance and takes energy in proportion to how well it is matched to the ball. A rod that is longer than the striker is a mismatch in time rather than in impedance: it keeps absorbing after the striker’s own reflected wave has told it to stop.

There is also a conservation law hiding here that is worth stating, because it is what rescues the usual textbook treatment. Momentum is conserved exactly in every case — which is why collisions are easier than forces, even here: the ball’s lost momentum is in the plate’s motion or the rod’s, shared between the parts but still totalled. Energy is conserved exactly too. What is not conserved is the kinetic energy of the bodies’ motion as wholes, because some of it has gone into their internal motion — waves. A coefficient of restitution is a statement about the motions as wholes, and it is below one whenever the collision leaves either body vibrating, whatever the reason.

Where Zener’s plate stops being the plate #

An infinite plate. The dashpot description holds only while the waves launched by the impact have not returned from the plate’s edges or supports. For a steel plate a centimetre thick the bending waves an impact of that length excites travel at about a kilometre a second, so within a sixty-microsecond impact they cover several centimetres. A plate much larger than that is effectively infinite; a small plate, or one struck near its edge, gets some of the energy back during the contact, and its restitution depends on where it is struck.

Hertz’s contact. The contact law assumes small deformations of smooth, elastic surfaces. Above an impact speed that for many steels is well under a metre per second, the metal in the contact yields a little, and plastic flow adds a loss the model does not have; that is why real restitution coefficients on thick blocks are still below one.

A thin plate. Zener’s dashpot is the response of a thin plate in bending. For a plate thick compared with the contact’s size the relevant waves are compressional and shear waves in the bulk, radiated as from a point on a half-space, and they carry off much less energy. That is the regime where the ball’s own vibrations dominate the loss instead, and it is small: a fraction of a per cent of the energy for a steel sphere, as Rayleigh estimated, because the impact lasts many times longer than the sphere’s lowest vibration period.

What the curves do not show #

Every curve here is a single impact. A ball bouncing repeatedly on a plate launches a new wave each time, and if the plate is finite those waves reflect and ring, so that the plate is still vibrating when the ball returns — and the next impact meets a surface that is moving. That can raise or lower the next rebound depending on the phase, and the bouncing then becomes irregular in a way no single coefficient describes. Nor do the curves show the sound. The ringing of the plate is audible, and its spectrum carries the plate’s resonances; the pitch of a dropped ball’s clatter says what it landed on, and a drum’s lack of harmonics is the same bending physics heard afterwards.

Still open: predicting restitution from first principles #

For elastic wave radiation the problem is solved: Zener’s plate and Saint-Venant’s rods are exact within their assumptions, and simulations handle any shape. For the other losses — plastic flow at the contact, viscoelastic friction in rubber or plastics, adhesion, and above all the combination of all of them in real materials with rough surfaces — no general theory predicts the coefficient from material properties well enough to replace measuring it. Engineers who design granular flows, sports equipment and machines that handle parts still measure restitution pair by pair, speed by speed. Which loss dominates in a given impact can be estimated, and the estimate says whether a thicker plate, a harder ball or a different polymer will help, but the number itself still comes from dropping something and watching it bounce.

The habit worth carrying away is to ask where energy went before calling it lost. A collision that returns less than all of its kinetic energy has put the rest somewhere, and between perfectly elastic bodies the only place available is their vibrations — waves that either leave, as in a thin plate, or stay, as in a long rod — so the coefficient of restitution is set by geometry and timing as much as by what the bodies are made of.