A wave is a shape that travels, and nothing else does

Drop a stone in a pond and something spreads outward. It is not the water. A leaf floating a metre away bobs up and down as the ripple passes and stays exactly where it was, and if the leaf does not travel then neither does the water underneath it.

What travelled was a shape. That sentence is easy to nod at and surprisingly hard to hold onto, because everything about the visual impression says that something is moving outward at speed — and something is, just not any of the material.

Two motions at right angles #

A travelling wave has two speeds in it and they are unrelated.

There is the speed at which a piece of the medium moves — up and down, in the figure above, at a rate that depends on how large the wave is. And there is the speed at which the pattern advances, which does not depend on the size of the wave at all.

Follow a single point on the horizontal axis and watch it over time: it goes up, comes back, goes down, comes back — a full oscillation, and it is the same oscillation a pendulum performs, governed by the same equation, for the same reason. Each bit of the medium is pulled back toward where it was by a restoring force proportional to how far it has been displaced.

What makes it a wave rather than a collection of independent oscillators is that each bit is coupled to its neighbours. A bit that is displaced pulls the next bit along with it, slightly later. That delay is the whole mechanism: the pattern appears to advance because each oscillator is running a little behind the one before it.

This is why the wave speed is a property of the medium and not of whatever made the wave. It is set by two competing quantities: how stiffly the medium pulls displaced material back, and how much inertia there is to move. Stiffer means faster; heavier means slower. For a stretched string the ratio is explicit,

$$v = \sqrt{T/\mu},$$

tension over mass per unit length, and every other medium has its own version of the same ratio. Sound in air runs at the square root of pressure over density. Sound in steel, which is far stiffer and only eight times denser, runs about fifteen times faster.

Shouting louder does not make sound arrive sooner. That is a genuinely non-obvious prediction, and it is the first thing to check about any claim that something is a wave.

The one equation everything obeys #

Since the medium fixes $v$, and since the shape repeats every wavelength $\lambda$ in space and every period $T$ in time, one crest must advance exactly one wavelength in exactly one period. So

$$v = \frac{\lambda}{T} = f\lambda.$$

This is arithmetic rather than physics, but it has real consequences, because it says frequency and wavelength are not independent. Fixing the medium fixes their product. A source that vibrates faster produces shorter waves and not faster ones.

Which of the two is the property of the source and which of the medium is worth getting the right way round. Frequency belongs to the source. A tuning fork at 440 Hz produces 440 Hz in air, in water, and in a steel bar. Wavelength belongs to the pairing, and adjusts. Sound entering water from air keeps its pitch and its wavelength quadruples, because the speed does.

The same asymmetry rules refraction: light crossing into glass keeps its colour and shortens its wavelength, and the bending at the boundary is a geometric consequence of exactly that.

Two kinds of shaking #

The figure draws displacement perpendicular to the direction of travel, which is a choice and not a requirement.

A wave on a string is transverse: the string moves across the direction the wave goes. So is light, and so is the ripple on the pond, roughly. Sound is longitudinal: air moves back and forth along the direction of travel, and what propagates is alternating compression and rarefaction rather than sideways displacement.

Plotting a longitudinal wave as a sine curve — as physics does constantly — is a translation, not a picture. The vertical axis means displacement along the direction of travel, and a crest means the material there has shifted forward, not upward. The mathematics is identical and the mental image is completely different, which is a good thing to notice early, because a great many wave figures are of this translated kind.

The distinction matters more than a bookkeeping detail. Only transverse waves can be polarised, since only they have a choice of shaking direction to make — which is why sound cannot be polarised and light can, and why the discovery that light could be polarised was the argument that settled it as a transverse wave.

Solids carry both kinds. That is how the Earth’s interior was mapped: an earthquake sends out a fast longitudinal wave and a slower transverse one, and the transverse wave cannot cross the outer core. The shadow it casts on the far side of the planet is the evidence that the outer core is liquid — a conclusion about the middle of the Earth, drawn entirely from which sort of shaking arrived where.

Why a medium can carry a wave at all #

Two ingredients are needed, and a medium missing either one carries nothing.

The first is inertia: displaced material has to resist being moved, or the disturbance would be over instantly. The second is a restoring influence: displaced material has to be pulled back, or it would simply stay displaced and nothing would return. Waves are what happens when those two take turns — inertia carrying the material past the resting position, the restoring force hauling it back, and the neighbours being dragged into the same cycle slightly late.

That is why the speed is always a ratio of the two, with stiffness on top and density underneath, and why the square root is there: the speed comes out of an equation in which each derivative has been taken twice.

The picture also says which two reservoirs the energy sits in. At the instant a piece of the medium passes through its resting position it is moving fastest and is not displaced at all, so all its energy is kinetic. At the crest it is momentarily still and maximally displaced, so all of it is stored in the restoring mechanism. In between it is divided. This is exactly the exchange a pendulum performs, running independently at every point of the medium, a quarter-cycle out of step with its neighbours — and that phase lag between neighbours is what carries energy along, because a bit that is displaced does work on the bit ahead of it.

Anything with inertia and a restoring force therefore has waves, whether or not anybody was looking for them. A stretched membrane, a column of air, a crystal lattice, the surface of a liquid, a magnetic field under tension, the pressure in an artery, a chain of coupled clocks, the interior of a star. The list is not a list of analogies. It is a list of systems obeying the same equation because they satisfy the same two conditions, which is why the results transfer without translation.

What a wave costs to send #

A wave carries energy, and the accounting is unforgiving in a way the figures do not show, because a drawing of a sine wave of constant amplitude is a drawing of a wave that costs nothing to maintain.

The first charge is geometric. A wave spreading from a point into three dimensions distributes its power over a sphere whose area grows as $r^2$, so intensity falls as $1/r^2$ and amplitude as $1/r$. This is the same counting argument that fixes the falloff of a field, reached without mentioning charge — a spreading disturbance and a static field share an exponent because they share a geometry. In practical terms every doubling of distance costs six decibels, so a voice that is comfortable at one metre is twenty decibels down at ten and has lost nothing to absorption at all.

The second charge is that intensity goes as the square of amplitude, which makes the currency non-linear in the obvious variable. Doubling the height of a wave quadruples the power needed to make it. That is the reason loudspeakers become inefficient at the bottom of their range, where large excursions are required, and the reason the last few decibels of anything are so expensive.

The third charge is absorption, and it is the one that decides what a wave can be used for. Media convert a fraction of the passing energy into heat, and for most of them the fraction rises steeply with frequency. Soft tissue absorbs ultrasound at roughly half a decibel per centimetre per megahertz, and that single number sets the shape of an entire technology: resolution improves with frequency, because the smallest resolvable detail is about a wavelength, while penetration gets worse with frequency at the same time. A 10 MHz probe resolves a fifth of a millimetre and is exhausted after four or five centimetres; a 3 MHz probe reaches a foetus and cannot see the fine structure. There is no setting that does both, and the trade is a property of the medium rather than of the equipment.

Air runs the same trade, which is why distant thunder is a rumble and near thunder is a crack. The high frequencies were in both; only the low ones survived the journey.

What one snapshot cannot tell #

A single figure of a wave contains an ambiguity that no amount of drawing can remove: it does not say which way the wave is going. The same picture is consistent with motion in either direction, and it is also consistent with a wave that is not going anywhere at all.

The way out is superposition, and it is worth naming the property being used: waves add point by point, value to value, with no interaction between them. That is the whole content of the superposition principle and it is what makes the rest of wave physics possible.

The reason the ambiguity is interesting is that superposition allows two waves to occupy the same medium at once. Adding a wave to its own mirror image gives a shape that oscillates in place and does not travel — a standing wave — and a snapshot of one at the right moment is indistinguishable from a snapshot of a travelling wave.

Superposition also allows something less obvious. Since sums of sines are still solutions, and since almost any repeating shape can be built out of sines, the sine wave is not one special case among many. It is the alphabet: a triangular pulse, a square edge, a plucked string’s kink and a spoken vowel are all sums of sine waves, and analysing them means finding out which ones.

The clearest demonstration that a sum can masquerade as a single thing is two frequencies close together. Added, they give a rapid oscillation at their average, swelling and fading at their difference — one wave to look at and to listen to, two in construction, and the throb is the only evidence of the second. A listener hears a single note wavering rather than two notes, which is exactly the ambiguity above appearing in the ear instead of on the page.

What happens when it runs out of room #

A travelling wave needs somewhere to travel. Confine it and something new appears, without any new physics being added.

A wave reaching a fixed end reflects, the reflection runs back through the incoming wave, and the sum oscillates in place. Every pattern on a clamped string is a travelling wave added to its own reflection, and not one of them goes anywhere at all — which is the ambiguity of the opening section arriving as a physical arrangement rather than as a difficulty about snapshots. Only certain wavelengths survive it, because the boundary insists on a node at each end and nodes are half a wavelength apart. A continuous medium ends up producing a discrete list of frequencies, which is the oldest and most mechanical example of quantisation there is.

Two other consequences of confinement are worth noting because they recur far outside acoustics. A confined wave stores energy without transporting any, which is what makes a resonant cavity useful — in a laser, in a microwave oven, in the body of a violin. And confinement in a region of size $L$ produces a lowest frequency proportional to $1/L$, so smaller means higher: the reason a piccolo sits above a bassoon, and, transposed into quantum mechanics, the reason a tightly confined particle has a large minimum energy.

The leaf does not quite stay put #

The page opened with a leaf bobbing up and down while the ripple passed beneath it, and that is a simplified account of what a floating object actually does. The correction is worth making, because it is where the clean separation between “the shape moves” and “the material does not” starts to leak — and because it is measurable.

In a deep-water wave, a particle of water does not move up and down. It moves in a circle, forward at the crest and backward in the trough, returning very nearly to where it began after each period. The transverse picture is right about the surface height and wrong about the trajectory, because a water wave is neither purely transverse nor purely longitudinal: gravity restores the vertical displacement and the water has to flow horizontally to supply it.

The radius of that circle falls off exponentially with depth, by a factor of about twenty-three for every wavelength down. Half a wavelength below the surface the motion is a few per cent of what it is on top, which is why a submarine a modest distance down rides out a storm that is punishing on the surface, and why a swell of long wavelength is felt much deeper than a short chop of the same height.

And the circles do not quite close. A particle is slightly higher, and so in slightly faster water, at the top of its orbit than at the bottom, so each cycle leaves it a little further forward than it started. That residue is the Stokes drift, it is second order in the wave’s steepness, and it is why floating debris gradually comes ashore in a swell that is transporting no water in any first-order sense. The leaf stays where it was to the accuracy of this page’s argument, and creeps forward beyond it.

A wave that travels the wrong way #

The most convincing demonstration that a shape and its medium are independent is one in which they move in opposite directions, and it happens on any busy motorway.

A driver brakes. The car behind brakes a moment later, and the one behind that a moment after — so a region of slow, closely-spaced traffic propagates backwards along the road at fifteen or twenty kilometres an hour, while every car in it is moving forwards at whatever the traffic allows. Nothing travels upstream. The shape does.

It has the two ingredients this page requires. The inertia is the drivers’ reaction time and their cars’ braking distance, which is what stops the disturbance being over instantly. The restoring influence is the tendency to close a gap that has opened ahead, which pulls the spacing back toward its preferred value. The propagation speed comes out of the ratio, which is why it depends on the density of traffic and not on how fast anybody is going.

It also produces the jam with no cause. Sugiyama’s experiment in 2008 put twenty-two cars on a circular track with instructions to drive at a constant speed and keep a fixed distance, and a stop-and-go wave appeared within minutes with no obstacle, no accident and no lane change — the medium was above the density at which the uniform state is stable, and the smallest fluctuation grew. Every driver in that jam had been through it and out the other side; the jam stayed where it was, moving slowly the other way.

Where the model stops #

Everything above rests on two assumptions doing heavy and mostly invisible work.

Linearity. Superposition is not a general property of waves; it is a property of small waves. It holds when the restoring force is proportional to displacement, and that proportionality is the first term of an expansion — exactly the same approximation as the small-angle one that makes a pendulum simple. Push a wave hard enough and the medium stops responding linearly: crests travel faster than troughs, the shape steepens as it goes, and a sine wave turns into a sawtooth. Ocean waves break because of this. So does the shock wave from an explosion, which is a sound wave that has steepened until it has no width left.

No dispersion. The relation $v = f\lambda$ was written as if $v$ were a single number for the medium, and for many media it very nearly is. Where it is not, different frequencies travel at different speeds, a compound pulse spreads out as it goes, and the concept of “the” wave speed splits into two — the speed of the crests and the speed of the packet, which are not equal. Deep-water waves are strongly dispersive, which is why a distant storm arrives as a swell that sorts itself by wavelength, longest first. Glass is mildly dispersive, which is why a prism makes a spectrum and why a rainbow has its colours in that order.

A third assumption is that a medium exists at all. It does for sound and for water, and it does not for light — the electric and magnetic fields oscillate with nothing oscillating, which is the fact that took longest to accept and that eventually forced the whole reconstruction of space and time. The wave equation does not care: it was derived for a medium and it holds without one, and every result on this page transfers unchanged.

Two smaller assumptions are visible in the figure itself. It draws a wave of unchanging amplitude, so there is no loss; every real wave dies out, and the medium is where the energy goes. And it draws a wave of unbounded extent, which no real wave has. A real pulse is finite, and a finite pulse is necessarily a mixture of frequencies rather than a single one — a fact with an exact statement, and one that reappears in quantum mechanics with much more dramatic consequences.

The ladder from here #

Later rungs: the wave equation derived from Newton’s laws applied to a string, and why a second derivative in space equals a second derivative in time. Energy transport, which goes as the square of amplitude and is the reason loudness and brightness are not linear in it. The Doppler shift, which is what happens when the source will not stay still. Dispersion and group velocity. Reflection at a boundary, and why the reflection from a fixed end is inverted. Impedance matching, which explains the horn on a trumpet and the coating on a lens. Solitons, which are nonlinear waves that hold their shape anyway. And the Fourier decomposition that makes the sine wave the alphabet rather than an example.

Everything here was worked out for strings and water long before anybody suspected that light and matter would turn out to obey the same equations. The debt runs entirely one way.