Illustrated Physics
physics, drawn
Waves

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

In a wave on water, no water goes anywhere. What moves is the shape — and separating the two motions is the whole of wave physics.
11 min read 6 figures The same equation againWhat stays the same

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.

A travelling wave, caught at one instantA sine wave plotted against position at a fixed moment. The wavelength is the distance between repeats; the ghosted curve is the same wave a moment later.00.511.522.5-101positionone wavelength
Fig. 1 A sine wave plotted against position at one instant. Every point of the medium moves only up and down; the pattern moves sideways. The ghosted curve is the same wave a moment later.

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.

A travelling wave, caught at one instantA sine wave plotted against position at a fixed moment. The wavelength is the distance between repeats; the ghosted curve is the same wave a moment later.00.511.522.5-101positionone wavelengthghost: the same wave, slightly later
Fig. 2 The same wave at two instants a short time apart. Every particle has moved vertically; the crest has moved horizontally. The two displacements are perpendicular and independent.

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=T/μ,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 vv, and since the shape repeats every wavelength λ\lambda in space and every period TT in time, one crest must advance exactly one wavelength in exactly one period. So

v=λT=fλ.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.

A travelling wave, caught at one instantA sine wave plotted against position at a fixed moment. The wavelength is the distance between repeats; the ghosted curve is the same wave a moment later.01234-101positionone wavelength
Fig. 3 A shorter, smaller wave in the same medium. More cycles fit into the same distance, so this wave has a higher frequency — but its crests advance at exactly the speed of the longer wave’s crests.

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.

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.

Two waves 0° out of step, and their sumTwo sine waves differing in phase by 0 degrees, drawn faintly, with their sum drawn solid. The sum is computed point by point.00.511.52-2-1012positionphase difference 0°peak 2.00× one wave
Fig. 4 Two waves in step, and their sum. Adding waves is done point by point, which is the entire content of the superposition principle — and the property that makes everything in 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.

Beats between 10 and 11 cyclesTwo tones a little apart in frequency, added together. The rapid oscillation is the average frequency; the slow swelling is the difference, heard as a throb.00.511.52-202timebeat rate = 1 per unit timethe difference of the two frequencies
Fig. 5 Two frequencies close together, added. The rapid oscillation is their average and the slow swelling is their difference — one wave in appearance, two in construction.

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.

Harmonics on a fixed stringThe first four standing-wave patterns on a string clamped at both ends. Only whole numbers of half-wavelengths fit, which is why the allowed frequencies are discrete.n = 1fundamentaln = 22× the frequencyn = 33× the frequency
Fig. 6 The first three patterns on a string clamped at both ends. Each is a travelling wave added to its own reflection, and the pattern does not go anywhere at all.

A wave reaching a fixed end reflects, the reflection runs back through the incoming wave, and the sum oscillates in place. Only certain wavelengths survive the arrangement, 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 LL produces a lowest frequency proportional to 1/L1/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.

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=fλv = f\lambda was written as if vv 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. 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.

Where this goes next

sourcesourcesolid: waves arrive in stepbetween them they arrive opposed When two waves meet, they simply add Waves pass through each other unchanged and their displacements add point by point. From that one impoverished-sounding rule comes interference, beats, and the evidence that light is a wave at all. n = 1fundamentaln = 22× the frequencyn = 33× the frequencyn = 44× the frequency Only some notes fit, and that is where discreteness comes from A string clamped at both ends can vibrate at some frequencies and not others. A continuous object producing a whole-number list is the oldest quantisation in physics. −π−π/2π/2π-22angleangular velocityat rest, hanging downbalanced upside downclosed: swingingopen: going over the topdashed: the boundary The pendulum, and the small lie that makes it simple A pendulum's period does not depend on how far it swings. This is one of the most useful false statements in physics, and it is worth knowing exactly how false.