Wave motion — the series · Illustrated Physics

A travelling wave, caught at one instant. A 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.

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.

part 1 · waves

50 Hz on two strings: 5.66 m and 2.83 m. The same 50 hertz note driven onto 2 strings at the same tension of 80 newtons but different thicknesses. The frequency is identical — it is the source's — while the wavelengths are 5.66 metres and 2.83 metres, because each string carries the wave at its own speed.

The medium decides the speed, and the source only decides the note

A wave's speed is not chosen by whatever made it. It is a property of the material the wave is crossing, fixed before the wave arrives, and the wavelength is whatever is left over after the division.

part 2 · waves

Three geometries, three exponents. Amplitude against distance for a wave spreading in one, two and three dimensions, on logarithmic axes, over 3 decades. The same power crosses every surface round the source, so the intensity falls as one over the area of that surface and the amplitude as one over its square root. The fitted slopes are 0.000 along a line, -0.500 over a cylinder, -1.000 over a sphere, each fitted by least squares to the drawn curve rather than written on it. A ripple on water is the middle case and a sound in a room is the last, which is why a ripple stays visible so much further than a shout carries.

How a wave thins out

A wave gets weaker with distance for two quite different reasons, and only one of them is a loss. Geometry alone fixes the first exactly — three exponents for three dimensions, with nothing about the medium in them — and whatever is left over is the medium eating the wave.

part 3 · waves

Reflected upside down. A pulse arriving at a join where the impedance rises by a factor of 3, drawn at three moments. The amplitudes are read off the marched wave: the reflected pulse is -0.500 of the incident one and the transmitted pulse is 0.500, against (1−Z₂/Z₁)/(1+Z₂/Z₁) = -0.500 and 2/(1+Z₂/Z₁) = 0.500 from the two matching conditions. The reflection is inverted, which is the same fact as a pulse on a string flipping when it reaches a wall: a wall is a medium of infinite impedance, and the inversion is what keeps the displacement at the join equal to zero. Note that the transmitted amplitude exceeds one where the second medium is lighter, and that this is not a violation of anything: amplitude is not energy.

The equation that lets a shape travel

Newton's second law applied to a piece of string a millimetre long gives T·y″ = µ·ÿ, and the derivation never once asks what the string is made of. Two things fall out immediately: the speed is √(T/µ) and belongs to the medium, and the general solution holds two arbitrary functions rather than one. The second of them is the reflection, which is why a boundary condition can be met at all.

part 4 · waves

The correction that took a century

Newton derived the speed of sound in 1687 and got 290 metres a second against a measured 340. The arithmetic was right; the assumption was not. Heat cannot cross a wavelength in a period, so the compressions are adiabatic — and the factor that repairs the answer turns out to be a count of the ways a molecule can move.

part 5 · waves

One sharp kick, heard 6 pulse-lengths away. The signal arriving at a fixed distance from a point source that emits a single short pulse, in one, two and three dimensions, each normalised to its own peak. In three dimensions the arriving signal is the emitted pulse, unchanged in shape: it arrives, and then there is silence. In two dimensions the same kick arrives at the same moment and then keeps arriving — a tail falling as one over the time, which is still at a tenth of its peak 12.1 pulse-lengths after the front has passed. In one dimension it never comes back down at all; the medium is left displaced. The wave equation is the same equation in all three, and the source is the same source; what differs is only how many dimensions the disturbance has to spread into. Sharp arrival is the exception rather than the rule — it happens in three dimensions, and in five, and in seven, and in no even number of them — and every argument that treats a wavefront as the whole of the signal is an argument that has quietly used the fact that we live in three.

The arrival that keeps arriving

A clap heard across a field arrives and stops. The same clap in two dimensions arrives at the same instant and then goes on arriving for ever, fading as one over the time — and the difference is not absorption, or echo, or scattering. It is the number of dimensions, and sharp arrival happens in three of them and in no even number at all.

part 6 · waves

A sine on its way to a vertical face. One period of a sine, followed by the equation whose only nonlinearity is that the local speed depends on the local height. The profiles are at σ = t/t_b of 0, 0.3, 0.6, 0.9, 0.995, each obtained by solving the implicit relation u = u₀(x − (c₀+βu)t) for u at every point by bisection — and checked against the partial differential equation itself, which it satisfies to 2.6e-7. The crest travels faster than the trough, so the descending front leans forward and the ascending one leans back; the wave stays exactly as tall as it started and exactly as long, and only its shape changes. The steepest gradient grows as one over (1 − σ) — measured here as 9.83 times its initial value at σ = 0.9, against ten — so it is infinite at σ = 1 and the curve has a vertical tangent. The picture cannot be drawn past that point, which is not a failure of the drawing: the solution genuinely becomes three-valued, and what actually happens is a jump whose width is set by the dissipation this equation does not contain.

The front that steepens until it cannot

In a linear medium every wave keeps its shape, because every part of it travels at the same speed. Let the speed depend on the height by even a little and the crest overtakes the trough, the front leans forward, and after a time that can be written down the wave demands two values at one place — which is where the description ends and a shock begins.

part 7 · waves

Nine slabs, no symmetry, and one transmission. A stack of 9 slabs of random wavenumber and random thickness, with no symmetry anywhere in it. Send a wave in from the left and the amplitude that emerges on the right is 0.6694240937; turn the stack round and send it in from the right and the amplitude is 0.6694240937. The two agree to every digit the arithmetic has. This is not a property of the stack — it survives any arrangement, any number of layers, any amount of internal reflection — but of the equation, which is unchanged when the sign of time is reversed, and the phase is protected as well as the modulus, which is the stronger statement and the one an interferometer would notice. The reflections are a different matter. Their moduli are equal too, at 0.742880, but only because nothing here absorbs; their phases differ by 1.621 radians, because the wave meets a different first surface from each side. What the theorem protects is the pair of ends, and not what the wave does on its way between them.

Swap the ends and nothing changes

Put a source at one end of the most complicated arrangement of materials anybody can build and a detector at the other, then exchange them. The reading is identical — not approximately, not on average, but to every digit the arithmetic has. Two things in physics break it, and neither of them is a shape.

part 8 · waves

The speed below which nothing on water leaves a wake

Every wave on clean water travels at 23 centimetres a second or faster: long waves are fast because gravity drives them, short ones because surface tension does, and in between there is a slowest wave. A boat, a duck or a fishing line can only make waves that keep pace with it, so anything crossing the water more slowly than 23 centimetres a second makes no waves at all. Faster, it makes two kinds at once — short ripples that run ahead of it and long waves that trail behind — and the threshold between nothing and both is set by gravity, surface tension and density alone.

part 9 · waves

The shadow that shows the core is liquid

Nobody has been more than a few kilometres into the Earth, and its interior was mapped from the timing of earthquake waves alone. The decisive observations were of waves that do not arrive. Seismometers between about 100° and 140° from a large earthquake record no direct compressional wave, because something deep inside bends the rays away; and over two-fifths of the planet beyond 100° no direct shear wave arrives at all, because that something cannot be sheared. A slow, liquid core casts both shadows, and a few faint waves inside the first one revealed a solid core within it.

part 10 · waves

The sideways shaking that arrives longest waves first. The transverse ground motion 3000 km from a sudden shear source, synthesised from the computed dispersion of the fundamental Love wave in a 35 km crust, with periods from about 8 to 120 seconds. Nothing arrives before 11.0 minutes, the travel time at the mantle's shear speed. Then the longest periods come in first — 80 s waves at their group velocity of 4.23 km/s, after 11.8 min — and the period shortens steadily through the train, 30 s waves arriving at 14.3 min. The strongest shaking comes last, near 14.8 min, where the group velocity passes through its minimum and waves over a band of periods arrive together. Read off a real seismogram, the arrival time of each period is the crust's dispersion curve, and the crust's thickness with it.

The shaking a uniform Earth could not carry

Early seismographs recorded a large sideways shaking that arrived after the body waves and before the Rayleigh wave, and no theory of a uniform Earth allowed it: a horizontal shear wave at a free surface reflects as itself and nothing holds it there. Augustus Love showed in 1911 that a layer does. A crust slower than the rock beneath traps the shaking between a perfect mirror at the top and total reflection at the bottom, and because the long waves reach deeper and run faster, the order in which the periods arrive is a measurement of how thick the crust is.

part 11 · waves

Waves that turn to face the beach. The paths of crests' motion for a 10 s swell arriving at 50° on a plane beach of slope 1 in 50, seen from above, with depth contours dashed. Snell's law, sin α over speed is constant, holds along the ray because the depth changes only in the offshore direction, and the integrated paths were checked against it. The end of each ray is nearly normal to the shore: 5° from square-on by the point where it is about to break. In this sense all surf comes in nearly straight whatever direction the storm was in. The bend is complete because the speed falls almost to zero at the shore, whereas a prism's angles are bounded by its index. What remains of the angle drives the current along a beach.

The waves that always arrive square to the beach

A storm a thousand kilometres off sends out swell in whatever direction it pleases, and that swell arrives on the beach running almost straight in. It does so whichever way the coast faces. Waves slow down in shallow water, so the part of a crest nearest the shore lags and the whole line swings round, as light does in glass. It also explains why waves shrink before they grow, and why they gather on a headland and leave the bay alone.

part 12 · waves

The pulse behaves like a body falling at half of g. The height of a pulse going down a uniform rope 10 m long, hanging free, from the integrated wave equation with tension set by the weight below each point, against the path x = (√L − √g t/2)² that the local speed √(gx) gives. It is a parabola: constant acceleration of g/2, the same on a rope of any mass, any material. So a pulse takes 2.02 s to go down 10 m, against a stone's 1.43 s. A falling stone has speed √(2gx) after dropping x, and the pulse has √(gx) with x the height left: the parabola is the same with the acceleration halved, and the pulse is slowing rather than speeding up.

The pulse that climbs a rope at half of g

Flick the top of a hanging rope, and the pulse that travels down it slows at a steady rate of half of g, whatever the rope is made of and however heavy it is. Send it back up from the free end and it comes back altered: not the same way up and not upside down, but something else, with a lobe of each sign. A hanging rope is the simplest medium in which a wave's speed varies, and it has several surprises to give.

part 13 · waves