Waves 101: The Mathematics of Wiggles

Last session we met the five big ideas of quantum mechanics, and every one of them is written in the language of waves, so we must become fluent in it. A wave is a travelling disturbance: a pattern of change that moves from place to place carrying energy and momentum, while the medium it passes through merely oscillates about its resting position. The medium is whatever is disturbed: a rope, the surface of a pond, the air in a hall. Light is stranger still: its waves are patterns in the electromagnetic field, which needs no material substance and fills even the vacuum of space.

The crucial point is that the pattern travels and the stuff does not. A cork on a pond bobs and sways as ripples pass, but it does not ride them to the shore; in a stadium wave, the ripple of standing fans races around the arena while every fan keeps the same seat. In a transverse wave, such as a wave on a rope, the medium moves perpendicular to the direction of travel. In a longitudinal wave, such as sound, it moves back and forth along that direction, forming compressions and rarefactions, where the air is alternately squeezed and stretched.

Five quantities describe any smooth, repeating wave, and each will return later in quantum form. The amplitude, A, is the maximum displacement from the resting position: the height of a crest. The wavelength (symbol λ) is the distance from one crest to the next, or between any two neighbouring points moving in step. The period, T, is the time for one full cycle. The frequency, f, is the number of cycles per second, so f = 1/T; its unit, the hertz, honours the German physicist Heinrich Hertz. Finally, the phase says where in its cycle a wave is at a given place and time, measured as an angle: one full cycle is 360°, or 2π radians.

These quantities are linked by an argument so simple I could weep. In one period, each crest advances to where the next crest stood, exactly one wavelength, so the wave speed is v = λ/T, or equivalently v = fλ: crests passing per second times their spacing. For most waves the medium fixes the speed, through a string's tension and mass per metre or the air's temperature, while the source fixes the frequency. Concert A, at 440 Hz, in room-temperature air where sound travels at about 343 m/s, therefore has a wavelength of 343 ÷ 440 ≈ 0.78 metres. Send the same note into water, where sound moves at nearly 1,500 m/s, and it stays at 440 Hz while its wavelength stretches beyond three metres.

In a series of papers culminating in 1865, James Clerk Maxwell unified electricity and magnetism and found that his equations predict waves: a changing electric field generates a magnetic field, and a changing magnetic field generates an electric field, so the pair can leapfrog through empty space. Calculating their speed from laboratory measurements of electric and magnetic forces, Maxwell obtained a value very close to the measured speed of light, and concluded that light is an electromagnetic wave. In the late 1880s Heinrich Hertz generated and detected such waves with electric sparks, producing the first deliberate radio waves. In vacuum every electromagnetic wave travels at c = 299,792,458 m/s, a value exact by definition since 1983, so for light the wave relation reads c = fλ.

The electromagnetic spectrum is the full range of these waves, ordered by frequency. Radio waves are longer than about a metre; an FM station broadcasting at 101.1 MHz emits waves about 3 metres long. Microwaves run from about a metre down to a millimetre (a microwave oven uses 2.45 GHz, a wavelength of 12.2 cm), followed by infrared, which you feel as warmth. Visible light is a strikingly narrow band, from roughly 750 nm at the red end to 380 nm at the violet end, or about 4×10¹⁴ to 8×10¹⁴ hertz. Beyond violet lie ultraviolet, X-rays and gamma rays, the last with wavelengths typically far smaller than an atom. The boundaries are conventions, not walls: X-rays and gamma rays overlap, and are usually named by origin, electrons for X-rays and nuclei or other high-energy processes for gamma rays. Every band is the same phenomenon at a different frequency.

Here is the property that makes waves utterly unlike billiard balls. When two waves arrive at the same place, the displacement of the medium is simply the sum of the displacements each would have produced alone. This is the principle of superposition, and it holds whenever the governing equation is linear: to excellent accuracy for ordinary light and gentle sound, and exactly for the Schrödinger equation of quantum mechanics, as chapter 10 will show. Waves pass through each other unharmed, but while they overlap, they interfere. Crest on crest, the waves are in phase and their amplitudes add, A₁ + A₂: constructive interference. Crest on trough, they are half a cycle, or 180°, out of phase and their amplitudes subtract: destructive interference. Two equal waves in exact opposition cancel completely.

What decides whether two waves arrive in step? Very often, geometry. Two loudspeakers driven by the same signal emit identical waves in phase. Sources that hold a fixed phase relationship like this are called coherent; independent sources drift in and out of step, and their pattern washes out. At any listening point, the path difference, Δ, is how much farther the sound has travelled from one speaker than from the other. If Δ is a whole number of wavelengths, such as 0, λ or 2λ, the waves arrive crest on crest and reinforce; if Δ is an odd number of half-wavelengths, such as λ/2 or 3λ/2, they arrive crest on trough and cancel. Each wavelength of extra path adds one full cycle of phase, so the phase difference is Δφ = 2πΔ/λ radians. Walk past two such speakers playing a steady tone and the sound swells and fades as you cross alternating lines of reinforcement and cancellation.

In 1800 the nature of light was a live controversy. Isaac Newton's Opticks of 1704 favoured a picture of light as a stream of tiny corpuscles, and Newton's prestige kept that view dominant over the wave theory Christiaan Huygens had published in 1690. Thomas Young, an English physician and polymath who had studied the interference of sound, suspected that light behaves the same way. In a lecture read to the Royal Society on 12 November 1801, he argued that the colours of thin films, including the rings Newton himself had measured, arise from interfering light waves. In a twist I never tire of, he used Newton's own measurements to compute wavelengths, putting extreme red at 0.0000266 inch, about 676 nm, and extreme violet at 0.0000167 inch, about 424 nm: remarkably close to modern values.

In November 1803 Young described holding a slip of card, about a thirtieth of an inch wide, in a narrow beam of sunlight. Its shadow was striped with fringes, and its very centre, where a particle theory predicts the deepest darkness, was always bright. His 1807 Course of Lectures describes the cleaner version taught today: light from a single source, which keeps the two beams coherent, passes through two very small holes or slits, producing a row of bright and dark stripes called fringes. Young explained them with exactly our rule. The bright stripes lie where light from one aperture has travelled farther than light from the other by 'one, two, three, or more' wavelengths, and the dark spaces where the difference is 'half a supposed undulation, of one and a half, of two and a half, or more.' Light added to light makes darkness. No classical stream of particles could do that; waves do it effortlessly. Chapter 7 will complicate this story enormously.

What does a light meter, a camera sensor or the retina of your eye actually register? Not the amplitude of the light wave, whose field swings back and forth hundreds of trillions of times a second, but its intensity: the power a wave delivers per unit area, measured in W/m². The energy of an oscillation grows as the square of its amplitude (a spring stores energy equal to one half of kA², where k is its stiffness), and a wave is a travelling chain of oscillations. So, for a given kind of wave at a given frequency, intensity is proportional to amplitude squared: I ∝ A². For light the rule holds at every frequency, with A the amplitude of the oscillating electric field. Double the amplitude and the intensity quadruples.

Combine this with interference and something remarkable appears. Two equal waves, each delivering intensity I alone, meet in phase to make an amplitude of 2A, and therefore an intensity of 4I, not 2I; meeting in opposition, they make zero. No energy is created or destroyed. It is redistributed, and averaged over the whole pattern the intensity is exactly 2I, as conservation of energy demands. In general, two overlapping waves deliver their separate intensities plus an interference term, set by the cosine of their phase difference, that paints the fringes. Mind the order of operations: first add the amplitudes, keeping track of their phases, then square. Square first, and you would erase every fringe Young ever saw.

Adding wiggles point by point is tedious; a more elegant trick happens to be the native language of quantum mechanics. Picture an arrow of length A pinned at the centre of a clock face, rotating steadily once per period. Its shadow on the horizontal axis slides back and forth exactly like a bit of rope as a wave passes. The arrow's length is the amplitude, its rotation rate is the angular frequency ω = 2πf, and its angle at any instant is the phase. Such an arrow is called a phasor. To add two waves of the same frequency, place their arrows tip to tail. Both rotate at the same rate, so the angle between them never changes, and the length of the resultant arrow is the amplitude of the sum.

The natural home for such arrows is the complex numbers. A complex number z = a + bi has a real part, a, and an imaginary part, b, where i is defined by i² = −1. Plot a horizontally and b vertically, and z becomes an arrow. Its magnitude, the arrow's length, is √(a² + b²), and its phase is the arrow's angle, θ. Multiplying by i rotates any arrow by a quarter turn, which is why i² = −1: two quarter turns point it backwards. Leonhard Euler's formula, published in 1748, completes the picture: e^(iθ) is the arrow of length one at angle θ, whose horizontal and vertical shadows are the cosine and sine of θ. Richard Feynman called it 'our jewel'. A wave of amplitude A and phase θ is represented by the arrow A·e^(iθ).

Two facts will carry you through the course. First, multiplying complex numbers multiplies their lengths and adds their angles, so a phase shift is just a rotation, multiplication by e^(iθ), which leaves the length, and so the intensity, untouched. Second, the squared length, |z|² = a² + b², equals z times its complex conjugate, the same arrow reflected across the horizontal axis. Classically, the complex arrow is bookkeeping and the real wave is its shadow. In quantum mechanics the wavefunction ψ is complex from the start, and its squared magnitude, |ψ|², gives the probabilities that experiments record.

Now confine a wave. Pluck a guitar string, clamped at both ends, and waves race along it, reflect, and superpose with their own reflections. Only waves that fit the string build up into a steady pattern; at other wavelengths the reflections return out of step and undo one another. What survives is a standing wave, which oscillates in place without travelling. It has nodes, points that never move, and antinodes, points of maximum swing, a quarter of a wavelength apart. Because the clamped ends must be nodes, the string must hold a whole number of half-wavelengths: L = nλ/2, with n = 1, 2, 3 and so on. The allowed wavelengths are λₙ = 2L/n, and since f = v/λ, the allowed frequencies are fₙ = n·v/(2L): a fundamental and its whole-number multiples, the harmonics.

Pause here, because this is the hinge of the entire course. Nobody imposed integers on the string; the integer n emerged on its own from wave behaviour plus confinement. A wave on an endless string may have any wavelength at all, but trap it between two fixed points and only a discrete ladder of wavelengths survives. A guitar string tuned to 110 Hz rings at 110, 220 and 330 Hz and upward, but never at 150 Hz. Organ pipes, drumheads and microwave ovens obey the same logic.

In chapter 6 you will learn that electrons are waves too, and in chapter 11 you will trap one in a box and watch this very rule, λₙ = 2L/n, turn into discrete energy levels. That is the true seed of quantization. Be precise, though. A classical string's allowed frequencies are discrete, but each mode can vibrate with any amplitude, and therefore with any energy. The quantum twist, which arrives next session with Max Planck's reluctant hypothesis of 1900, is that the energy of an oscillator of frequency f can change only in whole steps of Planck's constant, h, times f.