Tilt your head while looking at a phone through polarised sunglasses and the screen fades to black, then returns. Nothing was switched off. The light is still coming out of the glass; your lenses are simply refusing to let that particular kind of light through. A soap bubble does something stranger: it has no pigment at all, yet it shows swirling colours that change as the film thins.
Noise-cancelling headphones are stranger still. They make silence by adding sound: a microphone listens to the noise, and the speaker plays the same wave upside down. Two sounds, added together, give nothing.
All three are one idea. A physics graduate looks at them and sees the same thing: waves add up, crest on crest or crest on trough, and almost everything light and sound do follows from that. It is also the doorway to quantum mechanics, where particles turn out to obey the very same arithmetic.
Who takes it: physics majors, usually in year 2 (MIT teaches it as 8.03 Vibrations and Waves; Cambridge's Natural Sciences physics route covers the same ground). Engineers meet the maths again as electromagnetics and signals; some programmes add a separate optics course for photonics or optometry.
What it's for: understanding anything that oscillates and travels: sound, light, radio, water, and, later, the wavefunction itself.
Comes after: Physics I: Mechanics (34/06) and Differential Equations (34/03).
Leads to: Quantum Mechanics (34/83), which is wave mechanics with a new meaning for the wave; Electromagnetics & Waves (34/38); Signals & Systems (34/37); Partial Differential Equations (34/79), where the wave equation is a headline act; and, from the series, Vibrations (34/17), which is the one-spring cousin of this subject.
Honest line: you won't be able to design a lens system after this. You will see why interference makes dark bands, why a telescope's size sets what it can resolve, and why a single photon is already a wave problem.
Throw a stone in a pond and the ripple moves outward, but the water mostly just bobs up and down. What travels is the pattern. A wave is a disturbance handed from neighbour to neighbour; each bit of the medium moves a little and returns.
Three numbers describe a steady wave: the wavelength λ (crest to crest), the frequency f (crests passing a point each second) and the speed v (how fast the pattern moves). They are tied by one line of arithmetic: in one second, f crests go by, each λ long.
The medium sets the speed; the source sets the frequency; the wavelength is whatever is left. Sound in air at 20 °C runs at about 343 m/s, so middle A (440 Hz) has a wavelength of 0.78 m. An FM station at 98 MHz rides on radio waves 3.06 m long. Wi-Fi at 2.4 GHz is 12.5 cm. Green light at 550 nm is about 545 trillion cycles per second.
Here is the subtlety a graduate never forgets. When a wave crosses into a new medium, its frequency stays put (the source and the boundary insist on it, crest for crest) and its speed changes, so its wavelength changes. Light in glass of refractive index n = 1.5 travels at c/1.5 and its wavelength shrinks by the same factor: 550 nm green becomes 367 nm inside the glass, with no change of colour. Colour belongs to the frequency.
Sound: 20 Hz to 20 kHz at 343 m/s; the strip is 2 m wide. Light: the visible range, 400 to 790 THz, with c = 299 792 458 m/s (exact) and n = 1.5 for glass, 1.333 for water. The drawing is slowed enormously. For light, the top wave is the same light in vacuum: same frequency, longer wavelength.
Two waves in the same place don't collide; they superpose. At every point you add the two displacements. Where a crest meets a crest the total is double; where a crest meets a trough it is zero. This is the single most useful fact in the subject, and it is a statement about sums, which is why the maths (and later, quantum mechanics) leans on linearity.
Take two sources side by side, each sending out circular ripples. A point is bright if its distances to the two sources differ by a whole number of wavelengths (the waves arrive in step), and dark if they differ by a half-integer number. That is the whole rule:
Far from the sources, the bright directions satisfy d sin θ = mλ, where d is the spacing. Thomas Young showed in 1801 that light itself forms such bands from two slits, which was the decisive evidence that light is a wave. In the lab, light of 550 nm through slits 0.5 mm apart gives bands 2.2 mm apart on a screen 2 m away (λL/d), spacing you can measure with a ruler.
A useful consequence: you can only get a bright direction if mλ/d is less than 1. If the sources are closer than one wavelength apart, there is only a single bright direction, straight ahead, and the pattern is just a ball of waves. Noise cancelling is the same arithmetic in time: add a wave to its inverted copy and the sum is zero. It works best for low, steady hums, where the wavelength is long enough that the speaker and your ear are in the same phase of the wave.
Each pixel adds the two waves using its true distances to the sources. Orange and blue are positive and negative displacement; the dark lines that don't move are where the waves always cancel. The graph below shows the average brightness far away, cos²(π d sin θ / λ). Try d smaller than λ.
Light slows down in glass, water and diamond. The ratio of its speed in vacuum to its speed in the material is the refractive index n: about 1.0003 for air, 1.33 for water, 1.5 for common glass, 2.42 for diamond. When a ray crosses from one to another at a slant, it bends. Snell's law gives the angles (measured from the line perpendicular to the surface):
Why does it bend? Pierre de Fermat's answer (1662) is the one that lasts: of all possible paths from A to B, light takes the one that needs the least time. (Strictly, a path of stationary time: usually a minimum.) Picture a lifeguard who can run fast on sand and swim slowly. The quickest route to a swimmer isn't the straight line; it bends at the shoreline to spend less time in the slow water. That bend is Snell's law. Fermat's idea is also a first glimpse of the principle that organises Classical Mechanics (34/82): nature finds a path that is stationary.
Fermat explains lenses with no extra rules. A lens is shaped so that every path from a point to its image takes exactly the same time. For a thin lens with focal length f, an object at distance u makes an image at distance v where 1/f = 1/u + 1/v. Worked number: a 50 mm lens focused on something 2 m away puts the image 51.3 mm behind the lens, and the image is 0.026 times the object's size, upside down. Focusing a camera is moving the lens 1.3 mm.
Going the other way matters too. From glass to air, Snell's law runs out of angle: when n1 sin θ1 would exceed n2, there is no refracted ray. This total internal reflection sets in beyond a critical angle of 41.8° for glass to air and 48.6° for water to air. It is how optical fibres trap light, and why a diamond sparkles.
Source 1 m above the surface, target 1 m below. Times are exact sums of distance ÷ speed, with c = 0.29979 m/ns. At the minimum, n₁ sin θ₁ = n₂ sin θ₂ to within the slider's step.
Waves bend around corners and spread through gaps. Make the opening about as small as the wavelength and a wave leaves it as a spreading ripple. Make it large and the wave mostly carries straight on, but the edges still blur it. That spreading is diffraction, and it is the same adding-up of waves, now from every point across the opening.
For a round opening (a telescope mirror, a camera lens, the pupil of your eye) a point of light does not make a point on the image. It makes a bright disc surrounded by faint rings: the Airy pattern. The first dark ring sits at an angle that depends only on the ratio of wavelength to opening:
Two points closer together than that blur into one. Lord Rayleigh's rule of thumb says two equal points are "just resolved" when the centre of one falls on the first dark ring of the other; between the two peaks the combined brightness then dips to 73.5% of the peak, a 26% dip. Worked numbers for green light, computed live below: a pupil 3 mm wide cannot separate things less than 46 arcseconds apart, so two car headlights 1.5 m apart merge at about 6.7 km (real eyes do worse, because the retina and the eye's flaws add their own blur). The Hubble Space Telescope's 2.4 m mirror reaches about 0.058 arcseconds. A phone camera's few-millimetre lens is limited to about 28 arcseconds by diffraction alone.
The lesson is that resolution is bought with width, not magnification. Enlarging a blurred image only enlarges the blur. That is why astronomers want 10 m mirrors, why radio telescopes are linked across continents, and why a microscope can never see an atom with visible light (atoms are far smaller than the wavelength).
Green light, 550 nm. Brightness is the exact Airy pattern (2J₁(x)/x)² for each point, added (the two points are separate lamps, so their intensities add). The slider is in units of the Rayleigh angle; the readout converts it into arcseconds and into 1.5 m headlights at a distance.
Sound waves push and pull along their direction of travel. Light's wave is a sideways wobble of the electric field, perpendicular to the direction it travels, and the wobble can lie along any direction in that sideways plane. Light with the wobble mostly along one line is polarised. Reflected glare off water and roads is partly polarised, which is why polarising sunglasses cut it, and why your phone (an LCD that sends polarised light) goes dark at the wrong angle.
A polarising filter lets through only the component of the wobble along its axis. If the light is already polarised at angle θ to the filter's axis, the fraction of intensity that gets through is cos²θ: Malus's law, from 1809. Unpolarised light (every angle at once) loses half on its first filter and comes out polarised along that filter's axis.
Now the surprise. Cross two filters (axes 90° apart) and no light gets through: cos²90° = 0. Slide a third filter between them, at 45°, and light appears: after the first filter the intensity is ½, after the 45° one it is ½ × cos²45° = ¼, and after the last, ¼ × cos²45° = ⅛, which is 12.5% of the original beam. Adding an obstacle let more light through, because the middle filter re-aims the wobble. This is the classical twin of a famous quantum result about spin measured in different directions, which is the reason it is in this course.
Ideal filters. The first filter turns unpolarised light into polarised light (losing half); each later filter multiplies by cos² of the angle between neighbouring axes.
Light is a wave: that was settled by interference in the early 1800s and by Maxwell's equations in the 1860s. But light is also detected in lumps. A sensitive enough camera records single dots, one photon each. So what happens if you turn the lamp down until only one photon is in the apparatus at a time?
Each photon lands as a single dot, in a place that looks random. But dot by dot, the pattern builds up, and it is the interference pattern: bright bands where the wave's intensity is high, nothing where it cancels. Each photon, alone, behaves as if it went through both slits. Close one slit and the bands vanish, leaving the broad smear of a single slit. The wave doesn't say where the photon lands; it says how likely each place is.
The same thing happens with electrons: Akira Tonomura's team at Hitachi published a film of the pattern building up electron by electron in 1989 (the Merli–Missiroli–Pozzi group in Bologna had done a single-electron experiment earlier, in 1974). It happens with atoms, and with large molecules. Quantum mechanics (34/83) says this is not a puzzle to be explained away but the rule: a particle comes with a wave, and the wave's squared size is the probability of finding it. What the wave is is still argued about; Quantum Mechanics and 13/06 take that up.
Helium-neon laser light (633 nm), slits 0.25 mm apart and 0.06 mm wide, screen 1 m away: bands 2.5 mm apart under a broad envelope with its first zero at ±10.6 mm. Each dot is drawn at random from the exact wave intensity; the line is the prediction. Illustrative parameters, real physics.
Waves are patterns that travel, with v = fλ, and waves add: that one rule makes bright and dark bands, noise cancelling, and every blurry edge. Light takes the quickest path, which bends it and makes lenses; the width of an opening, measured in wavelengths, sets the finest detail it can show; and light is a transverse wave that filters can sort. Send photons one at a time and the same pattern emerges dot by dot, which is where waves hand over to quantum mechanics.
Try this: hold two pencils close together and look at a bright lamp through the narrow gap between them, squeezing the gap until it nearly closes. You will see dark bands appear inside the gap: that is diffraction, and the bands spread out as the gap narrows. Then take polarised sunglasses (or a pair of lenses) and look at a phone screen while you turn them.
Sources. Wave arithmetic, Snell's law, Fermat's principle, Rayleigh's criterion (1.21967 λ/D, from the first zero of J₁ at 3.8317) and Malus's law are standard textbook material (French; Hecht; Feynman Lectures I). Constants: c = 299 792 458 m/s is exact by definition of the metre (since 1983). Young's two-slit experiment: 1801, reported to the Royal Society in 1802–1803. Single-electron build-up: A. Tonomura, J. Endo, T. Matsuda, T. Kawasaki and H. Ezawa, "Demonstration of single-electron buildup of an interference pattern", American Journal of Physics 57, 117 (1989); Merli, Missiroli and Pozzi's earlier single-electron work (1974) is credited in Physics World, "The double-slit experiment". Textbook editions: Pearson's page for Hecht's Optics, 5th edition. Computed in node: the Airy dip at Rayleigh separation (73.5% of peak), all resolution distances, the thin-lens image distance, critical angles, and the Fermat minimum (checked against Snell's law).