When two waves meet, they simply add
Two people talking in a room produce sound waves that cross the space between them constantly. Neither voice is scrambled by the other. They pass straight through, and each arrives intact.
That is the extraordinary part, and it is so familiar that it takes an effort to see it as strange. Two thrown balls that met in mid-air would not do this. Waves do, and the rule governing what happens where they overlap is as simple as a rule can be: at every point, the displacements add.
Why adding is allowed
Superposition is not a definition of what waves do. It is a consequence, and it can fail.
It holds whenever the equation governing the medium is linear — meaning that if two functions each solve it, their sum solves it too. The wave equation has that property because the restoring force in the medium is proportional to displacement: double the displacement and exactly double the force comes back, with no extra term.
Where the restoring force acquires a squared term — which is what happens to a pendulum swung too far — superposition fails, and it fails in a specific way: the two waves start to affect each other. That is not a curiosity: it is the entire basis of nonlinear optics, in which one intense beam of light changes the medium enough to alter another. It takes a laser to do it, which is a fair measure of how well the linear approximation holds otherwise.
So the correct statement is: small waves add. Everything below is a description of the small-wave regime, which covers almost every wave anybody encounters and none of the interesting exceptions.
Cancellation is the surprising half
Adding two waves in step gives a bigger wave, which surprises nobody. Adding two waves out of step gives nothing at all, which should.
The energy has not been destroyed — it has been moved. Complete cancellation at one place is always accompanied by reinforcement somewhere else, and the total is conserved when the whole pattern is accounted for. But at the point of cancellation, two sources of disturbance produce less disturbance than either alone, and there is no way to make that intuitive by thinking about particles.
In between the two extremes the result is a wave of intermediate size, and the size depends on the phase difference smoothly.
Two facts about that intermediate case deserve emphasis. First, the sum of two sine waves of the same frequency is always another sine wave of that frequency — the frequency is untouchable by superposition, only amplitude and phase change. Second, the amplitudes add as vectors rather than as numbers: two equal waves a quarter-cycle apart give times one of them, not two times, and not one and a half.
The vector-addition rule is not a metaphor either. Representing each wave by an arrow whose length is its amplitude and whose angle is its phase turns superposition into arrow addition, and the resultant’s length is the amplitude of the sum. This is the same device that resolves a weight on a slope into two components, applied to a different quantity, and it is why phasor diagrams work in acoustics and in circuits alike.
Turning phase into geometry
Phase difference on its own is abstract. It becomes a picture when the two waves have travelled different distances to arrive.
Two sources vibrating together send out crests at the same moments. At a point equidistant from both, the crests arrive together and always will — that is the central line, and it is a straight line of maximum disturbance running away from the midpoint.
Move off that line and one path becomes longer than the other. When the extra distance reaches half a wavelength, one wave arrives crest-first while the other arrives trough-first, and the disturbance cancels. At a full wavelength of extra distance they are back in step, and there is another line of maximum. The pattern is therefore a fan of alternating loud and quiet directions, entirely determined by the source separation and the wavelength.
Moving the sources apart makes the pattern finer, not coarser, which is worth pausing on because the intuition usually runs the other way. The number of reinforcement lines is roughly the separation divided by the wavelength, so a wider pair packs more directions into the same fan. This inverse relationship — bigger apparatus, finer detail — is the whole principle behind interferometry, and the reason radio astronomers spread dishes across continents rather than building one large one.
The same effect, laid out in time
Two sources at slightly different frequencies produce the identical mathematics with time in place of distance.
Because the frequencies differ, the phase relationship drifts. At one moment the two are in step and the sum is loud; a little later they are opposed and it is quiet. The cycle repeats at a rate equal to the difference of the two frequencies, which is why two strings tuned a hertz apart produce one beat per second.
This is the most sensitive tuning method available and it costs nothing. Comparing two pitches by ear is accurate to a few percent at best. Counting beats is accurate to the fraction of a hertz, because the ear is being asked to detect a slow throb rather than to judge an interval — and the closer the tuning gets, the slower the beat, so the method becomes more sensitive exactly where sensitivity is needed.
Instrument tuners, radio receivers and atomic clocks all exploit the same move: compare an unknown frequency against a known one and measure the difference instead of the quantity. Heterodyne detection is beats with an electronic ear, and the reason a radio can pick out a station at 100.3 MHz from one at 100.1 is that the difference between them is a much easier number to work with than either.
The strategy generalises well past waves. Measuring a small difference between two large quantities is nearly always easier than measuring either — it is why the null field inside a conductor is a better test of the inverse-square law than any direct measurement, and why the most precise clock comparisons in relativity are differential rather than absolute.
What the pattern requires
The interference figure assumes something the picture cannot show: that the two sources keep a fixed phase relationship. Sources that do this are coherent, and coherence is a much stronger condition than it sounds.
Two loudspeakers driven by the same amplifier are coherent. Two separate violins playing the same note are not: their relative phase wanders unpredictably, so the pattern of loud and quiet directions moves around and averages away within moments. Light from two ordinary lamps is spectacularly incoherent — the phase from an ordinary source stays fixed for perhaps a hundred-millionth of a second — which is why room lighting shows no interference fringes and why the effect went unnoticed for so long.
Young’s solution in 1801 was to make two sources from one, by sending light from a single small source through two nearby slits. Whatever the phase does, it does the same thing at both slits, so the difference between them stays fixed and the pattern stands still. That experiment produced fringes where particles would have produced two bright bars, and it was the strongest single argument for the wave nature of light for the next century.
The same experiment, run much later with electrons, produced the same fringes — which is the point at which superposition stopped being a fact about media and became something stranger. Whatever an electron is, its description obeys a linear equation, and linear equations permit cancellation.
The requirement for coherence explains something else that is otherwise puzzling: why interference is so rarely seen despite waves being everywhere. Two conditions have to hold at once — a fixed phase relationship and a comparable amplitude — and ordinary sources satisfy neither. The pattern is not delicate physics; it is ordinary physics that ordinary conditions destroy, which is why demonstrating it took an apparatus and an argument rather than a glance.
Where the model stops
The interference figure has four assumptions built into it, each of which fails somewhere useful.
Equal amplitudes. The figure’s cancellation is complete because the two waves are the same size. Unequal waves leave a residue, and the quiet directions are quiet rather than silent. In practice, sources at different distances have different amplitudes when they arrive, so perfect nulls exist only in the drawing.
Point sources. Real sources have extent, and each part of an extended source produces its own pattern. Summing them washes the fringes out, which is why an interference pattern is a good way to measure the size of a source that is too small or too distant to resolve — the fringe contrast falls as the source grows, and Michelson used exactly that to measure the diameter of Betelgeuse in 1920.
Two sources. With many sources the picture changes character: the maxima stay in the same directions but grow much sharper, since cancellation now requires only that the contributions fail to line up rather than that two of them oppose. That is the difference between two slits and a diffraction grating, and it is why a grating can separate wavelengths a fraction of a nanometre apart.
One frequency. A source emitting a band of frequencies produces a pattern whose spacing depends on wavelength, so the patterns overlap and blur — visibly, since the central maximum is common to all wavelengths and the outer ones are not. White-light fringes are coloured at the edges and vanish after a few orders, for the same reason a prism separates colours: any wavelength-dependent geometry turns a mixture into a spread.
The ladder from here
Later rungs: the double-slit calculation with its small-angle geometry, and the fringe spacing that follows. Diffraction from a single slit, where a source interferes with itself. The grating, and resolving power. Thin-film interference — the colours of oil, soap and beetles — which is superposition between two surfaces of the same film. The Michelson interferometer, and the null result that broke the aether. Interference in electromagnetic fields, where the quantity being added is a vector rather than a height. And the standing wave, which is interference between a wave and its own reflection and is the mechanism behind every musical instrument.
Young’s paper was reviewed with contempt: one critic called it destitute of merit. The fringes had been on the wall the whole time.