Where two waves overlap, their heights simply add, point by point. Crest on crest makes a bigger crest; crest on trough cancels. Afterwards each wave goes on as if nothing had happened.
Send equal waves in opposite directions and they freeze into a pattern that does not travel: points that never move (nodes) between points that swing twice as far (antinodes).
A held end must be a node, so only vibrations with a whole number of half-wavelengths survive. Those are the harmonics: the notes a string or pipe can sing.
Shake a string at one of its harmonics and each push arrives in step with the wave bouncing back, so the swing builds up. Between the harmonics almost nothing happens.
A string or an open pipe sings at its lowest frequency f1 and at every whole-number multiple of it: the octave (2×), a fifth above that (3×), two octaves (4×), a major third above those (5×). A violin and a flute playing the same note differ in how much of each harmonic they carry, which is most of what you hear as the colour of an instrument.
A closed end has to be a node and an open end an antinode, and that fits an odd number of quarter-wavelengths. So the lowest note is v/4L, half what the same pipe gives open, and only the odd harmonics (1, 3, 5, 7) are there. The missing even harmonics are why a clarinet sounds hollow and reedy next to a flute of the same length, and why it plays an octave lower than you might expect.
A guitarist who rests a finger on the twelfth fret, exactly halfway, forces a node at the middle. That kills every mode with an antinode there, the odd harmonics, and leaves the even ones: the lowest that survives is the second, an octave above. It is why these notes sound pure and bell-like. Touch at a third of the way and you get the third harmonic, a twelfth up.
A microwave oven works at 2.45 GHz, a wavelength of 12.2 cm, and the metal box makes a standing wave of it. The nodes, where the field is zero, are half a wavelength apart, 6.1 cm. Take out the turntable, cover the floor with cheese and watch where it melts: the spots are 6.1 cm apart. It is a standard way to measure the speed of light in a kitchen.
Where two waves cancel, the energy has not vanished; it has gone somewhere else. In a standing wave it sloshes between the motion of the nodes’ neighbours and the stretch of the string, twice each cycle. Noise-cancelling headphones play a copy of the drone you hear, upside down: the two cancel in your ear canal and the energy goes into the headphone and the air outside. The same sum explains the colours of a soap film.
Two strings that are nearly in tune beat at the difference of their frequencies: 440 Hz against 444 Hz swells and fades four times a second. As you tighten one the beats slow, and when they stop the notes are the same. Pianos are tuned this way.
The pitch of a pipe is proportional to the speed of sound, which rises with the air’s temperature by about 0.2% for each degree; that is about 3.1 cents, a few hundredths of a semitone. A flute blown for ten minutes warms by several degrees and goes audibly sharp. A string is not affected by the air but by its own temperature, which changes its tension.
These are ideal waves in one dimension. A real string is slightly stiff, which pushes its upper harmonics sharp (the piano’s stretched tuning). A real pipe sounds as if it were a little longer than it is, by about 0.6 of its radius at each open end, and has finger holes, a bore that flares and a mouthpiece. A drum or a bell is a standing wave in two or three dimensions, with harmonics that are not whole multiples. The damping here is the simplest kind, a fixed Q at the frequency you choose, and the picture of the air is exaggerated by about ten thousand times so that you can see it.
The rule that when two waves are in the same place, the result is the sum of their heights. It holds for waves on a string, sound, water at small amplitude and light, and it means waves pass through each other without changing: they only add while they overlap.
A wave pattern that stays in place instead of travelling. It is what you get when two equal waves travel opposite ways, for instance a wave and its reflection from a fixed end. The pattern has nodes, points that never move, and antinodes, where the swing is largest, and only the amplitude changes with time.
A node is a point on a standing wave where the displacement is always zero; an antinode is where it is greatest. Neighbouring nodes are half a wavelength apart, and an antinode sits halfway between each pair. In a pipe, a displacement node is a pressure antinode and the other way round.
Because both ends are held. A wave that goes down the string, reflects and comes back must reproduce itself, and that is possible only if a whole number of half-wavelengths fits the length. That gives the frequencies f = nv/2L. Other frequencies die away because the reflections cancel instead of adding.
With one end closed, the longest wave that fits is four times the length, not twice. So the pipe’s lowest note is half the frequency, an octave lower than the same pipe open at both ends, and only the odd harmonics fit.
A big response when something is pushed at its natural frequency. Each push arrives just as the last wave comes back, so the effect adds up instead of cancelling, until damping balances it. A string or a pipe has a whole series of resonances, at its harmonics; a swing has one.
The slow swelling and fading you hear when two notes of nearly the same frequency are played together. The waves drift in and out of step, adding up and then cancelling, once for every cycle of the difference between them: 440 Hz and 444 Hz make four beats a second.
No. Waves cancel in some places and reinforce in others, and the energy is moved, not lost. In a standing wave it sloshes back and forth between the motion and the stretching; with noise-cancelling headphones it goes into the headphone and the surrounding air. Total energy is conserved.
A small tiled room is a closed box with a few resonances in the range of the voice, typically one or two hundred hertz. Notes near those frequencies are reinforced by standing waves between the walls, which is why a singer there seems to have a richer low end.
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