Illustrated Physics
physics, drawn
Concept

Helmholtz resonator — where it appears

A volume of fluid connected to its surroundings through a narrow neck, whose trapped fluid acts as a spring and whose neck plug as a mass. It rings at one low frequency set by the volume, the neck's area and its length, as a bottle does when blown across.

Named by 2 essays across one field — each of them below, with the objects they name alongside it.

A bottle as a mass on a spring. A 750 ml bottle with a neck 8 cm long and 19 mm across, and its equivalent: the plug of air in the neck, 32 milligrams, moving in and out as a piston on the air in the body, which acts as a spring of stiffness 15 N/m because squeezing it raises its pressure. The plug's mass and the spring's stiffness give a frequency of 109 Hz, the note heard when air is blown across the mouth. The effective neck length includes about one and a half radii of air beyond its two ends that moves with the plug. Nothing in the body's air moves much — it is compressed and released almost uniformly, because the wavelength, three metres, is far larger than the bottle. That is what makes the resonator lumped: one mass, one spring, one note.

The note a bottle sings whatever its shape

Blow across the mouth of an empty wine bottle and it sounds a low note, about 109 hertz for a 750-millilitre bottle. Its wavelength is three metres, ten times the bottle's height, so it is not a standing wave in the bottle the way a flute's note is a standing wave in the flute. It is a mass bouncing on a spring: the plug of air in the neck, thirty milligrams of it, riding on the springiness of the air in the body. The note depends on the body's volume and the neck's size and on nothing about the body's shape, which is why a sphere, a cube and a tall cylinder of the same volume all sing the same note — and why a car driven with one window open throbs at eighteen hertz.

waves · Resonance The standing shapes of a lake and of a harbour. The water surface at its greatest tilt in the slowest standing waves of a basin of length L, exaggerated vertically. In a closed lake the water must be still at both ends' walls, which are antinodes of the surface: the first mode tilts the whole lake about a node in the middle, the second has two nodes. In a harbour open to the sea at one end, the sea holds the mouth's level fixed, so the mouth is a node: the first mode has a quarter of a wave in the harbour, the second three quarters. Their periods are 2L/(n√(gh)) for the lake, Merian's formula, and 4L/((2n + 1)√(gh)) for the harbour — a harbour of a given length rings at twice the period of a lake of the same length and depth.

The tide a lake makes by itself

On Lake Geneva the water at Geneva rises and falls by a few centimetres every seventy-three minutes, and the water at the far end does the opposite, as though the whole lake were a bath slopping back and forth. It is a standing wave, set ringing by wind and air pressure, and its period follows from the lake's length and depth alone. A harbour rings the same way with its mouth as a node, and it does something odder: narrowing its entrance, the obvious way to keep waves out, makes its resonance stronger.

waves · Standing waves

Named alongside it

The objects these essays reach for when they reach for this one.

Quality factorResonanceStanding waveAcoustic impedanceAdiabatic compressionLumped elementNormal modeRadiation dampingShallow water waveSound

All concepts