Acoustic resonance

Acoustic resonance is an important consideration for instrument builders, as most acoustic instruments use resonators, such as the strings and body of a violin, the length of tube in a flute, and the shape of a drum membrane.

Those frequencies that are not one of the resonances are quickly filtered out—they are attenuated—and all that is left is the harmonic vibrations that we hear as a musical note.

Like strings, vibrating air columns in ideal cylindrical or conical pipes also have resonances at harmonics, although there are some differences.

However, a cylinder closed at both ends can also be used to create or visualize sound waves, as in a Rubens Tube.

The resonance properties of a cylinder may be understood by considering the behavior of a sound wave in air.

At the open end of the tube, air molecules can move freely, producing a displacement antinode.

In the first harmonic, the closed tube contains exactly half of a standing wave (node-antinode-node).

The intuition for this boundary condition ∂(Δp)/∂x = 0 at x = 0 and x = L is that the pressure of the closed ends will follow that of the point next to them.

In the first harmonic, the open tube contains exactly half of a standing wave (antinode-node-antinode).

Note that the diagrams in this reference show displacement waves, similar to the ones shown above.

These stand in sharp contrast to the pressure waves shown near the end of the present article.

[3] Open cylindrical tubes resonate at the approximate frequencies: where n is a positive integer (1, 2, 3...) representing the resonance node, L is the length of the tube and v is the speed of sound in air (which is approximately 343 metres per second [770 mph] at 20 °C [68 °F]).

The reflection ratio is slightly less than 1; the open end does not behave like an infinitesimal acoustic impedance; rather, it has a finite value, called radiation impedance, which is dependent on the diameter of the tube, the wavelength, and the type of reflection board possibly present around the opening of the tube.

Adjusting the taper of this cylinder for a decreasing cone can tune the second harmonic or overblown note close to the octave position or 8th.

The intuition for this boundary condition ∂(Δp)/∂x = 0 at x = L is that the pressure of the closed end will follow that of the point next to it.

In the two diagrams below are shown the first three resonances of the pressure wave in a cylindrical tube, with antinodes at the closed end of the pipe.

The resonant frequencies of a stopped conical tube — a complete cone or frustum with one end closed — satisfy a more complicated condition: where the wavenumber k is and x is the distance from the small end of the frustum to the vertex.

In words, a complete conical pipe behaves approximately like an open cylindrical pipe of the same length, and to first order the behavior does not change if the complete cone is replaced by a closed frustum of that cone.

Sound waves in a rectangular box include such examples as loudspeaker enclosures and buildings.

is the equivalent length of the neck with end correction For a spherical cavity, the resonant frequency formula becomes where For a sphere with just a sound hole, L=0 and the surface of the sphere acts as a flange, so In dry air at 20 °C, with d and D in metres, f in hertz, this becomes This is a classic demonstration of resonance.

To do it reliably for a science demonstration requires practice and careful choice of the glass and loudspeaker.

Alvin Lucier has used acoustic instruments and sine wave generators to explore the resonance of objects large and small in many of his compositions.

Pauline Oliveros and Stuart Dempster regularly perform in large reverberant spaces such as the 2-million-US-gallon (7,600 m3) cistern at Fort Worden, WA, which has a reverb with a 45-second decay.

Malmö Academy of Music composition professor and composer Kent Olofsson's "Terpsichord, a piece for percussion and pre-recorded sounds, [uses] the resonances from the acoustic instruments [to] form sonic bridges to the pre-recorded electronic sounds, that, in turn, prolong the resonances, re-shaping them into new sonic gestures.