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Acoustic resonance is the tendency of an acoustic system to absorb more energy when it is forced or driven at a frequency that matches one of its own natural frequencies of vibration (its resonance frequency) than it does at other frequencies. As such, acoustic resonance is a branch of mechanical resonance that is concerned with the mechanical vibrations in the frequency range of human hearing, in other words sound. For humans, hearing is normally limited to frequencies between about 20 Hz and 20,000 Hz (20 kHz)[1], although these limits are not definite.

An acoustically resonant object usually has more than one resonance frequency, especially at harmonics of the strongest resonance. It will easily vibrate at those frequencies, and vibrate less strongly at other frequencies. It will "pick out" its resonance frequency from a complex excitation, such as an impulse or a wideband noise excitation. In effect, it is filtering out all frequencies other than its 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. Acoustic resonance is also important for hearing. For example, resonance of a stiff structural element, called the basilar membrane within the cochlea of the inner ear allows hair cells on the membrane to detect sound. (For mammals the membrane by having different resonance on either end so that high frequencies are concentrated on one end and low frequencies on the other.)

Like mechanical resonance, acoustic resonance can result in catastrophic failure of the vibrator. The classic example of this is breaking a wine glass with sound at the precise resonant frequency of the glass; although this is difficult in practice.[2]

Contents

Resonance of a string

Strings under tension, as in instruments such as lutes, harps, guitars, pianos, violins and so forth, have resonant frequencies directly related to the mass, length, and tension of the string. The wavelength that will create the first resonance on the string is equal to twice the length of the string. Higher resonances correspond to wavelengths that are integer divisions of the fundamental wavelength. The corresponding frequencies are related to the speed v of a wave traveling down the string by the equation

f = {nv \over 2L}

where L is the length of the string (for a string fixed at both ends) and n = 1, 2, 3... The speed of a wave through a string or wire is related to its tension T and the mass per unit length ρ:

v = \sqrt {T \over \rho}

So the frequency is related to the properties of the string by the equation

f = {n\sqrt {T \over \rho} \over 2 L} = {n\sqrt {T \over m / L} \over 2 L}

where T is the tension, ρ is the mass per unit length, and m is the total mass.

Higher tension and shorter lengths increase the resonant frequencies. When the string is excited with an impulsive function (a finger pluck or a strike by a hammer), the string vibrates at all the frequencies present in the impulse (an impulsive function theoretically contains 'all' frequencies). 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.

String resonance in music instruments

String resonance occurs on string instruments. Strings or parts of strings may resonate at their fundamental or overtone frequencies when other strings are sounded. For example, an A string at 440 Hz will cause an E string at 330 Hz to resonate, because they share an overtone of 1320 Hz (3rd overtone of A and 4th overtone of E).

Resonance of a tube of air

The resonance of a tube of air is related to the length of the tube, its shape, and whether it has closed or open ends. Musically useful tube shapes are conical and cylindrical (see bore). A pipe that is closed at one end is said to be stopped while an open pipe is open at both ends. Modern orchestral flutes behave as open cylindrical pipes; clarinets and lip-reed instruments (brass instruments) behave as closed cylindrical pipes; and saxophones, oboes, and bassoons as closed conical pipes. Vibrating air columns also have resonances at harmonics, like strings.

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Cylinders

By convention a rigid cylinder that is open at both ends is referred to as an "open" cylinder; whereas, a rigid cylinder that is open at one end and has a rigid surface at the other end is referred to as a "closed" cylinder.

The first three resonances in an open cylindrical tube. The horizontal axis is pressure.
The first three resonances in a closed cylindrical tube. The horizontal axis is pressure.

Open

Open cylindrical tubes resonate at the approximate frequencies

f = {nv \over 2L}

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 meters per second at 20 °C and at sea level).

A more accurate equation considering an end correction is given below:

f = {nv \over 2(L+0.8d)}

where d is the diameter of the resonance tube. This equation compensates for the fact that the exact point at which a sound wave is reflecting at an open end is not perfectly at the end section of the tube, but a small distance outside the tube.

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.

Closed

A closed cylinder will have approximate resonances of

f = {nv \over 4L}

where "n" here is an odd number (1, 3, 5...). This type of tube produces only odd harmonics and has its fundamental frequency an octave lower than that of an open cylinder (that is, half the frequency).

A more accurate equation is given below:

f = {nv \over 4(L+0.4d)}.

Cones

An open conical tube, that is, one in the shape of a frustum of a cone with both ends open, will have resonant frequencies approximately equal to those of an open cylindrical pipe of the same length.

The resonant frequencies of a stopped conical tube — a complete cone or frustum with one end closed — satisfy a more complicated condition:

kL = nπ − tan − 1kx

where the wavenumber k is

k = 2πf / v

and x is the distance from the small end of the frustum to the vertex. When x is small, that is, when the cone is nearly complete, this becomes

k(L+x) \approx n\pi

leading to resonant frequencies approximately equal to those of an open cylinder whose length equals L + x. 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.

Rectangular box

For a rectangular box, the resonant frequencies are given by

f = {v \over 2} \sqrt{\left({\ell \over L_x}\right)^2 + \left({m \over L_y}\right)^2 + \left({n \over L_z}\right)^2}

where v is the speed of sound, Lx and Ly and Lz are the dimensions of the box, and \scriptstyle\ell, n, and m are the nonnegative integers. However, \scriptstyle\ell, n, and m cannot all be zero.

Resonance of a sphere of air

Sphere with sound hole.gif

The diameter of a sphere with a sound hole is given by

D=17.87\sqrt[3]{\frac{d}{f^2}}

where: (in meters)

D = diameter of sphere
d = diameter of sound hole
f = frequency


Sphere with a neck.gif

The diameter of a sphere with a necked sound hole is given by

D=\sqrt[3]{\frac{3d^2C^2}{8Lf^2\pi^2}}

where: (in meters)

D = diameter of sphere
d = diameter of sound hole
C = speed of sound
L = length of neck
f = frequency


False tones

Some large conical instruments like tubas have a strong and useful resonance that is not in the well-known harmonic series. For example, most large B-flat tubas have a strong resonance at low E-flat (E-flat1, 39 Hz), which is between the fundamental and the second harmonic (an octave higher than the fundamental). These alternative resonances are often known as false tones or privileged tones.

The most convincing explanation for false-tones is that the horn is acting as a 'third of a pipe' rather than as a half-pipe. The bell remains an anti-node, but there would then be a node 1/3 of the way back to the mouthpiece. If so, it seems that the fundamental would be missing entirely, and would only be inferred from the overtones. However, the node and the anit-node collide in the same spot and cancel out the fundamental.

Resonance in musical composition

Several composers have begun to make resonance the subject of compositions. Alvin Lucier has used acoustic instruments and sine wave generators to explore the resonance of objects large and small in many of his compositions. The complex inharmonic partials of a swell shaped crescendo and decrescendo on a tam tam or other percussion instrument interact with room resonances in James Tenney's Koan: Having Never Written A Note For Percussion. Pauline Oliveros and Stuart Dempster regularly perform in large reverberant spaces such as the two million gallon cistern at Fort Warden, WA, which has a reverb with a 45-second decay.

References

  1. ^ http://books.google.com/books?id=RUDTFBbb7jAC&pg=PA248
  2. ^ http://www.physics.ucla.edu/demoweb/demomanual/acoustics/effects_of_sound/breaking_glass_with_sound.html
  • Nederveen, Cornelis Johannes, Acoustical aspects of woodwind instruments. Amsterdam, Frits Knuf, 1969.
  • Rossing, Thomas D., and Fletcher, Neville H., Principles of Vibration and Sound. New York, Springer-Verlag, 1995.

See also


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