- Acoustic resonance
**Acoustic resonance**is the tendency of an acoustic system to absorb more energy when thefrequency of its oscillations matches the system's natural frequency of vibration (its "resonance frequency") than it does at other frequencies.A resonant object will probably have 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

resonator s, such as the strings and body of aviolin , the length of tube in aflute , and the shape of a drum membrane.**Resonance of a string**Strings under tension, as in instruments such as

lute s,harp s,guitar s,piano s,violin s 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:

:$v\; =\; sqrt\; \{T\; over\; ho\}$

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

:$f\; =\; \{nsqrt\; \{T\; over\; ho\}\; over\; 2\; L\}\; =\; \{nsqrt\; \{T\; over\; m\; /\; L\}\; over\; 2\; L\}$

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.

**tring resonance in music instruments**String resonance occurs onstring instruments . Strings or parts of strings may resonate at theirfundamental orovertone 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

flute s behave as open cylindrical pipes;clarinet s and lip-reed instruments (brass instrument s) behave as closed cylindrical pipes; andsaxophone s,oboe s, andbassoon s as closed conical pipes. Vibrating air columns also have resonances at harmonics, like strings.**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.

**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 th the length of the tube and "v" is the

speed of sound in air (which is approximately 344 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 infinite

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:

:$kL\; =\; npi\; -\; an^\{-1\}\; kx$

where the

wavenumber k is:$k\; =\; 2pi\; f/v$

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

:$k(L+x)\; approx\; npi$

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

:$f\; =\; \{v\; over\; 2\}\; sqrt\{left(\{ell\; over\; L\_x\}\; ight)^2\; +\; left(\{m\; over\; L\_y\}\; ight)^2\; +\; left(\{n\; over\; L\_z\}\; ight)^2\}$

where "v" is the speed of sound, "L

_{x}" and "L_{y}" and "L_{z}" are the dimensions of the box, and $scriptstyleell,$ "n", and "m" are the nonnegative integers. However, $scriptstyleell$, "n", and "m" cannot all be zero.**Resonance in musical composition**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 complexinharmonic partial s of a swell shapedcrescendo and decrescendo on atam tam or other percussion instrument interact with room resonances inJames Tenney 's "Koan: Having Never Written A Note For Percussion".Pauline Oliveros andStuart 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*** 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***

Harmony

*Music theory

*Resonance

*Sympathetic string

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