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Acoustic resonance

Acoustic resonance is a phenomenon in which an acoustic system amplifies sound waves whose frequency matches one of its own natural frequencies of vibration (its resonance frequencies).

Experiment using two tuning forks oscillating at the same frequency. One of the forks is being hit with a rubberized mallet. Although the first tuning fork hasn't been hit, the other fork is visibly excited due to the oscillation caused by the periodic change in the pressure and density of the air by hitting the other fork, creating an acoustic resonance between the forks. However, if a piece of metal is placed on a prong, the effect dampens, and the excitations become less and less pronounced as resonance isn't achieved as effectively.

The term "acoustic resonance" is sometimes used to narrow mechanical resonance to the frequency range of human hearing, but since acoustics is defined in general terms concerning vibrational waves in matter,[1] acoustic resonance can occur at frequencies outside the range of human hearing.

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 has tapering resonances across its length 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.

Vibrating string edit

 
String resonance of a bass guitar A note with fundamental frequency of 110 Hz.

In musical instruments, strings under tension, as in 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

 

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

 

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

 

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 edit

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 edit

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. Many musical instruments resemble tubes that are conical or cylindrical (see bore). A pipe that is closed at one end and open at the other is said to be stopped or closed while an open pipe is open at both ends. Modern orchestral flutes behave as open cylindrical pipes; clarinets behave as closed cylindrical pipes; and saxophones, oboes, and bassoons as closed conical pipes,[2] while most modern lip-reed instruments (brass instruments) are acoustically similar to closed conical pipes with some deviations (see pedal tones and false tones). Like strings, vibrating air columns in ideal cylindrical or conical pipes also have resonances at harmonics, although there are some differences.

Cylinders edit

Any cylinder resonates at multiple frequencies, producing multiple musical pitches. The lowest frequency is called the fundamental frequency or the first harmonic. Cylinders used as musical instruments are generally open, either at both ends, like a flute, or at one end, like some organ pipes. 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. Sound travels as a longitudinal compression wave, causing air molecules to move back and forth along the direction of travel. Within a tube, a standing wave is formed, whose wavelength depends on the length of the tube. At the closed end of the tube, air molecules cannot move much, so this end of the tube is a displacement node in the standing wave. At the open end of the tube, air molecules can move freely, producing a displacement antinode. Displacement nodes are pressure antinodes and vice versa.

Closed at both ends edit

The table below shows the displacement waves in a cylinder closed at both ends. Note that the air molecules near the closed ends cannot move, whereas the molecules near the center of the pipe move freely. In the first harmonic, the closed tube contains exactly half of a standing wave (node-antinode-node). Considering the pressure wave in this setup, the two closed ends are the antinodes for the change in pressure Δp; Therefore, at both ends, the change in pressure Δp must have the maximal amplitude (or satisfy ∂(Δp)/∂x = 0 in the form of the Sturm–Liouville formulation), which gives the equation for the pressure wave:  . 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. Applying the boundary condition ∂(Δp)/∂x = 0 at x = L gives the wavelengths of the standing waves:

 

And the resonant frequencies are

 
Frequency Order Name 1 Name 2 Name 3 Wave representation Molecular representation
1 · f =   440 Hz n = 1 1st partial fundamental tone 1st harmonic    
2 · f =   880 Hz n = 2 2nd partial 1st overtone 2nd harmonic    
3 · f = 1320 Hz n = 3 3rd partial 2nd overtone 3rd harmonic    
4 · f = 1760 Hz n = 4 4th partial 3rd overtone 4th harmonic    

Open at both ends edit

In cylinders with both ends open, air molecules near the end move freely in and out of the tube. This movement produces displacement antinodes in the standing wave. Nodes tend to form inside the cylinder, away from the ends. In the first harmonic, the open tube contains exactly half of a standing wave (antinode-node-antinode). Thus the harmonics of the open cylinder are calculated in the same way as the harmonics of a closed/closed cylinder.

The physics of a pipe open at both ends are explained in Physics Classroom. 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.

By overblowing an open tube, a note can be obtained that is an octave above the fundamental frequency or note of the tube. For example, if the fundamental note of an open pipe is C1, then overblowing the pipe gives C2, which is an octave above C1.[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]). This equation comes from the boundary conditions for the pressure wave, which treats the open ends as pressure nodes where the change in pressure Δp must be zero.

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

 

where r is the radius 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.

So when n is 1:

 
 
 
 

where v is the speed of sound, L is the length of the resonant tube, r is the radius of the tube, f is the resonant sound frequency, and λ is the resonant wavelength.

Closed at one end edit

When used in an organ a tube which is closed at one end is called a "stopped pipe". Such cylinders have a fundamental frequency but can be overblown to produce other higher frequencies or notes. These overblown registers can be tuned by using different degrees of conical taper. A closed tube resonates at the same fundamental frequency as an open tube twice its length, with a wavelength equal to four times its length. In a closed tube, a displacement node, or point of no vibration, always appears at the closed end and if the tube is resonating, it will have a displacement antinode, or point of greatest vibration at the Phi point (length × 0.618) near the open end.

By overblowing a cylindrical closed tube, a note can be obtained that is approximately a twelfth above the fundamental note of the tube, or a fifth above the octave of the fundamental note. For example, if the fundamental note of a closed pipe is C1, then overblowing the pipe gives G2, which is one-twelfth above C1. Alternatively we can say that G2 is one-fifth above C2 — the octave above C1. 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.[4] Opening a small "speaker hole" at the Phi point, or shared "wave/node" position will cancel the fundamental frequency and force the tube to resonate at a 12th above the fundamental. This technique is used in a recorder by pinching open the dorsal thumb hole. Moving this small hole upwards, closer to the voicing will make it an "Echo Hole" (Dolmetsch Recorder Modification) that will give a precise half note above the fundamental when opened. Note: Slight size or diameter adjustment is needed to zero in on the precise half note frequency.[3]

A closed tube will have approximate resonances of:

 

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). This equation comes from the boundary conditions for the pressure wave, which treats the closed end as pressure antinodes where the change in pressure Δp must have the maximal amplitude, or satisfy ∂(Δp)/∂x = 0 in the form of the Sturm–Liouville formulation. 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.

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

 .

Again, when n is 1:

 
 
 
 

where v is the speed of sound, L is the length of the resonant tube, d is the diameter of the tube, f is the resonant sound frequency, and λ is the resonant wavelength.

Pressure wave edit

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. In diagram 1, the tube is open at both ends. In diagram 2, it is closed at one end. The horizontal axis is pressure. Note that in this case, the open end of the pipe is a pressure node while the closed end is a pressure antinode.

Cones edit

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:

 

where the wavenumber k is

 

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

 

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.

Closed rectangular box edit

Sound waves in a rectangular box include such examples as loudspeaker enclosures and buildings. Rectangular buildings have resonances described as room modes. For a rectangular box, the resonant frequencies are given by[5]

 

where v is the speed of sound, Lx and Ly and Lz are the dimensions of the box.  ,  , and   are nonnegative integers that cannot all be zero. If the small loudspeaker box is airtight, the frequency low enough and the compression is high enough, the sound pressure (decibel level) inside the box will be the same anywhere inside the box, this is hydraulic pressure.

Resonance of a sphere of air (vented) edit

The resonant frequency of a rigid cavity of static volume V0 with a necked sound hole of area A and length L is given by the Helmholtz resonance formula[6][7]

 

where   is the equivalent length of the neck with end correction

   for an unflanged neck[8]
   for a flanged neck
 

For a spherical cavity, the resonant frequency formula becomes

 

where

D = diameter of sphere
d = diameter of sound hole
 

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

 

Breaking glass with sound via resonance edit

 
Breaking glass with sound using resonance

This is a classic demonstration of resonance. A glass has a natural resonance, a frequency at which the glass will vibrate easily. Therefore the glass needs to be moved by the sound wave at that frequency. If the force from the sound wave making the glass vibrate is big enough, the size of the vibration will become so large that the glass fractures. To do it reliably for a science demonstration requires practice and careful choice of the glass and loudspeaker.[9]

In musical composition edit

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 tamtam 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 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."[10]

See also edit

References edit

  1. ^ Kinsler L.E., Frey A.R., Coppens A.B., Sanders J.V., "Fundamentals of Acoustics", 3rd Edition, ISBN 978-0-471-02933-5, Wiley, New York, 1982.
  2. ^ Wolfe, Joe. "Saxophone acoustics: an introduction". University of New South Wales. Retrieved 1 January 2015.
  3. ^ a b Kool, Jaap. Das Saxophon. J. J. Weber, Leipzig. 1931. Translated by Lawrence Gwozdz in 1987, discusses "open" and "closed" tubes.
  4. ^ Horns, Strings and Harmony, by Arthur H. Benade
  5. ^ Kuttruff, Heinrich (2007). Acoustics: An Introduction. Taylor & Francis. p. 170. ISBN 978-0-203-97089-8.
  6. ^ Wolfe, Joe. "Helmholtz Resonance". University of New South Wales. Retrieved 1 January 2015.
  7. ^ Greene, Chad A.; Argo IV, Theodore F.; Wilson, Preston S. (2009). A Helmholtz resonator experiment for the Listen Up project. Proceedings of Meetings on Acoustics. ASA. p. 025001. doi:10.1121/1.3112687.
  8. ^ Raichel, Daniel R. (2006). The Science and Applications of Acoustics. Springer. pp. 145–149. ISBN 978-0387-26062-4.
  9. ^ Acoustics research centre (14 January 2019). "How to break a glass with sound". University of Salford. Retrieved 17 January 2019.
  10. ^ Olofsson, Kent (4 February 2015). "Resonances and Responses". Divergence Press (4). University of Haddersfield Press. doi:10.5920/divp.2015.48.
  • 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.

External links edit

  • Standing Waves Applet

acoustic, resonance, this, article, about, mechanical, resonance, sound, including, musical, instruments, general, description, mechanical, resonance, physics, engineering, mechanical, resonance, general, description, resonance, resonance, phenomenon, which, a. This article is about mechanical resonance of sound including musical instruments For a general description of mechanical resonance in physics and engineering see Mechanical resonance For a general description of resonance see Resonance Acoustic resonance is a phenomenon in which an acoustic system amplifies sound waves whose frequency matches one of its own natural frequencies of vibration its resonance frequencies source source source source source source source source Experiment using two tuning forks oscillating at the same frequency One of the forks is being hit with a rubberized mallet Although the first tuning fork hasn t been hit the other fork is visibly excited due to the oscillation caused by the periodic change in the pressure and density of the air by hitting the other fork creating an acoustic resonance between the forks However if a piece of metal is placed on a prong the effect dampens and the excitations become less and less pronounced as resonance isn t achieved as effectively The term acoustic resonance is sometimes used to narrow mechanical resonance to the frequency range of human hearing but since acoustics is defined in general terms concerning vibrational waves in matter 1 acoustic resonance can occur at frequencies outside the range of human hearing 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 has tapering resonances across its length 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 Contents 1 Vibrating string 1 1 String resonance in music instruments 2 Resonance of a tube of air 2 1 Cylinders 2 1 1 Closed at both ends 2 1 2 Open at both ends 2 1 3 Closed at one end 2 1 4 Pressure wave 2 2 Cones 2 3 Closed rectangular box 3 Resonance of a sphere of air vented 4 Breaking glass with sound via resonance 5 In musical composition 6 See also 7 References 8 External linksVibrating string editFurther information Vibrating string nbsp String resonance of a bass guitar A note with fundamental frequency of 110 Hz In musical instruments strings under tension as in 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 n v 2 L displaystyle f nv over 2L nbsp where L is the length of the string for a string fixed at both ends and n 1 2 3 Harmonic in an open end pipe that is both ends of the pipe are open The speed of a wave through a string or wire is related to its tension T and the mass per unit length r v T r displaystyle v sqrt T over rho nbsp So the frequency is related to the properties of the string by the equation f n T r 2 L n T m L 2 L displaystyle f n sqrt T over rho over 2L n sqrt T over m L over 2L nbsp where T is the tension r 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 edit Main article String resonance music 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 editThe resonance of a tube of air is related to the length of the tube its shape and whether it has closed or open ends Many musical instruments resemble tubes that are conical or cylindrical see bore A pipe that is closed at one end and open at the other is said to be stopped or closed while an open pipe is open at both ends Modern orchestral flutes behave as open cylindrical pipes clarinets behave as closed cylindrical pipes and saxophones oboes and bassoons as closed conical pipes 2 while most modern lip reed instruments brass instruments are acoustically similar to closed conical pipes with some deviations see pedal tones and false tones Like strings vibrating air columns in ideal cylindrical or conical pipes also have resonances at harmonics although there are some differences Cylinders edit See also Standing wave Standing wave in a pipe Any cylinder resonates at multiple frequencies producing multiple musical pitches The lowest frequency is called the fundamental frequency or the first harmonic Cylinders used as musical instruments are generally open either at both ends like a flute or at one end like some organ pipes 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 Sound travels as a longitudinal compression wave causing air molecules to move back and forth along the direction of travel Within a tube a standing wave is formed whose wavelength depends on the length of the tube At the closed end of the tube air molecules cannot move much so this end of the tube is a displacement node in the standing wave At the open end of the tube air molecules can move freely producing a displacement antinode Displacement nodes are pressure antinodes and vice versa Closed at both ends edit The table below shows the displacement waves in a cylinder closed at both ends Note that the air molecules near the closed ends cannot move whereas the molecules near the center of the pipe move freely In the first harmonic the closed tube contains exactly half of a standing wave node antinode node Considering the pressure wave in this setup the two closed ends are the antinodes for the change in pressure Dp Therefore at both ends the change in pressure Dp must have the maximal amplitude or satisfy Dp x 0 in the form of the Sturm Liouville formulation which gives the equation for the pressure wave D p x t p max cos 2 p x l cos w t displaystyle Delta p x t p text max cos left 2 pi x over lambda right cos omega t nbsp The intuition for this boundary condition Dp 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 Applying the boundary condition Dp x 0 at x L gives the wavelengths of the standing waves l 2 L n n 1 2 3 displaystyle lambda frac 2L n n 1 2 3 nbsp And the resonant frequencies are f v l n v 2 L displaystyle f frac v lambda frac nv 2L nbsp Frequency Order Name 1 Name 2 Name 3 Wave representation Molecular representation 1 f 440 Hz n 1 1st partial fundamental tone 1st harmonic nbsp nbsp 2 f 880 Hz n 2 2nd partial 1st overtone 2nd harmonic nbsp nbsp 3 f 1320 Hz n 3 3rd partial 2nd overtone 3rd harmonic nbsp nbsp 4 f 1760 Hz n 4 4th partial 3rd overtone 4th harmonic nbsp nbsp Open at both ends edit In cylinders with both ends open air molecules near the end move freely in and out of the tube This movement produces displacement antinodes in the standing wave Nodes tend to form inside the cylinder away from the ends In the first harmonic the open tube contains exactly half of a standing wave antinode node antinode Thus the harmonics of the open cylinder are calculated in the same way as the harmonics of a closed closed cylinder The physics of a pipe open at both ends are explained in Physics Classroom 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 By overblowing an open tube a note can be obtained that is an octave above the fundamental frequency or note of the tube For example if the fundamental note of an open pipe is C1 then overblowing the pipe gives C2 which is an octave above C1 3 Open cylindrical tubes resonate at the approximate frequencies f n v 2 L displaystyle f nv over 2L nbsp 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 This equation comes from the boundary conditions for the pressure wave which treats the open ends as pressure nodes where the change in pressure Dp must be zero A more accurate equation considering an end correction is given below f n v 2 L 0 6 r displaystyle f nv over 2 L 0 6r nbsp where r is the radius 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 So when n is 1 f v 2 L 0 6 r displaystyle f v over 2 L 0 6r nbsp f 2 L 0 6 r v displaystyle f 2 L 0 6r v nbsp f l v displaystyle f lambda v nbsp l 2 L 0 6 r displaystyle lambda 2 L 0 6r nbsp where v is the speed of sound L is the length of the resonant tube r is the radius of the tube f is the resonant sound frequency and l is the resonant wavelength Closed at one end edit When used in an organ a tube which is closed at one end is called a stopped pipe Such cylinders have a fundamental frequency but can be overblown to produce other higher frequencies or notes These overblown registers can be tuned by using different degrees of conical taper A closed tube resonates at the same fundamental frequency as an open tube twice its length with a wavelength equal to four times its length In a closed tube a displacement node or point of no vibration always appears at the closed end and if the tube is resonating it will have a displacement antinode or point of greatest vibration at the Phi point length 0 618 near the open end By overblowing a cylindrical closed tube a note can be obtained that is approximately a twelfth above the fundamental note of the tube or a fifth above the octave of the fundamental note For example if the fundamental note of a closed pipe is C1 then overblowing the pipe gives G2 which is one twelfth above C1 Alternatively we can say that G2 is one fifth above C2 the octave above C1 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 4 Opening a small speaker hole at the Phi point or shared wave node position will cancel the fundamental frequency and force the tube to resonate at a 12th above the fundamental This technique is used in a recorder by pinching open the dorsal thumb hole Moving this small hole upwards closer to the voicing will make it an Echo Hole Dolmetsch Recorder Modification that will give a precise half note above the fundamental when opened Note Slight size or diameter adjustment is needed to zero in on the precise half note frequency 3 A closed tube will have approximate resonances of f n v 4 L displaystyle f nv over 4L nbsp 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 This equation comes from the boundary conditions for the pressure wave which treats the closed end as pressure antinodes where the change in pressure Dp must have the maximal amplitude or satisfy Dp x 0 in the form of the Sturm Liouville formulation The intuition for this boundary condition Dp x 0 at x L is that the pressure of the closed end will follow that of the point next to it A more accurate equation considering an end correction is given below f n v 4 L 0 3 d displaystyle f nv over 4 L 0 3d nbsp Again when n is 1 f v 4 L 0 3 d displaystyle f v over 4 L 0 3d nbsp f 4 L 0 3 d v displaystyle f 4 L 0 3d v nbsp f l v displaystyle f lambda v nbsp l 4 L 0 3 d displaystyle lambda 4 L 0 3d nbsp where v is the speed of sound L is the length of the resonant tube d is the diameter of the tube f is the resonant sound frequency and l is the resonant wavelength Pressure wave edit 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 In diagram 1 the tube is open at both ends In diagram 2 it is closed at one end The horizontal axis is pressure Note that in this case the open end of the pipe is a pressure node while the closed end is a pressure antinode nbsp 1 nbsp 2 Cones edit 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 k L n p tan 1 k x displaystyle kL n pi tan 1 kx nbsp where the wavenumber k is k 2 p f v displaystyle k 2 pi f v nbsp 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 n p displaystyle k L x approx n pi nbsp 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 Closed rectangular box edit Sound waves in a rectangular box include such examples as loudspeaker enclosures and buildings Rectangular buildings have resonances described as room modes For a rectangular box the resonant frequencies are given by 5 f v 2 ℓ L x 2 m L y 2 n L z 2 displaystyle 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 nbsp where v is the speed of sound Lx and Ly and Lz are the dimensions of the box ℓ displaystyle ell nbsp m displaystyle m nbsp and n displaystyle n nbsp are nonnegative integers that cannot all be zero If the small loudspeaker box is airtight the frequency low enough and the compression is high enough the sound pressure decibel level inside the box will be the same anywhere inside the box this is hydraulic pressure Resonance of a sphere of air vented editThe resonant frequency of a rigid cavity of static volume V0 with a necked sound hole of area A and length L is given by the Helmholtz resonance formula 6 7 f v 2 p A V 0 L e q displaystyle f frac v 2 pi sqrt frac A V 0 L eq nbsp where L e q displaystyle L eq nbsp is the equivalent length of the neck with end correction L e q L 0 75 d displaystyle L eq L 0 75d nbsp for an unflanged neck 8 L e q L 0 85 d displaystyle L eq L 0 85d nbsp for a flanged neck nbsp For a spherical cavity the resonant frequency formula becomes f v d p 3 8 L e q D 3 displaystyle f frac vd pi sqrt frac 3 8L eq D 3 nbsp where D diameter of sphere d diameter of sound hole dd nbsp For a sphere with just a sound hole L 0 and the surface of the sphere acts as a flange so f v p 3 d 8 0 85 D 3 displaystyle f frac v pi sqrt frac 3d 8 0 85 D 3 nbsp In dry air at 20 C with d and D in metres f in hertz this becomes f 72 6 d D 3 displaystyle f 72 6 sqrt frac d D 3 nbsp Breaking glass with sound via resonance edit nbsp Breaking glass with sound using resonance This is a classic demonstration of resonance A glass has a natural resonance a frequency at which the glass will vibrate easily Therefore the glass needs to be moved by the sound wave at that frequency If the force from the sound wave making the glass vibrate is big enough the size of the vibration will become so large that the glass fractures To do it reliably for a science demonstration requires practice and careful choice of the glass and loudspeaker 9 In musical composition editSeveral 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 tamtam 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 2 million US gallon 7 600 m3 cistern at Fort Worden WA which has a reverb with a 45 second decay Malmo 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 10 See also editHarmony Music theory Resonance Reverberation Standing wave Sympathetic string Reflection phase changeReferences edit Kinsler L E Frey A R Coppens A B Sanders J V Fundamentals of Acoustics 3rd Edition ISBN 978 0 471 02933 5 Wiley New York 1982 Wolfe Joe Saxophone acoustics an introduction University of New South Wales Retrieved 1 January 2015 a b Kool Jaap Das Saxophon J J Weber Leipzig 1931 Translated by Lawrence Gwozdz in 1987 discusses open and closed tubes Horns Strings and Harmony by Arthur H Benade Kuttruff Heinrich 2007 Acoustics An Introduction Taylor amp Francis p 170 ISBN 978 0 203 97089 8 Wolfe Joe Helmholtz Resonance University of New South Wales Retrieved 1 January 2015 Greene Chad A Argo IV Theodore F Wilson Preston S 2009 A Helmholtz resonator experiment for the Listen Up project Proceedings of Meetings on Acoustics ASA p 025001 doi 10 1121 1 3112687 Raichel Daniel R 2006 The Science and Applications of Acoustics Springer pp 145 149 ISBN 978 0387 26062 4 Acoustics research centre 14 January 2019 How to break a glass with sound University of Salford Retrieved 17 January 2019 Olofsson Kent 4 February 2015 Resonances and Responses Divergence Press 4 University of Haddersfield Press doi 10 5920 divp 2015 48 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 External links editStanding Waves Applet Retrieved from https en wikipedia org w index php title Acoustic resonance amp oldid 1211653043, 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