Acoustic resonance

Acoustic resonance is a phenomenon in which an acoustic system responds strongly to sound waves whose frequency matches one of its own natural frequencies of vibration.
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, 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.
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

In musical instruments, strings under tension, as in lutes, harps, guitars, pianos, violins and so forth, have resonant frequencies determined primarily by string length, mass per unit length, and tension. In the ideal string model, the fundamental resonance has a wavelength equal to twice the string length. 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 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:
So the frequency is related to the properties of the string by the equation
Under these assumptions, the resonant frequencies form an exact harmonic series. Higher tension and shorter lengths increase the resonant frequencies. When the string is excited with an impulsive function, the string vibrates at all the frequencies present in the impulse 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.
Measurements of piano strings show that their partials are not exact integer multiples of the fundamental frequency. Robert W. Young measured the inharmonicity of plain steel piano strings in situ in six pianos of different sizes and makes. He showed that the frequency deviation increases with mode number and depends on string diameter and vibrating length. The results follow stiff-string theory, which predicts a quadratic dependence on mode number and an inverse fourth-power dependence on string length. Young found that inharmonicity near middle C is similar across pianos but is lower in larger instruments and increases rapidly at higher pitches.

String resonance in music instruments

occurs on string instruments. In some cases, a string may be excited indirectly by another sounding string if their harmonic frequencies are closely related, producing sympathetic vibration. For example, an A string at 440 Hz can excite an E string at 330 Hz because both produce a harmonic near 1320 Hz, allowing energy transfer between the strings.

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. Many musical instruments resemble tubes that are conical or cylindrical. 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,
while most modern lip-reed instruments are acoustically similar to closed conical pipes with some deviations.
Like strings, vibrating air columns in ideal cylindrical or conical pipes also have resonances at harmonics, although there are some differences.

Cylinders

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.
In real cylinders, resonances deviate from the ideal harmonic series. Effects such as end correction, radiation impedance, and interaction between the air jet and the resonator shift the higher modes. Practical pipes are only approximately harmonic.

Closed at both ends

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. 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, which gives the equation for the pressure wave:. The intuition for this boundary condition at and is that the pressure of the closed ends will follow that of the point next to them. Applying the boundary condition at 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

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. 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 . 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.
Open cylindrical tubes resonate at the approximate frequencies:
where n is a positive integer representing the resonance node, L is the length of the tube and v is the speed of sound in air. 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

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 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. 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" 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.
A closed tube will have approximate resonances of:
where "n" here is an odd number. This type of tube produces only odd harmonics and has its fundamental frequency an octave lower than that of an open cylinder. 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 in the form of the Sturm–Liouville formulation. The intuition for this boundary condition at 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.