When something close to the natural frequency of an object strikes it, you get an amplification of its vibration. The easiest way to think of resonance is pushing someone on a swing. You push them at exactly the same frequency at the swing is swinging -- the swing's natural frequency. Net effect: the person goes higher and higher. if you were pushing them slightly slower or slightly faster than the swing's frequency, you'd be out of sync, and the person wouldn't go as high.

You get the same phenomenon with sound, too. If you hum a perfect E, then you can actually see the string on a guitar start to vibrate at the same frequency. This is because it is resonating with your voice.

You can use resonance to do lots of curiously destructive things. For example, you know that stunt where the woman sings a note and the wineglass shatters? You can actually do it, and it is caused by resonance. The wineglass' natural frequency is the same as the opera singer's voice (you can work out the natural frequency by tapping the glass) and the vibrations caused by an opera singer's voice are violent enough to break the glass. There was a bridge made, called the Tacoma Narrows Bridge, whose natural frequency just happened to match the winds that used to strike it. This eventually caused the bridge to collapse, barely three months after it was completed. Finally, using infrasound, you can actually cause a person's organs to rupture and kill them if you cause them to vibrate at their natural frequency which is 3 to 7 Hz. This has happened.

once I thought we were born here with no clues
no path, no means, no scent of home
like a cellist without a bow,
grappling with an arcane instrument
before a vast audience of laughter

as if they knew better than me -
"Tabula rasa", as if babies come into being
with no brain or heart, no feeling,
nothing that might have been carried
from a lighter, timeless world

look at her fingers tremble on the strings -
she's not afraid of the sound
but of the audience, what they'll do
when the sound wakes their hearts -
one single note, to kiss, to destroy -

something to rise out of the brain
into the early evening skyline
they know the trees are shaking in the wind
they saw the constellations appearing
like diamonds sifted out of the sandy clouds

take care - they never asked to be reminded -
"I'll know when I fall in love" - how are you so sure -
except that you are a singing wineglass,
a bell that hums when a voice speaks underneath,
that knows the truth because you feel it making you true -

by its teeth, she'll put you to shame -
while you wander through glaciers, mazes
like endless Inca cities, stepped and geometric,
unable to escape the memory of death

except that you hear the violinist -
she doesn't know what she does, but the sound
is not bound by her knowledge - if you cry
when the crescendo takes hold of her hands,
what is it in you that moves, that resonates,

what did you recognise, that you feel so ruined,
devastated by happiness, reduced to nothing by love,
like an empty evening sky for seeing comets,
like wind for laughing, roads for the feeling of distance -
an empty peace in your clearlight bedroom.

In the realm of electricity, a circuit is said to be in resonance if the inductive and capacitive reactances of the circuit cancel out, leaving only the resistance. This state only occurs at one frequency called the resonant frequency. This frequency can be determined by using the following formula:

Frequency = 1 / 2pi * sqrt (capacitance * inductance)

If the capacitance is given in farads, and the inductance in henrys, the frequency will come out as hertz. This frequency not only is the frequency that the circuit will pass with the least impedance, it is also the fundmental frequency the circuit will assume if the circuit is charged, then allowed to discharge through itself.

The basic building blocks of the universe seem to be either waves or vibrating strings, and most of the things they make up move in bigger waves and vibrations. If we hope to understand much about the physical workings of the universe, then, we need to have some idea about the way waves and vibrations work. The details of wave motion vary, but many of the principles are universal.

Among the most important concepts to grasp are resonance and standing waves; these are fundamental to countless phenomena in almost every branch of physics. They also underlie the production and perception of speech and music, and have countless applications in engineering. Resonance is what allows gentle pushes to propel a child ever higher on a swing, and it is what allows whipping winds or marching armies to tear asunder seemingly solid bridges.

Broadly speaking, resonance is the reinforcement or creation of an oscillation by an in-coming wave. The energy delivered by the wave will generally be strongest if the wave is at the same frequency as the oscillation, because this allows the two of them to maintain the same phase relationship, so that the direction of the push always matches that of the oscillatory motion. In cases where the vibration is caused by the wave in the first place, the 'natural' frequency is what is important - the frequency at which an object will naturally vibrate if it is excited, as discussed below. If the wave and the oscillation have different frequencies, then sooner or later they will drift out of phase and the motion of one will work against that of the other - however, if the wave is at a multiple of the frequency of the oscillation, a net reinforcement can still result.

The nature of standing waves is closely tied up with resonance, and it is not possible to fully understand one without grasping the other. Standing waves occur whenever a steady wave hits a reflecting barrier. The reflected wave travels at the same speed as the incoming wave, but in the opposite direction; this means that the peaks and troughs of each interfere with those of the other to make a pattern of 'nodes' and 'anti-nodes' - still points, and points which alternate between being crests and troughs. The strongest standing waves occur when the waves are reflected back again, and fit snugly inside a space which is just the right size and shape to allow incoming waves to be in phase with their own reflections and re-reflections; the frequencies at which this occurs are the resonant frequencies of the object the waves are in.

This effect, in which reflected waves are bounced back again after a whole number of wavelengths, is one of the most important kinds of resonance, and is the reason why tuning forks, for example, ring at a particular pitch. The fundamental or 'natural' frequency of anything which we ring or pluck to produce tones is generally the main pitch it makes. It amounts to the number of times a sound wave can travel from one end of the object to the other and back again in a second.

It is only waves of this frequency, or multiples thereof (harmonics), which consistently interfere constructively with their reflections and re-reflections. Anything else will soon be out of phase with the incoming wave, so the wave will actually reduce the energy of the system through destructive interference.

Conversely, an incoming sound matching one of the resonant frequencies of an object will cause larger and larger vibrations, limited only by damping - hence the supposed ability of some opera singers to shatter wine glasses, and also the possibility of tuning a guitar by watching the strings carefully.

There are many different types of resonance, and they are important in an endless variety of contexts. The following list covers many, but by no means all...

### Types of Resonance, and their Applications

• Acoustic Resonance - the sound of a musical instrument is always the result of one or more kind of acoustic resonance; the types of resonance involved affect which harmonics we hear, and hence the timbre of the note:
• Helmholtz Resonance - a cavity with an opening resonates at a frequency which depends only on its volume and the dimensions of the opening - in principle, the shape of the hollow makes no difference. The classic example of Helmholtz resonance is the sound made when you blow across the top of a bottle; the effect is also significant in string instruments, where the air vibrating inside the body boosts certain notes.
• Resonating strings - the strings of string instruments and pianos resonate simply when whole numbers of wavelengths fit into their length. The speed of the waves in a string (and hence its fundamental frequency) depends on their weight and tension, which is why these instruments have strings of different thicknesses, with pegs to adjust the tension.
• Tuning forks - like strings, tuning forks and the like carry waves along their length. Since they are fixed at one end, though, they only resonate with waves at odd multiples of their length.
• Drum-skins and sounding-boards - resonate in two dimensions, making more resonant frequencies possible. The mathematics of this are more complex, but the principles are the same. You can see pictures illustrating the resonances of the front and back plates of a violin at http://www.phys.unsw.edu.au/~jw/patterns1.html - and there is a very good applet at http://www.falstad.com/circosc/ which demonstrates many of the possible modes of vibration.
• 3D acoustic resonance - designers of concert halls and so on need to be very careful about their acoustics, because resonance can reinforce certain frequencies at the expense of others - sometimes this is desirable, but in many cases great efforts are made to reduce resonance as much as possible.
• The human vocal system uses a combination of these effects to produce speech. The details of this process are the subject of the branch of linguistics called articulatory phonetics; understanding how we make speech-sounds helps us to understand the ways that language can evolve.
• Atomic-level resonance
• One of the great advances made possible by quantum mechanics was the ability of physical chemists to explain the properties of the chemical elements in terms of standing waves made by the electrons encircling their nuclei. Atoms only emit and absorb radiation at particular frequencies, which depend on the energies of different electron 'orbitals'. Electrons, like all subatomic particles, act like waves in most circumstances; their orbitals correspond to patterns of standing waves around the atom. The absorption of light by atoms therefore fits under the definition of resonance given above.
• Magnetic Resonance Imaging (MRI) is a hugely important medical imaging technique, which works by applying a magnetic field to a person's body and detecting the magnetic resonance of molecules, chiefly hydrogen, in order to determine their distribution.
• Electromagnetic resonance
• Aerials work by resonating with incoming electromagnetic waves (radio, TV, microwave, etc.) - typically they are tuned to a particular frequency, and sensitive to a range of frequencies around it. They may also pick up higher harmonics of that frequency.
• Electrical resonance
• An electrical circuit including an inductor and a capacitor will resonate at a frequency depending on the inductance and capacitance involved, with energy oscillating between the magnetic field of the inductor and the electric field of the capacitor. Circuits like this are useful for generating waves electronically, and for filtering unwanted frequencies in what is called a band-pass filter. They are also the most important component of the highly entertaining Tesla coil, which creates artificial lightning effects - although it never fulfilled its inventor's original dream - the mass-distribution of electrical power without the need for wires.
• Orbital resonance
• The orbit of a planet or a satellite can be more stable when its period matches a simple ratio of the orbit of another body nearby; for example, the orbits of Jupiter's moons Ganymede, Europa and Io are stabilised by being in the ratio 1:2:4. This is very reminiscent of the ancient idea of the Music of the Spheres; it seems that Pythagoras and his followers got the idea at least partly right, however wildly wrong they were about the details. A separate but related phenomenon is tidal locking, the effect which has caused Earth's Moon always to face towards the planet.
• Synthesisers
• Sophisticated synths (electronic sound synthesisers) generally have a setting called resonance, which gives a boost to sounds of certain frequencies (specifically, those which are close to the cut-off point for band-pass filters).
• Psychological resonance
• People often talk of ideas, stories, poems and so on resonating with them, or with other ideas. The analogy is that just as a sound can set off sympathetic vibrations in something with a matching resonant frequency, one idea can excite other ideas with something in common, reinforcing the original idea rather like a sounding-board reinforces the sound of a guitar string. While this is not obviously the same kind of resonance as I have discussed elsewhere, it is an enormously important concept in the arts, being one of the key features which can make a work of art powerful, and as such it deserves a mention.

### Dangers of Resonance

• Machines
• Many of the physical limitations of machines, including vehicles and manufacturing plants, are determined by their susceptibility and resistance to vibrations. The most destructive vibrations are often those at resonant frequencies of some part of the machine, and avoiding these is crucial in high-powered machinery.
• Buildings
• The buildings which are most damaged by earthquakes are often those unfortunate enough to have a resonant frequency matching the frequency of the quake. Tall buildings in earthquake zones are often built with ingenious systems of dampers to absorb the vibrations of the incoming earthquake waves, chiefly to reduce the danger of this happening.
• Wind can also be a problem for tall buildings, when their geometry causes the wind to buffet them at a resonant frequency. Although it is rare for any building to get blown right over, people always feel unsafe when the floor they are standing on sways; for this reason, tall buildings in windy areas often employ the same sort of dampers as those in earthquake zones.
• Other sources of vibration - such as heavy machinery, or many people walking or dancing - can cause floors to resonate disconcertingly.
• Bridges
• Armies are trained to break step when crossing bridges; if they all marched in time, and their pace happened to match a resonant frequency of the bridge, there would be a serious danger of collapse.
• The Tacoma Narrows Bridge was ripped apart by a complicated resonance effect, with the 40mph wind forming periodic vortices which mutually reinforced the turning motion of the bridge. This is the classic textbook example of forced resonance, but the full explanation is more complicated than you may have been led to believe - and more complicated than I have space to explain here.
• London's Millennium Bridge was closed after a few days because it wobbled alarmingly as 80,000 people walked across on its opening day. The engineers had made allowances for the well-known danger of resonant effects from the vertical motion of people stomping across, but they had not realised the danger posed by people swaying from side to side as they walk, with a frequency half that of their footfalls. Once the bridge began to sway, people walking across sub-consciously fell into step with it, swaying along with the bridge and amplifying its motion in a sort of vicious circle. Expensive hydraulic dampers were put in place to control the motion, and the bridge was finally re-opened more than a year later.
• The Human Body
• Certain frequencies of vibration can cause people to feel unwell, and sometimes do real damage, by causing resonant vibrations of their abdominal cavity, head, eyes or other parts. This is something which designers of heavy machinery and vehicles - especially race cars - need to be aware of, in order to minimise discomfort caused to their users.
• Possible military applications of such sounds have long been talked about. Although these do appear to be technically feasible, their actual military usefulness is questionable.
• Wolf notes
Unwanted resonance in musical instruments can result in a wolf note - a note which causes the whole instrument to resonate, much louder than other notes.
• Feedback
Microphones which are in range of the speakers amplifying them can produce howling or screeching sounds, rising in volume, as they reproduce the sound waves they picked up just on the other side of the room, one or more wavelengths behind. These can usually be avoided either by changing the distance between the two, or reducing the amplification.
• Plumbing
• The strange sounds produced by plumbing are often the result of resonance within the system, with various sources of vibration causing humming, howling, whistling and banging as they hit resonant frequencies of pipes, boilers and radiators. The precise dynamics of this are largely mysterious.

This writeup was produced partly to complement a physics toy which I have made, Resonata - you can see this at http://oolong.co.uk/resonata.htm. I am hoping that this - both the text and the toy - will be useful for anyone learning or teaching about waves. For now you will need Java to view it, but a Flash version will follow shortly.

I would like to thank unperson, wrinkly, redbaker, Calast, tdent and quantumlemur for their helpful suggestions.

Resonances are the cornerstone of experimental high-energy particle physics. A resonance is the primary signature of a short-lived exotic particle, and they are relatively easy to detect and measure. Careful study of a resonance can yield a measurement of a particle's mass and lifetime.

### Where Resonances Appear

Particle physics experiments, large or small, involve observing reactions between particles. Resonances are cleanest and easiest to see in simple reactions, and the simplest higher-energy reactions in particle physics come from lepton-antilepton collisions. In this writeup, I will primarily consider electron-positron collisions as they are simple but frequently used in particle physics experiments; the simplicity allows for relatively high precision. The mechanism for finding resonances is much the same for more complicated reactions, such as proton-antiproton annihilation.

For a particular choice of reaction, such as e+e- -> μ+μ-, the probability of occurence is quantified with a number called the cross-section, which is a function of the incoming particle energies. The variation of the cross-section contains vast amounts of information about the mechanism of the reaction.

Now in modern particle theory, a reaction like the above is not a direct conversion:

``` μ+     μ-    ^
\     /    / \
\   /      |
\ /       |
.        | increasing time
/ \       |
/   \      |
/     \     |
e+     e-
```

Rather, the particle interaction is mediated by virtual particles, in this case usually a photon. This reaction can then be written as e+e- -> γ -> μ+μ-, or shown schematically as:

```μ+     μ-
\     /
\   /
\ /
.
|
| γ (virtual photon)
|
.
/ \
/   \
/     \
e+     e-
```

The cross-section for this reaction is relatively constant with energy, and can be calculated from theory to high precision.

However, this mechanism is not the only reaction of the form e+e- -> stuff -> μ+μ-. If the total energy of the two particles is close to the mass of a real particle, such as a J/ψ meson (whose double name is due to its near-simultaneous discovery by two groups each with their own ideas on naming), the e+e- collision can produce a real particle rather than a virtual photon that decays to two muons, i.e.

```μ+     μ-
\     /
\   /
\ /
O
|
| J/ψ (real meson)
|
O
/ \
/   \
/     \
e+     e-
```
The probability of J/ψ production adds to the probability of the photon-mediated reaction, but all that is seen is the initial particles and final particles. So, the cross-section for the reaction e+e- -> μ+μ- increases in the area where J/ψ production is likely, namely with energy close to the mass of the J/ψ. Thus, a peak appears in the graph of cross-section versus energy at the J/ψ mass; this is the J/ψ resonance. (This is actually how the J/ψ particle was discovered.)

### What Resonances Mean

Resonances will appear in virtually any reaction, so long as the available quantum numbers match that of the produced particle. An electron-positron collision will never see the resonance of a (single) charged particle, since the total charge available is zero. Similar constraints apply to other particle characteristics ('quantum numbers') such as spin and parity, and so the appearance or non-appearance of resonances in the cross-sections of different reactions will reveal information about the quantum numbers of the corresponding particle.

The centre of a resonance peak will be found at the rest mass of the corresponding particle. This is the best way to measure the masses of the legions of short-lived, massive particles that exist. Furthermore, due to the energy-time uncertainty principle, δEδt ≥ h/4π, the width of a resonance peak is inversely proportional to the particle's lifetime. Thus, careful plotting of a resonance peak can be used to determing both the mass and lifetime of a particle.

Each distinct resonance appears to be a distinct particle. A particle will have the same mass as its antiparticle, but the two particles will have opposite quantum numbers; aside from this every resonance of the same central energy can be said to arise from the same particle. Thus, we classify every set of measured resonances with the same quantum numbers and energy as a particle and assign it a particle name.

### Resonance Classification and Particle Structure

In the early days of particle physics, each particle classified in this way was interpreted as its own fundamental particle. Soon, the bewildering array of new particles caused one particle physicist to remark, "Once, the discoverer of a new fundamental particle would earn a Nobel Prize; now, it seems that it should be punishable by a \$10,000 fine."

With Murray Gell-Mann's work on the Eightfold Way and the ensuing development of the quark model, it became clear that the vast majority of observed particles are not fundamental particles, but rather various composites of mostly unobserved particles called quarks. The Standard Model of particle physics contains fewer than two dozen particles, plus their corresponding antiparticles, six of which are quarks. All of the composite particles fall into two classes: baryons, which are composites of three quarks or three antiquarks (an antibaryon), and mesons, which are composed of one quark and one antiquark.

Having chosen a compliment of quark and/or antiquark types (whimsically called 'flavours'), there are multiple ways in which they can be combined. The most basic difference that can be applied is different alignment of spin. Quarks all have a spin of 1/2, which can be set up to point in one of two directions. For a meson, the spins of the two particles can be aligned, in which case the spins add giving the meson a spin of 1, or the spins can be anti-aligned, in which case they subtract and the meson has a spin of 0. Baryons work much the same way: either all three are aligned giving a spin of 3/2, or one is opposite the other two giving spin 1/2.

These different spin states apppear, given the classification scheme above, as different particles, even though they have the same constituents. Aligned spins have higher potential energy; spin-bearing charged particles have magnetic moments that cause them to act somewhat like bar magnets, and particles with the same spin direction effectively have their north poles pointing the same direction. Thus, since energy is mass, the higher-spin mesons and baryons will have higher masses.

The physics of a composite particle is even more complicated than spin alignments. Two classical objects can have any given choice of relative energy; a satellite can be orbited around the Earth at any altitude the launching rocket can put it at. For quantum objects such as quarks, and electrons in atoms, there are discrete steps of relative potential energy. Thus, when building a meson, the two quarks can be placed at a number of different 'separations' each having a different potential energy. These different potential energies lead to different masses, and so the energy levels of the two-quark (or three-quark) system show up as different particles, which depending on the arrangement of the quarks may even have different quantum numbers. Borrowing from atomic physics, the lowest-energy of these states is called the ground state and all of the others are called excited states.

This all brings into question exactly what we mean when we declare two sets of resonances to be 'different particles'. For one, the nomenclature of different spin-states is often held over from before the quark model. For example, the spin-0 combination of an up quark and an anti-down quark is called a π+, while the spin-1 combination of the same quarks is called a ρ+. Similarly, we call a spin-1/2 up-up-down baryon a proton (p+), while the spin-3/2 combination is a Δ+. Nomenclature for excited states reflects more modern knowledge; the first excited state of the pion is called π(1300), the second π(1800). The number in parentheses is the resonance energy in MeV (mega-electronvolts).

In my opinion, the convention most consistent with both history and the quark model is to consider all resonances of the same quark content and spin states to be the same particle, i.e. there is a particle called π+ whose ground state is at 140 MeV, and has excited states at 1300 MeV, 1800 MeV and so on. One might say that the π+ and ρ+ are 'really the same particle' too, and that is valid, but this is entirely a matter of taste.

Thanks to IWhoSawTheFace for inspiration. Particle physics knowledge taken from my undergraduate and graduate particle physics courses, the textbook Introduction to Elementary Particles by David J. Griffiths, and the Particle Data Group website at http://pdg.lbl.gov/ .
(CC)
This writeup is copyright 2005 D.G. Roberge and is released under the Creative Commons Attribution-NoDerivs-NonCommercial licence. Details can be found at http://creativecommons.org/licenses/by-nd-nc/2.5/ .

Res"o*nance (r?z"?-nans), n. [Cf. F. résonance, L. resonantia an echo.]

1.

The act of resounding; the quality or state of being resonant.

2. (Acoustics)

A prolongation or increase of any sound, either by reflection, as in a cavern or apartment the walls of which are not distant enough to return a distinct echo, or by the production of vibrations in other bodies, as a sounding-board, or the bodies of musical instruments.

Pulmonary resonance (Med.), the sound heard on percussing over the lungs. --
Vocal resonance (Med.), the sound transmitted to the ear when auscultation is made while the patient is speaking.

Res"o*nance, n.

An electric phenomenon corresponding to that of acoustic resonance, due to the existance of certain relations of the capacity, inductance, resistance, and frequency of an alternating circuit.

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