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.
- 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
- 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.
- 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.
- 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.
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.
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.
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.