Specific detectivity

2013-08-30T03:34:08Z

24.37.252.218: The definition of D* is incorrect as the units are not in agreement. D* has the units of m-Hz^1/2/W. I have added the missing frequency bandwith term.

{{Other uses|Hydraulic jump}}
{{Refimprove|date=September 2011}}

In [[physics]], a '''standing wave''' – also known as a '''stationary wave''' – is a [[wave]] that remains in a constant position.
[[File:standingwaves.svg|thumb|250px|Two opposing waves combine to form a standing wave.]]

This phenomenon can occur because the medium is moving in the opposite direction to the wave, or it can arise in a stationary medium as a result of [[Interference (wave propagation)|interference]] between two waves traveling in opposite directions. In the second case, for waves of equal [[amplitude]] traveling in opposing directions, there is on [[average]] no net [[Flux#Transport phenomena|propagation of energy]].

In a [[resonator]], standing waves occur during the phenomenon known as [[resonance]].

== Moving medium ==
As an example of the first type, under certain meteorological conditions standing waves form in the atmosphere in the [[Lee waves|lee]] of mountain ranges. Such waves are often exploited by [[gliding|glider pilots]].

Standing waves and [[hydraulic jump]]s also form on fast flowing [[Rapid|river rapids]] and tidal currents such as the [[Saltstraumen]] [[maelstrom]]. Many standing river waves are popular [[River_surfing#Standing_waves|river surfing]] breaks.

== Opposing waves ==
{|align="right" class="toccolours" style="margin:0 0 1em 1em;"
| style="background:#ccccff; text-align:center;"|
<center>
; Standing waves
</center>
|-
|
<gallery>
Image:Standing wave.gif|Standing wave in stationary medium. The red dots represent the wave [[Node (physics)|nodes]].
Image:Standing wave 2.gif|A standing wave (black) depicted as the sum of two propagating waves traveling in opposite directions (red and blue).
</gallery>
|-
|
<gallery>
File:Standing wave.svg|Electric force vector (E) and magnetic force vector (H) of a standing wave.
Image:Standing waves on a string.gif|Standing waves in a string — the [[fundamental frequency|fundamental]] mode and the first 5 [[overtone]]s.
</gallery>
|-
|
<gallery>
File:Drum vibration mode01.gif|A three-dimensional [[Vibrations of a circular drum|standing wave on a disk]]; this is the fundamental mode
Image:Drum vibration mode21.gif|A higher harmonic [[Vibrations of a circular drum|standing wave on a disk]] with two nodal lines crossing at the center.
</gallery>
|-
|}
As an example of the second type, a ''standing wave'' in a [[transmission line]] is a wave in which the distribution of [[current (electricity)|current]], [[voltage]], or [[field strength]] is formed by the [[Superposition principle|superposition]] of two waves of the same [[frequency]] propagating in opposite directions. The effect is a series of [[node (physics)|node]]s (zero [[particle displacement|displacement]]) and [[anti-node]]s (maximum [[particle displacement|displacement]]) at fixed points along the transmission line. Such a standing wave may be formed when a wave is transmitted into one end of a transmission line and is [[reflection (electrical)|reflected]] from the other end by an [[Electrical impedance|impedance]] [[Impedance matching|mismatch]], ''i.e.'', discontinuity, such as an [[open circuit]] or a [[short circuit|short]].<ref>{{FS1037C}}</ref> The failure of the line to transfer power at the standing wave frequency will usually result in [[attenuation distortion]].

In practice, losses in the transmission line and other components mean that a perfect reflection and a pure standing wave are never achieved. The result is a ''partial standing wave'', which is a superposition of a standing wave and a traveling wave. The degree to which the wave resembles either a pure standing wave or a pure traveling wave is measured by the [[standing wave ratio]] (SWR).<ref>{{citation | title=Fundamentals of Physical Acoustics | first=David T. | last=Blackstock | publisher=Wiley–IEEE | year=2000 | isbn=0-471-31979-1 }}, 568 pages. See page 141.</ref>

Another example is standing waves in the open [[ocean]] formed by waves with the same wave period moving in opposite directions. These may form near storm centres, or from reflection of a swell at the shore, and are the source of [[microbarom]]s and [[microseism]]s.

===Mathematical description===
In one dimension, two waves with the same frequency, wavelength and amplitude traveling in opposite directions will interfere and produce a standing wave or stationary wave. For example: a wave traveling to the right along a taut string and hitting the end will reflect back in the other direction along the string, and the two waves will superpose to produce a standing wave. The reflective wave has to have the same amplitude and frequency as the incoming wave.

If the string is held at both ends, forcing zero movement at the ends, the ends become zeroes or ''[[node (physics)|node]]s'' of the wave. The length of the string then becomes a measure of which waves the string will entertain: the longest wavelength is called the ''[[Fundamental frequency|fundamental]]''. Half a wavelength of the fundamental fits on the string. Shorter wavelengths also can be supported as long as multiples of half a wavelength fit on the string. The frequencies of these waves all are multiples of the fundamental, and are called ''[[harmonic]]s'' or ''[[overtone]]s''. For example, a guitar player can select an overtone by putting a finger on a string to force a node at the proper position between the ends of the string, suppressing all harmonics that do not share this node.

Harmonic waves travelling in opposite directions can be represented by the equations below:
:<math>
y_1\; =\; y_0\, \sin(kx - \omega t)\,
</math>

and
:<math>
y_2\; =\; y_0\, \sin(kx +\omega t)\,
</math>

where:
*''y0'' is the [[amplitude]] of the wave,
*''ω'' (called [[angular frequency]], measured in ''[[radian]]s per second'') is ''2π'' times the [[frequency]] (in ''[[hertz]]''),
*''k'' (called the [[wave number]] and measured in ''radians per metre'') is ''2π'' divided by the [[wavelength]] ''λ'' (in ''metres''), and
*''x'' and ''t'' are variables for longitudinal position and time, respectively.

So the resultant wave ''y'' equation will be the sum of ''y1'' and ''y2'':
:<math>
y\; =\; y_0\, \sin(kx - \omega t)\; +\; y_0\, \sin(kx + \omega t).\,
</math>

Using the [[Trigonometric_identity#Product-to-sum_and_sum-to-product_identities|trigonometric sum-to-product identity]] for 'sin(''u'') + sin(''v'')' to simplify:
:<math>
y\; =\; 2\, y_0\, \cos(\omega t)\; \sin(kx).\,
</math>

This describes a wave that oscillates in time, but has a spatial dependence that is stationary: sin(''kx''). At locations ''x'' = 0, ''λ''/2, ''λ'', 3''λ''/2, ... called the [[node (physics)|node]]s the amplitude is always zero, whereas at locations ''x'' = ''λ''/4, 3''λ''/4, 5''λ''/4, ... called the [[anti-node]]s, the amplitude is maximum. The distance between two conjugative nodes or anti-nodes is ''λ''/2.

Standing waves can also occur in two- or three-dimensional [[resonator]]s. With standing waves on two dimensional membranes such as [[drumhead]]s, illustrated in the animations above, the nodes become nodal lines, lines on the surface at which there is no movement, that separate regions vibrating with opposite phase. These nodal line patterns are called [[Chladni figure]]s. In three-dimensional resonators, such as musical instrument [[sound box]]es and microwave [[cavity resonator]]s, there are nodal surfaces.

==Examples==
One easy to understand example of standing waves is two people shaking either end of a [[jump rope]]. If they shake in sync the rope can form a regular pattern of waves oscillating up and down, with stationary points along the rope where the rope is almost still (nodes) and points where the arc of the rope is maximum (antinodes)

===Sound waves===
[[File:Rotatingsaturnhexagon gif.ogv|frame|right|The hexagonal cloud feature at the north pole of [[Saturn]] was initially thought to be standing [[Rossby wave]]s.<ref>[http://pubs.giss.nasa.gov/docs/1990/1990_Allison_etal.pdf A Wave Dynamical Interpretation of Saturn's Polar Region], M. Allison, D. A. Godfrey, R. F. Beebe, Science vol. 247, pg. 1061 (1990)</ref> This explanation has recently been disputed though.<ref>[http://dx.doi.org/10.1016/j.icarus.2009.10.022 A laboratory model of Saturn’s North Polar Hexagon], A. C. Barbosa Aguiar, P. L. Read, R. D. Wordsworth, T. Salter, Y. H. Yamazaki, Icarus, vol. 206 (2009)</ref>]]
Standing waves are also observed in physical media such as strings and columns of air. Any waves traveling along the medium will reflect back when they reach the end. This effect is most noticeable in musical instruments where, at various multiples of a [[vibrating string]] or [[air column]]'s [[natural frequency]], a standing wave is created, allowing [[harmonics]] to be identified. Nodes occur at fixed ends and anti-nodes at open ends. If fixed at only one end, only odd-numbered harmonics are available. At the open end of a pipe the anti-node will not be exactly at the end as it is altered by its contact with the air and so [[end correction]] is used to place it exactly. The density of a string will affect the frequency at which harmonics will be produced; the greater the density the lower the frequency needs to be to produce a standing wave of the same harmonic.

===Light===
Standing waves are also observed in optical media such as optical wave guides, [[optical cavity|optical cavities]], etc. [[Laser]]s use optical cavities in the form of a pair of facing mirrors. The gain medium in the cavity (such as a crystal) emits light coherently, exciting standing waves of light in the cavity. The wavelength of light is very short (in the range of [[nanometer]]s, 10-9 m) so the standing waves are microscopic in size. One use for standing light waves is to measure small distances, using [[optical flat]]s.

===Mechanical waves===
Standing waves can be mechanically induced into solid medium using resonance. One easy way to understand example is two people shaking either end of a jump rope. If they shake in sync, the rope will form a regular pattern with nodes and antinodes and appear to be stationary, hence the name standing wave. Similarly a cantilever beam can have a standing wave imposed on it by applying a base excitation. In this case the free end moves the greatest distance laterally compared to any location along the beam. Such a device can be used as a [[sensor]] to track changes in [[natural frequency|frequency]] or [[phase shift|phase]] of the resonance of the fiber. One application is as a measurement device for [[dimensional metrology]].<ref>{{citation | title= The development of a virtual probe tip with application to high aspect ratio microscale features| author=M.B. Bauza, R.J Hocken, S.T Smith, S.C Woody,| publisher=Rev. Sci Instrum, 76 (9) 095112 | year=2005 }} .</ref><ref>http://www.insitutec.com</ref>
===Seismic waves===
Standing surface waves on the Earth are observed as [[seismic wave#free oscillations of the Earth| free oscillations of the Earth]].

==See also==
*'''[[Index of wave articles]]''':
:[[Amphidromic point]], [[Clapotis]], [[Longitudinal mode]], [[Modelocking]], [[Metachronal rhythm]]. [[Resonant room modes]], [[Seiche]], [[Trumpet]], [[Voltage standing wave ratio]], [[Wave]], [[Kundt's tube]]
*'''[[List of electronics topics]]''':
:[[Cavity resonator]], [[Characteristic impedance]], [[Cymatics]], [[Electrical impedance|Impedance]], [[Normal mode]]

==References and notes==
{{reflist}}

==External links==
*[http://blog.kaistale.com/?p=889 Simulation of standing waves in a room using WebGL]
*[http://www.falstad.com/loadedstring/ Java applet of standing waves on a vibrating string]
*[http://www.phy.hk/wiki/englishhtm/TwaveStatA.htm Java applet of transverse standing wave]
*[http://www.phy.hk/wiki/englishhtm/StatWave.htm Java applet showing the production of standing wave on a string by adjusting frequency]
*[http://www.stmary.ws/highschool/physics/home/notes/waves/StandingWave.htm Standing Wave Lessons - Includes helpful graphics, flash animations, YouTube videos]

{{DEFAULTSORT:Standing Wave}}
[[Category:Wave mechanics]]

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Specific detectivity