- Standing wave
A

**standing wave**, also known as a**stationary wave**, is awave that remains in a constant position. 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 ofinterference between two waves traveling in opposite directions. In the second case, for waves of equalamplitude travelling in opposing directions, there is onaverage no net propagation of energy.**Moving medium**As an example of the first type, under certain meteorological conditions standing waves form in the atmosphere in the lee of mountain ranges. Such waves are often exploited by glider pilots.

Standing waves and

hydraulic jump s also form on fast flowing river rapids and tidal currents such as theSaltstraumen maelstrom .**Opposing waves** As an example of the second type, a "standing wave" in a

nodes.transmission line is a wave in which the distribution of current,voltage , orfield strength is formed by thesuperposition of two waves propagating in opposite directions. The effect is a series of nodes (zero displacement) andanti-node s (maximum 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 reflected from the other end by an impedance mismatch, "i.e.",discontinuity , such as anopen circuit or a short. The failure of the line to transfer power at the standing wave frequency will usually result inattenuation distortion .Another example are 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 ofmicrobarom s andmicroseism s.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 travelling wave. The degree to which the wave resembles either a pure standing wave or a pure travelling wave is measured by the

standing wave ratio (SWR).**Mathematical description**In one dimension, two waves with the same frequency, wavelength and amplitude travelling in opposite directions will interfere and produce a standing wave or stationary wave. For example: a harmonic wave travelling to the right and hitting the end of the string produces standing wave. The reflective wave has to have the same amplitude and frequency as the incoming wave.

Let the harmonic waves be represented by the equations below:

:$y\_1;\; =;\; y\_0,\; sin(kx\; -\; omega\; t)$and:$y\_2;\; =;\; y\_0,\; sin(kx\; +\; omega\; t)$

where:

*"y_{0}" is theamplitude of the wave,

*"ω" (calledangular frequency , measured in "radian s per second") is "2π" times thefrequency (in "hertz "),

*"k" (called thewave number and measured in "radians per metre") is "2π" divided by thewavelength "λ" (in "metres"), and

*"x" and "t" are variables for longitudinal position and time, respectively.

standing wave on a disk.So the resultant wave "y" equation will be the sum of "y

_{1}" and "y_{2}"::$y;\; =;\; y\_0,\; sin(kx\; -\; omega\; t);\; +;\; y\_0,\; sin(kx\; +\; omega\; t).$

Using a

trigonometric identity to simplify, the standing wave is described by::$y;\; =;\; 2,\; y\_0,\; cos(omega\; t);\; sin(kx).$

At nodes "x = 0, λ/2, λ, 3λ/2 ···" etc. whereas at

anti-node s "x = λ/4, 3λ/4, 5λ/4 ···" etc.The distance between two conjugative nodes or anti-nodes is "λ/2".Standing waves can also occur in more than one dimension, such as in a

resonator . The illustration on the right shows a standing wave on a disk.**Physical waves**[

thumb|right|180px|The_hexagonal_cloud_feature_at_the_north_pole_of_Saturn is thought to be some sort of standing wave pattern.] Standing waves are also observed in physical media such as strings and columns of air. Any waves travelling along the medium will reflect back when they reach the end. This effect is most noticeable in musical instruments where, at various multiples of avibrating string orair column 'snatural frequency , a standing wave is created, allowingharmonics 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 it's contact with the air and soend correction is used to place it exactly.**Optical waves**Standing waves are also observed in optical media such as optical wave guides, optical cavities, etc. In an optical cavity, the light wave from one end is made to reflect from the other. The transmitted and reflected waves superpose, and form a standing-wave pattern.

**ee also***

: :List of wave topics Amphidromic point ,Clapotis ,Longitudinal mode ,Modelocking ,Seiche ,Trumpet ,Voltage standing wave ratio ,Wave

*: :List of electronics topics Cavity resonator ,Characteristic impedance ,Cymatics , Impedance,Federal Standard 1037C ,Normal mode **External links*** [

*http://www.lightandmatter.com/html_books/3vw/ch04/ch04.html Vibrations and Waves*] - a chapter from an online textbook

* [*http://www.youtube.com/v/Zkox6niJ1Wc&autoplay=1 Standing Waves experiment*] Shows how the point moves with frequency change.

* [*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*]

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