# Standing wave

File:Overtone.jpg
Vibration and standing waves in a string, The fundamental and the first 6 overtones

A standing wave, also known as a stationary wave, is a wave 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 of interference between two waves travelling in opposite directions. In the second case, for waves of equal amplitude travelling in opposing directions, there is on average 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 jumps also form on fast flowing river rapids and tidal currents such as the Saltstraumen maelstrom.

## Opposing waves

File:Standing wave.gif
Standing wave in stationary medium. The red dots represent the wave nodes.
File:E eForceVector-H magForceVector-StandingWave-USP2106769.PNG
Electric force vector (E) and magnetic force vector (H) of a Standing Wave.

As an example of the second type, a standing wave in a transmission line is a wave in which the distribution of current, voltage, or field strength is formed by the superposition of two waves propagating in opposite directions. The effect is a series of nodes (zero displacement) and anti-nodes (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 an open circuit or a short. The failure of the line to transfer power at the standing wave frequency will usually result in attenuation 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 of microbaroms and microseisms.

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

The equation of a standing wave is represented as follows:
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:

${\displaystyle y_{1}\;=\;y_{0}\,\sin(kx-\omega t)}$  and
${\displaystyle y_{2}\;=\;y_{0}\,\sin(kx+\omega t).}$

where:

So the resultant wave y equation will be the sum of y1 and y2:

${\displaystyle y\;=\;y_{0}\,\sin(kx-\omega t)\;+\;y_{0}\,\sin(kx+\omega t).}$

Using a trigonometric identity to simplify, the standing wave is described by:

${\displaystyle y\;=\;2\,y_{0}\,\cos(\omega t)\;\sin(kx).}$

At nodes x = 0, λ/2, λ, 3λ/2 ··· etc. whereas at anti-nodes x = λ/4, 3λ/4, 5λ/4 ··· etc. The distance between two conjugative nodes or anti-nodes is λ/2.

## Physical waves

File:Rotatingsaturnhexagon.gif
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 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.

## 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.