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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. There is no net propagation of energy, so the receiver actually doesn't see any signal.

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.

Standing wave in stationary medium. The red dots represent the wave nodes

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 centers, or from reflection of a swell at the shore, and are the source of microbaroms and microsiems.

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).

The equation of a standing wave is represented as follows:
Two waves with the same frequency, wavelength and amplitude traveling in opposite directions will interfere and produce 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:

y₁ = y₀sin(kx − ωt)
and y₂ = y₀sin(kx + ωt)

where k=2π/λ
So the resultant wave equation will be:

y = y₀sin(kx − ωt) + y₀sin(kx + ωt)

Using a trigonometric identity to simplify,

y = 2y₀cos(ωt)sin(kx)

where y₀ is the amplitude of the wave, ω (called angular frequency, measured in radians per second) is 2π times the frequency (in Hz), k (called the wave number and measured in radians per metre) is 2π divided by the wavelength (in meters), and x and t are variables for longitudinal position and time, respectively.
At nodes x=0,λ/2,λ,3λ/2.... etc. whereas at antinodes x=λ/4,3λ/4,5λ/4.... etc. The distance between two conjugative nodes or antinodes is λ/2.

The hexagonal cloud feature at the north pole of Saturn is thought by most astronomers 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 antinodes at open ends. If fixed at only one end, only odd-numbered harmonics are available.

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.

• List of wave topics : Amphidromic point, Clapotis, Longitudinal mode, Modelocking, Seiche, Trumpet, Voltage standing wave ratio, Wave,
• List of electronics topics : Impedance, Federal Standard 1037C, Normal mode, Cavity resonator, Characteristic impedance, Cymatics
• People: Nikola Tesla

• Vibrations and Waves - a chapter from an online textbook
• Standing Waves experiment Shows how the point moves with frequency change.
• Java applet of standing waves on a vibrating string.
• Java applet of transverse standing wave
• Java applet showing the production of standing wave on a string by adjusting frequency