Standing waves in a string — the fundamental mode and the first 6 overtones.

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

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

A standing wave (black) depicted as the sum of two propagating waves traveling in opposite directions (red and blue).

Electric force vector (E) and magnetic force vector (H) of a standing wave.

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

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:

• y₀ is the amplitude of the wave,
• ? (called angular frequency, measured in radians per second) is 2? times the frequency (in hertz),
• k (called the wave number and measured in radians per metre) is 2? divided by the wavelength ? (in metres), and
• x and t are variables for longitudinal position and time, respectively.

A two-dimensional standing wave on a disk.

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

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

The hexagonal cloud feature at the north pole of Saturn is thought to be some sort of standing wave pattern.

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.

See 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

• 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

bg:?????? ????? ca:Ona estacionària cs:Stojaté vln?ní de:Stehende Welle et:Seisulaine el:??????? ???? es:Onda estacionaria fr:Onde stationnaire ko:??? hr:Stojni val it:Onda stazionaria he:?? ???? lv:St?vvilnis lt:Stovin?ioji banga hu:Állóhullám (hang) nl:Staande golf ja:??? pl:Fala stoj?ca pt:Onda estacionária ro:Und? sta?ionar? ru:??????? ????? fi:Seisova aalto sv:Stående våg th:????????? uk:?????? ????? zh:??