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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 traveling in opposite directions. In the second case, for waves of equal amplitude traveling in opposing directions, there is on average no net propagation of energy.

Standing waves in resonators are one cause of the phenomenon called resonance.

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 Saltstraumenmarker maelstrom. Many standing river waves are popular river surfing breaks.

Opposing waves

Standing waves
Image:Standing wave.gif|Standing wave in stationary medium. The red dots represent the wave 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).
File:Standing wave.svg|Electric force vector (E) and magnetic force vector (H) of a standing wave.Image:Harmonic partials on strings.svg|Standing waves in a string — the fundamental mode and the first 6 overtones.
File:Drum vibration mode01.gif|A two-dimensional standing wave on a disk; this is the fundamental modeImage:Drum vibration mode21.gif|A higher harmonic standing wave on a disk with a node at the center of the drum.
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 of the same frequency propagating in opposite directions. The effect is a series of node (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 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 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 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).

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 harmonic wave traveling to the right and hitting the end of the string produces 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 nodes 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. 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 harmonics or overtones. 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.

Let the harmonic waves be represented by the equations below:

y_1\; =\; y_0\, \sin(kx - \omega t)
y_2\; =\; y_0\, \sin(kx + \omega t)


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).

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, 3λ/2, ... called the node the amplitude is always zero, whereas at locations x = λ/4, 3λ/4, 5λ/4, ... called the anti-nodes, the amplitude is maximum. 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

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.

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.

Mechanical waves

Standing waves can be mechanically induced into solid medium using resonance. One easy 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 frequency or phase of the resonance of the fiber. One application is as a measurement device for dimensional metrology .

See also

Amphidromic point, Clapotis, Longitudinal mode, Modelocking, Seiche, Trumpet, Voltage standing wave ratio, Wave
Cavity resonator, Characteristic impedance, Cymatics, Impedance, Normal mode

References and notes

  1. , 568 pages. See page 141.
  2. .

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