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 Saltstraumen maelstrom. Many
standing river waves are popular
river surfing breaks.
Opposing waves
 Standing waves

Image:Standing wave.gifStanding wave in stationary medium. The
red dots represent the wave nodes.Image:Standing wave 2.gifA standing
wave (black) depicted as the sum of two propagating waves traveling
in opposite directions (red and blue). 
File:Standing wave.svgElectric force vector (E) and magnetic
force vector (H) of a standing wave.Image:Harmonic partials on
strings.svgStanding waves in a string — the fundamental mode and the first 6
overtones. 
File:Drum vibration mode01.gifA twodimensional standing wave on a disk; this
is the fundamental modeImage:Drum vibration mode21.gifA 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
antinodes (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)
and
 y_2\; =\; y_0\, \sin(kx + \omega t)
where:
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).
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
antinodes, the amplitude is maximum. The distance
between two conjugative nodes or antinodes 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 antinodes at open ends. If fixed at only one end, only
oddnumbered harmonics are available. At the open end of a pipe the
antinode 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 standingwave 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
 , 568 pages. See page 141.
 .
 http://www.insitutec.com
External links