Standing wave

In physics, a standing wave, also known as a stationary wave, is a wave that oscillates in time but whose peak amplitude profile does not move in space. The peak amplitude of the wave oscillations at any point in space is constant with respect to time, and the oscillations at different points throughout the wave are in phase. The locations at which the absolute value of the amplitude is minimum are called nodes, and the locations where the absolute value of the amplitude is maximum are called antinodes.
Standing waves were first described scientifically by Michael Faraday in 1831. Faraday observed standing waves on the surface of a liquid in a vibrating container. Franz Melde coined the term "standing wave" around 1860 and demonstrated the phenomenon in his classic experiment with vibrating strings.
This phenomenon can occur because the medium is moving in the direction opposite to the movement of the wave, or it can arise in a stationary medium as a result of interference between two waves traveling in opposite directions. The most common cause of standing waves is the phenomenon of resonance, in which standing waves occur inside a resonator due to interference between waves reflected back and forth at the resonator's resonant frequency.
For waves of equal amplitude traveling 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. A requirement for this in river currents is a flowing water with shallow depth in which the inertia of the water overcomes its gravity due to the supercritical flow speed and is therefore neither significantly slowed down by the obstacle nor pushed to the side. Many standing river waves are popular river surfing breaks.

Opposing waves

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 nodes and anti-nodes 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.
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.
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.

Mathematical description

This section considers representative one- and two-dimensional cases of standing waves. First, an example of an infinite length string shows how identical waves traveling in opposite directions interfere to produce standing waves. Next, two finite length string examples with different boundary conditions demonstrate how the boundary conditions restrict the frequencies that can form standing waves. Next, the example of sound waves in a pipe demonstrates how the same principles can be applied to longitudinal waves with analogous boundary conditions.
Standing waves can also occur in two- or three-dimensional resonators. With standing waves on two-dimensional membranes such as drumheads, illustrated in the animations above, the nodes become nodal lines, lines on the surface at which there is no movement, that separate regions vibrating with opposite phase. These nodal line patterns are called Chladni figures. In three-dimensional resonators, such as musical instrument sound boxes and microwave cavity resonators, there are nodal surfaces. This section includes a two-dimensional standing wave example with a rectangular boundary to illustrate how to extend the concept to higher dimensions.

Standing wave on an infinite length string

To begin, consider a string of infinite length along the x-axis that is free to be stretched transversely in the y direction.
For a harmonic wave traveling to the right along the string, the string's displacement in the y direction as a function of position x and time t is
The displacement in the y-direction for an identical harmonic wave traveling to the left is
where

y_max is the amplitude of the displacement of the string for each wave,
ω is the angular frequency or equivalently 2π times the frequency f,
λ is the wavelength of the wave.

For identical right- and left-traveling waves on the same string, the total displacement of the string is the sum of y_R and y_L,
Using the trigonometric sum-to-product identity,
Equation does not describe a traveling wave. At any position x, y simply oscillates in time with an amplitude that varies in the x-direction as. The animation at the beginning of this article depicts what is happening. As the left-traveling blue wave and right-traveling green wave interfere, they form the standing red wave that does not travel and instead oscillates in place.
Because the string is of infinite length, it has no boundary condition for its displacement at any point along the x-axis. As a result, a standing wave can form at any frequency.
At locations on the x-axis that are even multiples of a quarter wavelength,
the amplitude is always zero. These locations are called nodes. At locations on the x-axis that are odd multiples of a quarter wavelength
the amplitude is maximal, with a value of twice the amplitude of the right- and left-traveling waves that interfere to produce this standing wave pattern. These locations are called anti-nodes. The distance between two consecutive nodes or anti-nodes is half the wavelength, λ/2.

Standing wave on a string with two fixed ends

Next, consider a string with fixed ends at and. The string will have some damping as it is stretched by traveling waves, but assume the damping is very small. Suppose that at the fixed end a sinusoidal force is applied that drives the string up and down in the y-direction with a small amplitude at some frequency f. In this situation, the driving force produces a right-traveling wave. That wave reflects off the right fixed end and travels back to the left, reflects again off the left fixed end and travels back to the right, and so on. Eventually, a steady state is reached where the string has identical right- and left-traveling waves as in the infinite-length case and the power dissipated by damping in the string equals the power supplied by the driving force so the waves have constant amplitude.
Equation still describes the standing wave pattern that can form on this string, but now Equation is subject to boundary conditions where at and because the string is fixed at and because we assume the driving force at the fixed end has small amplitude. Checking the values of y at the two ends,
File:Standing waves on a string.gif|thumb|200px|upright|Standing waves in a string – the fundamental mode and the first 5 harmonics.
This boundary condition is in the form of the Sturm–Liouville formulation. The latter boundary condition is satisfied when. L is given, so the boundary condition restricts the wavelength of the standing waves to
Waves can only form standing waves on this string if they have a wavelength that satisfies this relationship with L. If waves travel with speed v along the string, then equivalently the frequency of the standing waves is restricted to
The standing wave with oscillates at the fundamental frequency and has a wavelength that is twice the length of the string. Higher integer values of n correspond to modes of oscillation called harmonics or overtones. Any standing wave on the string will have n + 1 nodes including the fixed ends and n anti-nodes.
To compare this example's nodes to the description of nodes for standing waves in the infinite length string, Equation can be rewritten as
In this variation of the expression for the wavelength, n must be even. Cross multiplying we see that because L is a node, it is an even multiple of a quarter wavelength,
This example demonstrates a type of resonance and the frequencies that produce standing waves can be referred to as resonant frequencies.

Standing wave on a string with one fixed end

Next, consider the same string of length L, but this time it is only fixed at. At, the string is free to move in the y direction. For example, the string might be tied at to a ring that can slide freely up and down a pole. The string again has small damping and is driven by a small driving force at.
In this case, Equation still describes the standing wave pattern that can form on the string, and the string has the same boundary condition of at. However, at where the string can move freely there should be an anti-node with maximal amplitude of y. Equivalently, this boundary condition of the "free end" can be stated as at, which is in the form of the Sturm–Liouville formulation. The intuition for this boundary condition at is that the motion of the "free end" will follow that of the point to its left.
Reviewing Equation, for the largest amplitude of y occurs when, or
This leads to a different set of wavelengths than in the two-fixed-ends example. Here, the wavelength of the standing waves is restricted to
Equivalently, the frequency is restricted to
In this example n only takes odd values. Because L is an anti-node, it is an odd multiple of a quarter wavelength. Thus the fundamental mode in this example only has one quarter of a complete sine cycle–zero at and the first peak at –the first harmonic has three quarters of a complete sine cycle, and so on.
This example also demonstrates a type of resonance and the frequencies that produce standing waves are called resonant frequencies.