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In physics, a standing wave, also known as a stationary wave, is a wave which 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 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 noticed by Michael Faraday in 1831. Faraday observed standing waves on the surface of a liquid in a vibrating container.[1][2] Franz Melde coined the term "standing wave" (German: stehende Welle or Stehwelle) around 1860 and demonstrated the phenomenon in his classic experiment with vibrating strings.[3][4][5][6]

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

## Opposing waves

(2)

${\displaystyle n=1,2,3,\ldots }$

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[10][11]

${\displaystyle n=1,2,3,\ldots }$

(2)

[10][11]

${\displaystyle f={\frac {v}{\lambda }}={\frac {nv}{2L}}.}The standing wave with$n = 1 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, note that Equation (2) can be rewritten as

${\displaystyle \lambda ={\frac {4L}{n}},}$
2) 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,

${\displaystyle L={\frac {n\lambda }{4}},}$
${\displaystyle n=2,4,6,\ldots }$

This example demonstrates a type of resonance and the frequencies that produce standing waves can be referred to as resonant frequencies.[10][12][13]

### 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 x = 0. At x = L, the string is free to move in the y direction. For example, the string might be tied at x = L 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 x = 0.

In this case, Equation (1) still describes the standing wave pattern that can form on the string, and the string has the same boundary condition of y = 0 at x = 0. However, at x = L where the string can move freely there should be an anti-node with maximal amplitude of y. Reviewing Equation (1), for x = L the largest amplitude of y occurs when