Standing wave

In one dimension, two waves with the same wavelength and amplitude, traveling in opposite directions will interfere and produce a standing wave. For example, a wave traveling to the right along a taut string held stationary at its right end will reflect back in the other direction along the string, and the two waves will superpose to produce a standing wave. To create a standing wave, the two oppositely directed waves must have the same amplitude and frequency. The phenomenon can be demonstrated mathematically by deriving the equation for the sum of two oppositely moving waves:

A harmonic wave traveling to the right along the x-axis is described by the equation

${\displaystyle y_{1}(x,t)=A\sin \left({2\pi x \over \lambda }-\omega t\right)\,}$

An identical harmonic wave traveling to the left is described by the equation

${\displaystyle y_{2}(x,t)=A\sin \left({2\pi x \over \lambda }+\omega t\right)\,}$

where:

• ${\displaystyle A\,}$ is the amplitude of the wave,
• ${\displaystyle \omega \,}$ (called the angular frequency and measured in radians per second) is 2π times the frequency (in hertz)
• ${\displaystyle \lambda \,}$ is the wavelength of the wave (in metres)
• ${\displaystyle x\,}$ and ${\displaystyle t\,}$ are variables for longitudinal position and time, respectively.

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

${\displaystyle y(x,t)=y_{1}+y_{2}=A\sin \left({2\pi x \over \lambda }-\omega t\right)+A\sin \left({2\pi x \over \lambda }+\omega t\right)\,}$

Using the trigonometric sum-to-product identity ${\displaystyle \sin a+\sin b=2\sin \left({a+b \over 2}\right)\cos \left({a-b \over 2}\right)\,}$ to simplify:

${\displaystyle y=2A\sin \left({2\pi x \over \lambda }\right)\cos(\omega t)\,}$

This equation describes a wave that oscillates in time, but has a spatial dependence that is stationary; at any point x the amplitude of the oscillations is constant with value ${\displaystyle 2A\sin \left({2\pi x \over \lambda }\right)\,}$. At locations which are even multiples of a quarter wavelength

${\displaystyle x=\ldots ,-{3\lambda \over 2},\;-\lambda ,\;-{\lambda \over 2},\;0,\;{\lambda \over 2},\;\lambda ,\;{3\lambda \over 2},\ldots }$

called the nodes, the amplitude is always zero, whereas at locations which are odd multiples of a quarter wavelength

${\displaystyle x=\ldots ,-{5\lambda \over 4},\;-{3\lambda \over 4},\;-{\lambda \over 4},\;{\lambda \over 4},\;{3\lambda \over 4},\;{5\lambda \over 4},\ldots }$

called the anti-nodes, the amplitude is maximum, with a value of twice the amplitude of the original waves. The distance between two consecutive nodes or anti-nodes is λ/2.

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.

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.

Standing waves are also observed in optical media such as optical waveguides and optical cavities. Lasers use optical cavities in the form of a pair of facing mirrors, which constitute a Fabry–Pérot interferometer. The gain medium in the cavity (such as a crystal) emits light coherently, exciting standing waves of light in the cavity.[13] The wavelength of light is very short (in the range of nanometers, 10⁻⁹ m) so the standing waves are microscopic in size. One use for standing light waves is to measure small distances, using optical flats.

Interference between X-ray beams can form an X-ray standing wave (XSW) field.[14] Because of the short wavelength of X-rays (less than 1 nanometer), this phenomenon can be exploited for measuring atomic-scale events at material surfaces. The XSW is generated in the region where an X-ray beam interferes with a diffracted beam from a nearly perfect single crystal surface or a reflection from an X-ray mirror. By tuning the crystal geometry or X-ray wavelength, the XSW can be translated in space, causing a shift in the X-ray fluorescence or photoelectron yield from the atoms near the surface. This shift can be analyzed to pinpoint the location of a particular atomic species relative to the underlying crystal structure or mirror surface. The XSW method has been used to clarify the atomic-scale details of dopants in semiconductors,[15] atomic and molecular adsorption on surfaces,[16] and chemical transformations involved in catalysis.[17]

Standing waves can be mechanically induced into a 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.[18][19]

Standing surface waves on the Earth are observed as free oscillations of the Earth.

The Faraday wave is a non-linear standing wave at the air-liquid interface induced by hydrodynamic instability. It can be used as a liquid-based template to assemble microscale materials.[20]