# Standing wave

Last updated

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 amplitude is minimum are called nodes, and the locations where the amplitude is maximum are called antinodes.

Physics is the natural science that studies matter, its motion and behavior through space and time, and that studies the related entities of energy and force. Physics is one of the most fundamental scientific disciplines, and its main goal is to understand how the universe behaves.

In physics, mathematics, and related fields, a wave is a disturbance of a field in which a physical attribute oscillates repeatedly at each point or propagates from each point to neighboring points, or seems to move through space.

The amplitude of a periodic variable is a measure of its change over a single period. There are various definitions of amplitude, which are all functions of the magnitude of the difference between the variable's extreme values. In older texts the phase is sometimes called the amplitude.

## Contents

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]

Michael Faraday FRS was an English scientist who contributed to the study of electromagnetism and electrochemistry. His main discoveries include the principles underlying electromagnetic induction, diamagnetism and electrolysis.

Franz Emil Melde was a German physicist and professor. A graduate of the University of Marburg under Christian Ludwig Gerling, he later taught there, focusing primarily on acoustics, also making contributions to fields including fluid mechanics and meteorology. He began in 1860 as Gerling's assistant at the University's Mathematical and Physical Institute, succeeding him in 1864.

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.

In mechanical systems, resonance is a phenomenon that only occurs when the frequency at which a force is periodically applied is equal or nearly equal to one of the natural frequencies of the system on which it acts. This causes the system to oscillate with larger amplitude than when the force is applied at other frequencies.

A resonator is a device or system that exhibits resonance or resonant behavior. That is, it naturally oscillates with greater amplitude at some frequencies, called resonant frequencies, than at other frequencies. The oscillations in a resonator can be either electromagnetic or mechanical. Resonators are used to either generate waves of specific frequencies or to select specific frequencies from a signal. Musical instruments use acoustic resonators that produce sound waves of specific tones. Another example is quartz crystals used in electronic devices such as radio transmitters and quartz watches to produce oscillations of very precise frequency.

For waves of equal amplitude traveling in opposing directions, there is on average no net propagation of energy.

In colloquial language, an average is a single number taken as representative of a list of numbers. Different concepts of average are used in different contexts. Often "average" refers to the arithmetic mean, the sum of the numbers divided by how many numbers are being averaged. In statistics, mean, median, and mode are all known as measures of central tendency, and in colloquial usage any of these might be called an average value.

## 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.

Gliding is a recreational activity and competitive air sport in which pilots fly unpowered aircraft known as gliders or sailplanes using naturally occurring currents of rising air in the atmosphere to remain airborne. The word soaring is also used for the sport.

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.

A hydraulic jump is a phenomenon in the science of hydraulics which is frequently observed in open channel flow such as rivers and spillways. When liquid at high velocity discharges into a zone of lower velocity, a rather abrupt rise occurs in the liquid surface. The rapidly flowing liquid is abruptly slowed and increases in height, converting some of the flow's initial kinetic energy into an increase in potential energy, with some energy irreversibly lost through turbulence to heat. In an open channel flow, this manifests as the fast flow rapidly slowing and piling up on top of itself similar to how a shockwave forms.

Saltstraumen is a small strait with one of the strongest tidal currents in the world. It is located in the municipality of Bodø in Nordland county, Norway. It is located about 10 kilometres (6.2 mi) southeast of the town of Bodø. The narrow channel connects the outer Saltfjorden to the large Skjerstad Fjord between the islands of Straumøya and Knaplundsøya. The Saltstraumen Bridge on Norwegian County Road 17 crosses Saltstraumen.

## Opposing waves

 Standing waves Standing wave in stationary medium. The red dots represent the wave nodes. A standing wave (black) depicted as the sum of two propagating waves traveling in opposite directions (red and blue). Electric force vector (E) and magnetic force vector (H) of a standing wave. Standing waves in a string – the fundamental mode and the first 5 overtones. A standing wave on a circular membrane, an example of standing waves in two dimensions. This is the fundamental mode. A higher harmonic standing wave on a disk with two nodal lines crossing at the center.

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 (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. [7] 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 (SWR). [8]

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

In one dimension, two waves with the same wavelength and amplitude, traveling in opposite directions will interfere and produce a standing wave or stationary 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 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 wave ratio, phase, and energy transfer

If the two oppositely moving traveling waves are not of the same amplitude, they will not cancel completely at the nodes, the points where the waves are 180° out of phase, so the amplitude of the standing wave will not be zero at the nodes, but merely a minimum. Standing wave ratio (SWR) is the ratio of the amplitude at the antinode (maximum) to the amplitude at the node (minimum). A pure standing wave will have an infinite SWR. It will also have a constant phase at any point in space (but it may undergo a 180° inversion every half cycle). A finite, non-zero SWR indicates a wave that is partially stationary and partially travelling. Such waves can be decomposed into a superposition of two waves: a travelling wave component and a stationary wave component. An SWR of one indicates that the wave does not have a stationary component – it is purely a travelling wave, since the ratio of amplitudes is equal to 1. [9]

A pure standing wave does not transfer energy from the source to the destination. [10] However, the wave is still subject to losses in the medium. Such losses will manifest as a finite SWR, indicating a travelling wave component leaving the source to supply the losses. Even though the SWR is now finite, it may still be the case that no energy reaches the destination because the travelling component is purely supplying the losses. However, in a lossless medium, a finite SWR implies a definite transfer of energy to the destination.

## Examples

One easy example to understand standing waves is two people shaking either end of a jump rope. If they shake in sync the rope can form a regular pattern of waves oscillating up and down, with stationary points along the rope where the rope is almost still (nodes) and points where the arc of the rope is maximum (antinodes)

### Sound 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.

### Visible light

Standing waves are also observed in optical media such as optical wave guides, optical cavities, etc. Lasers use optical cavities in the form of a pair of facing mirrors. The gain medium in the cavity (such as a crystal) emits light coherently, exciting standing waves of light in the cavity. 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.

### X-rays

Interference between X-ray beams can form an X-ray standing wave (XSW) field. [13] 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, [14] atomic and molecular adsorption on surfaces, [15] and chemical transformations involved in catalysis. [16]

### Mechanical waves

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. [17] [18]

### Seismic waves

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. [19]

## References and notes

1. Alwyn Scott (ed), Encyclopedia of Nonlinear Science, p. 683, Routledge, 2006 ISBN   1135455589.
2. Theodore Y. Wu, "Stability of nonlinear waves resonantly sustained", Nonlinear Instability of Nonparallel Flows: IUTAM Symposium Potsdam, New York, p. 368, Springer, 2012 ISBN   3642850847.
3. Melde, Franz. Ueber einige krumme Flächen, welche von Ebenen, parallel einer bestimmten Ebene, durchschnitten, als Durchschnittsfigur einen Kegelschnitt liefern: Inaugural-Dissertation... Koch, 1859.
4. Melde, Franz. "Ueber die Erregung stehender Wellen eines fadenförmigen Körpers." Annalen der Physik 185, no. 2 (1860): 193–215.
5. Melde, Franz. Die Lehre von den Schwingungscurven...: mit einem Atlas von 11 Tafeln in Steindruck. JA Barth, 1864.
6. Melde, Franz. "Akustische Experimentaluntersuchungen." Annalen der Physik 257, no. 3 (1884): 452–470.
7.  This article incorporates  public domain material from the General Services Administration document "Federal Standard 1037C" .
8. Blackstock, David T. (2000), "Fundamentals of Physical Acoustics", Acoustical Society of America Journal, 109 (4): 1274–1276, Bibcode:2001ASAJ..109R1274B, doi:10.1121/1.1354982, ISBN   978-0-471-31979-5 , 568 pages. See page 141.
9. R S Rao, Microwave Engineering, pp. 153–154, PHI Learning, 2015 ISBN   8120351592.
10. K A Tsokos, Physics for the IB Diploma, p. 251, Cambridge University Press, 2010 ISBN   0521138213.
11. A Wave Dynamical Interpretation of Saturn's Polar Region Archived 2011-10-21 at the Wayback Machine , M. Allison, D. A. Godfrey, R. F. Beebe, Science vol. 247, pg. 1061 (1990)
12. Barbosa Aguiar, Ana C. (2010). "A laboratory model of Saturn's North Polar Hexagon". Icarus. 206 (2): 755–763. Bibcode:2010Icar..206..755B. doi:10.1016/j.icarus.2009.10.022.
13. Batterman, Boris W.; Cole, Henderson (1964). "Dynamical Diffraction of X Rays by Perfect Crystals". Reviews of Modern Physics. 36 (3): 681–717. Bibcode:1964RvMP...36..681B. doi:10.1103/RevModPhys.36.681.
14. Batterman, Boris W. (1969). "Detection of Foreign Atom Sites by Their X-Ray Fluorescence Scattering". Physical Review Letters. 22 (14): 703–705. Bibcode:1969PhRvL..22..703B. doi:10.1103/PhysRevLett.22.703.
15. Golovchenko, J. A.; Patel, J. R.; Kaplan, D. R.; Cowan, P. L.; Bedzyk, M. J. (1982). "Solution to the Surface Registration Problem Using X-Ray Standing Waves" (PDF). Physical Review Letters. 49 (8): 560–563. Bibcode:1982PhRvL..49..560G. doi:10.1103/PhysRevLett.49.560.
16. Feng, Z.; Kim, C.-Y.; Elam, J.W.; Ma, Q.; Zhang, Z.; Bedzyk, M.J. (2009). "Direct Atomic-Scale Observation of Redox-Induced Cation Dynamics in an Oxide-Supported Monolayer Catalyst: WOx/α-Fe2O3(0001)". J. Am. Chem. Soc. 131 (51): 18200–18201. doi:10.1021/ja906816y. PMID   20028144.
17. Bauza, Marcin B.; Hocken, Robert J.; Smith, Stuart T.; Woody, Shane C. (2005). "Development of a virtual probe tip with an application to high aspect ratio microscale features". Review of Scientific Instruments. 76 (9): 095112–095112–8. Bibcode:2005RScI...76i5112B. doi:10.1063/1.2052027.
18. "Precision Engineering and Manufacturing Solutions – IST Precision". www.insitutec.com. Archived from the original on 31 July 2016. Retrieved 28 April 2018.
19. Chen, Pu (2014). "Microscale Assembly Directed by Liquid-Based Template". Advanced Materials. 26 (34): 5936–5941. doi:10.1002/adma.201402079. PMC  . PMID   24956442.

## Related Research Articles

Frequency is the number of occurrences of a repeating event per unit of time. It is also referred to as temporal frequency, which emphasizes the contrast to spatial frequency and angular frequency. The period is the duration of time of one cycle in a repeating event, so the period is the reciprocal of the frequency. For example: if a newborn baby's heart beats at a frequency of 120 times a minute, its period—the time interval between beats—is half a second. Frequency is an important parameter used in science and engineering to specify the rate of oscillatory and vibratory phenomena, such as mechanical vibrations, audio signals (sound), radio waves, and light.

In physics, interference is a phenomenon in which two waves superpose to form a resultant wave of greater, lower, or the same amplitude. Constructive and destructive interference result from the interaction of waves that are correlated or coherent with each other, either because they come from the same source or because they have the same or nearly the same frequency. Interference effects can be observed with all types of waves, for example, light, radio, acoustic, surface water waves, gravity waves, or matter waves. The resulting images or graphs are called interferograms.

The phase velocity of a wave is the rate at which the phase of the wave propagates in space. This is the velocity at which the phase of any one frequency component of the wave travels. For such a component, any given phase of the wave will appear to travel at the phase velocity. The phase velocity is given in terms of the wavelength λ (lambda) and time period T as

In physics, the wavelength is the spatial period of a periodic wave—the distance over which the wave's shape repeats. It is thus the inverse of the spatial frequency. Wavelength is usually determined by considering the distance between consecutive corresponding points of the same phase, such as crests, troughs, or zero crossings and is a characteristic of both traveling waves and standing waves, as well as other spatial wave patterns. Wavelength is commonly designated by the Greek letter lambda (λ). The term wavelength is also sometimes applied to modulated waves, and to the sinusoidal envelopes of modulated waves or waves formed by interference of several sinusoids.

Longitudinal waves are waves in which the displacement of the medium is in the same direction as, or the opposite direction to, the direction of propagation of the wave. Mechanical longitudinal waves are also called compressional or compression waves, because they produce compression and rarefaction when traveling through a medium, and pressure waves, because they produce increases and decreases in pressure.

A sine wave or sinusoid is a mathematical curve that describes a smooth periodic oscillation. A sine wave is a continuous wave. It is named after the function sine, of which it is the graph. It occurs often in pure and applied mathematics, as well as physics, engineering, signal processing and many other fields. Its most basic form as a function of time (t) is:

In thermodynamics and solid state physics, the Debye model is a method developed by Peter Debye in 1912 for estimating the phonon contribution to the specific heat (heat capacity) in a solid. It treats the vibrations of the atomic lattice (heat) as phonons in a box, in contrast to the Einstein model, which treats the solid as many individual, non-interacting quantum harmonic oscillators. The Debye model correctly predicts the low temperature dependence of the heat capacity, which is proportional to – the Debye T3 law. Just like the Einstein model, it also recovers the Dulong–Petit law at high temperatures. But due to simplifying assumptions, its accuracy suffers at intermediate temperatures.

In physics, a wave vector is a vector which helps describe a wave. Like any vector, it has a magnitude and direction, both of which are important: Its magnitude is either the wavenumber or angular wavenumber of the wave, and its direction is ordinarily the direction of wave propagation.

In physical sciences and electrical engineering, dispersion relations describe the effect of dispersion in a medium on the properties of a wave traveling within that medium. A dispersion relation relates the wavelength or wavenumber of a wave to its frequency. From this relation the phase velocity and group velocity of the wave have convenient expressions which then determine the refractive index of the medium. More general than the geometry-dependent and material-dependent dispersion relations, there are the overarching Kramers–Kronig relations that describe the frequency dependence of wave propagation and attenuation.

A vibration in a string is a wave. Resonance causes a vibrating string to produce a sound with constant frequency, i.e. constant pitch. If the length or tension of the string is correctly adjusted, the sound produced is a musical tone. Vibrating strings are the basis of string instruments such as guitars, cellos, and pianos.

A node is a point along a standing wave where the wave has minimum amplitude. For instance, in a vibrating guitar string, the ends of the string are nodes. By changing the position of the end node through frets, the guitarist changes the effective length of the vibrating string and thereby the note played. The opposite of a node is an anti-node, a point where the amplitude of the standing wave is at maximum. These occur midway between the nodes.

In fluid dynamics, dispersion of water waves generally refers to frequency dispersion, which means that waves of different wavelengths travel at different phase speeds. Water waves, in this context, are waves propagating on the water surface, with gravity and surface tension as the restoring forces. As a result, water with a free surface is generally considered to be a dispersive medium.

Acoustic resonance is a phenomenon in which an acoustic system amplifies sound waves whose frequency matches one of its own natural frequencies of vibration.

Diffraction processes affecting waves are amenable to quantitative description and analysis. Such treatments are applied to a wave passing through one or more slits whose width is specified as a proportion of the wavelength. Numerical approximations may be used, including the Fresnel and Fraunhofer approximations.

An ultrasonic grating is a type of diffraction grating produced by interfering ultrasonic waves in a medium altering the physical properties of the medium, and hence the refractive index, in a grid-like pattern. The term acoustic grating is a more general term that includes operation at audible frequencies.

In physics and engineering, the envelope of an oscillating signal is a smooth curve outlining its extremes. The envelope thus generalizes the concept of a constant amplitude. The figure illustrates a modulated sine wave varying between an upper and a lower envelope. The envelope function may be a function of time, space, angle, or indeed of any variable.

In optics, the Fraunhofer diffraction equation is used to model the diffraction of waves when the diffraction pattern is viewed at a long distance from the diffracting object, and also when it is viewed at the focal plane of an imaging lens.

A damped sine wave is a sinusoidal function whose amplitude approaches zero as time increases.

In fluid dynamics, a trochoidal wave or Gerstner wave is an exact solution of the Euler equations for periodic surface gravity waves. It describes a progressive wave of permanent form on the surface of an incompressible fluid of infinite depth. The free surface of this wave solution is an inverted (upside-down) trochoid – with sharper crests and flat troughs. This wave solution was discovered by Gerstner in 1802, and rediscovered independently by Rankine in 1863.