Standing wave

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Animation of a standing wave (red) created by the superposition of a left traveling (blue) and right traveling (green) wave Waventerference.gif
Animation of a standing wave (red) created by the superposition of a left traveling (blue) and right traveling (green) wave
Longitudinal standing wave Standing.gif
Longitudinal 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. [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 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 (Froude number: 1.7 – 4.5, surpassing 4.5 results in direct standing wave [7] ) 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

Standing 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 (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. [8] 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). [9]

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 [10]

The displacement in the y-direction for an identical harmonic wave traveling to the left is


For identical right- and left-traveling waves on the same string, the total displacement of the string is the sum of yR and yL,

Using the trigonometric sum-to-product identity ,






Note that Equation ( 1 ) does not describe a traveling wave. At any position x, y(x,t) simply oscillates in time with an amplitude that varies in the x-direction as . [10] 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 x = 0 and x = L. The string will have some damping as it is stretched by traveling waves, but assume the damping is very small. Suppose that at the x = 0 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 ( 1 ) still describes the standing wave pattern that can form on this string, but now Equation ( 1 ) is subject to boundary conditions where y = 0 at x = 0 and x = L because the string is fixed at x = L and because we assume the driving force at the fixed x = 0 end has small amplitude. Checking the values of y at the two ends,

Standing waves in a string - the fundamental mode and the first 5 harmonics. Standing waves on a string.gif
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 [11]






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 [11] [12]

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

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. [11] [13] [14]

Standing wave on a string with one fixed end

Transient analysis of a damped traveling wave reflecting at a boundary Transient to standing wave.gif
Transient analysis of a damped traveling wave reflecting at a boundary

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. Equivalently, this boundary condition of the "free end" can be stated as ∂y/∂x = 0 at x = L, which is in the form of the Sturm–Liouville formulation. The intuition for this boundary condition ∂y/∂x = 0 at x = L is that the motion of the "free end" will follow that of the point to its left.

Reviewing Equation ( 1 ), for x = L the largest amplitude of y occurs when ∂y/∂x = 0, 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

Note that 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 x = 0 and the first peak at x = L–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.

Standing wave in a pipe

Consider a standing wave in a pipe of length L. The air inside the pipe serves as the medium for longitudinal sound waves traveling to the right or left through the pipe. While the transverse waves on the string from the previous examples vary in their displacement perpendicular to the direction of wave motion, the waves traveling through the air in the pipe vary in terms of their pressure and longitudinal displacement along the direction of wave motion. The wave propagates by alternately compressing and expanding air in segments of the pipe, which displaces the air slightly from its rest position and transfers energy to neighboring segments through the forces exerted by the alternating high and low air pressures. [15] Equations resembling those for the wave on a string can be written for the change in pressure Δp due to a right- or left-traveling wave in the pipe.


If identical right- and left-traveling waves travel through the pipe, the resulting superposition is described by the sum

Note that this formula for the pressure is of the same form as Equation ( 1 ), so a stationary pressure wave forms that is fixed in space and oscillates in time.

If the end of a pipe is closed, the pressure is maximal since the closed end of the pipe exerts a force that restricts the movement of air. This corresponds to a pressure anti-node (which is a node for molecular motions, because the molecules near the closed end can't move). If the end of the pipe is open, the pressure variations are very small, corresponding to a pressure node (which is an anti-node for molecular motions, because the molecules near the open end can move freely). [16] [17] The exact location of the pressure node at an open end is actually slightly beyond the open end of the pipe, so the effective length of the pipe for the purpose of determining resonant frequencies is slightly longer than its physical length. [18] This difference in length is ignored in this example. In terms of reflections, open ends partially reflect waves back into the pipe, allowing some energy to be released into the outside air. Ideally, closed ends reflect the entire wave back in the other direction. [18] [19]

First consider a pipe that is open at both ends, for example an open organ pipe or a recorder. Given that the pressure must be zero at both open ends, the boundary conditions are analogous to the string with two fixed ends,

which only occurs when the wavelength of standing waves is [18]

or equivalently when the frequency is [18] [20]

where v is the speed of sound.

Next, consider a pipe that is open at x = 0 (and therefore has a pressure node) and closed at x = L (and therefore has a pressure anti-node). The closed "free end" boundary condition for the pressure at x = L can be stated as ∂(Δp)/∂x = 0, which is in the form of the Sturm–Liouville formulation. The intuition for this boundary condition ∂(Δp)/∂x = 0 at x = L is that the pressure of the closed end will follow that of the point to its left. Examples of this setup include a bottle and a clarinet. This pipe has boundary conditions analogous to the string with only one fixed end. Its standing waves have wavelengths restricted to [18]

or equivalently the frequency of standing waves is restricted to [21] [20]

Note that for the case where one end is closed, n only takes odd values just like in the case of the string fixed at only one end.

Molecular representation of a standing wave with n = 2 for a pipe that is closed at both ends. Considering longitudinal displacement, note that the molecules at the ends and the molecules in the middle are not displaced by the wave, representing nodes of longitudinal displacement. Halfway between the nodes there are longitudinal displacement anti-nodes where molecules are maximally displaced. Considering pressure, note that the molecules are maximally compressed and expanded at the ends and in the middle, representing pressure anti-nodes. Halfway between the anti-nodes are pressure nodes where the molecules are neither compressed nor expanded as they move. Molecule2.gif
Molecular representation of a standing wave with n = 2 for a pipe that is closed at both ends. Considering longitudinal displacement, note that the molecules at the ends and the molecules in the middle are not displaced by the wave, representing nodes of longitudinal displacement. Halfway between the nodes there are longitudinal displacement anti-nodes where molecules are maximally displaced. Considering pressure, note that the molecules are maximally compressed and expanded at the ends and in the middle, representing pressure anti-nodes. Halfway between the anti-nodes are pressure nodes where the molecules are neither compressed nor expanded as they move.

So far, the wave has been written in terms of its pressure as a function of position x and time. Alternatively, the wave can be written in terms of its longitudinal displacement of air, where air in a segment of the pipe moves back and forth slightly in the x-direction as the pressure varies and waves travel in either or both directions. The change in pressure Δp and longitudinal displacement s are related as [22]

where ρ is the density of the air. In terms of longitudinal displacement, closed ends of pipes correspond to nodes since air movement is restricted and open ends correspond to anti-nodes since the air is free to move. [18] [23] A similar, easier to visualize phenomenon occurs in longitudinal waves propagating along a spring. [24]

We can also consider a pipe that is closed at both ends. In this case, both ends will be pressure anti-nodes or equivalently both ends will be displacement nodes. This example is analogous to the case where both ends are open, except the standing wave pattern has a π2 phase shift along the x-direction to shift the location of the nodes and anti-nodes. For example, the longest wavelength that resonates–the fundamental mode–is again twice the length of the pipe, except that the ends of the pipe have pressure anti-nodes instead of pressure nodes. Between the ends there is one pressure node. In the case of two closed ends, the wavelength is again restricted to

and the frequency is again restricted to

A Rubens tube provides a way to visualize the pressure variations of the standing waves in a tube with two closed ends. [25]

2D standing wave with a rectangular boundary

Next, consider transverse waves that can move along a two dimensional surface within a rectangular boundary of length Lx in the x-direction and length Ly in the y-direction. Examples of this type of wave are water waves in a pool or waves on a rectangular sheet that has been pulled taut. The waves displace the surface in the z-direction, with z = 0 defined as the height of the surface when it is still.

In two dimensions and Cartesian coordinates, the wave equation is


To solve this differential equation, let's first solve for its Fourier transform, with

Taking the Fourier transform of the wave equation,

This is an eigenvalue problem where the frequencies correspond to eigenvalues that then correspond to frequency-specific modes or eigenfunctions. Specifically, this is a form of the Helmholtz equation and it can be solved using separation of variables. [26] Assume

Dividing the Helmholtz equation by Z,

This leads to two coupled ordinary differential equations. The x term equals a constant with respect to x that we can define as

Solving for X(x),

This x-dependence is sinusoidal–recalling Euler's formula–with constants Akx and Bkx determined by the boundary conditions. Likewise, the y term equals a constant with respect to y that we can define as

and the dispersion relation for this wave is therefore

Solving the differential equation for the y term,

Multiplying these functions together and applying the inverse Fourier transform, z(x,y,t) is a superposition of modes where each mode is the product of sinusoidal functions for x, y, and t,

The constants that determine the exact sinusoidal functions depend on the boundary conditions and initial conditions. To see how the boundary conditions apply, consider an example like the sheet that has been pulled taut where z(x,y,t) must be zero all around the rectangular boundary. For the x dependence, z(x,y,t) must vary in a way that it can be zero at both x = 0 and x = Lx for all values of y and t. As in the one dimensional example of the string fixed at both ends, the sinusoidal function that satisfies this boundary condition is

with kx restricted to

Likewise, the y dependence of z(x,y,t) must be zero at both y = 0 and y = Ly, which is satisfied by

Restricting the wave numbers to these values also restricts the frequencies that resonate to

If the initial conditions for z(x,y,0) and its time derivative ż(x,y,0) are chosen so the t-dependence is a cosine function, then standing waves for this system take the form

So, standing waves inside this fixed rectangular boundary oscillate in time at certain resonant frequencies parameterized by the integers n and m. As they oscillate in time, they do not travel and their spatial variation is sinusoidal in both the x- and y-directions such that they satisfy the boundary conditions. The fundamental mode, n = 1 and m = 1, has a single antinode in the middle of the rectangle. Varying n and m gives complicated but predictable two-dimensional patterns of nodes and antinodes inside the rectangle. [27]

Note from the dispersion relation that in certain situations different modes–meaning different combinations of n and m–may resonate at the same frequency even though they have different shapes for their x- and y-dependence. For example if the boundary is square, Lx = Ly, the modes n = 1 and m = 7, n = 7 and m = 1, and n = 5 and m = 5 all resonate at

Recalling that ω determines the eigenvalue in the Helmholtz equation above, the number of modes corresponding to each frequency relates to the frequency's multiplicity as an eigenvalue.

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. [28]

A pure standing wave does not transfer energy from the source to the destination. [29] 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.


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).

Acoustic resonance

The hexagonal cloud feature at the north pole of Saturn was initially thought to be standing Rossby waves. [30] However, this explanation has recently been disputed. [31]

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 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. [32] 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. [33] 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, [34] atomic and molecular adsorption on surfaces, [35] and chemical transformations involved in catalysis. [36]

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. [37] [38]

Seismic waves

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

Faraday waves

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. [39]


A seiche is an example of a standing wave in an enclosed body of water. It is characterised by the oscillatory behaviour of the water level at either end of the body and typically has a nodal point near the middle of the body where very little change in water level is observed. It should be distinguished from a simple storm surge where no oscillation is present. In sizeable lakes, the period of such oscillations may be between minutes and hours, for example Lake Geneva's longitudinal period is 73 minutes and its transversal seiche has a period of around 10 minutes, [40] while Lake Huron can be seen to have resonances with periods between 1 and 2 hours. [41] See Lake seiches. [42] [43] [44]

See also




  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. Dietsche, Daniela (2014-12-31). "Surfbare Wechselsprünge | Espazium". (in German). Retrieved 2022-01-13.
  8. PD-icon.svg This article incorporates public domain material from Federal Standard 1037C. General Services Administration. Archived from the original on 2022-01-22.
  9. Blackstock, David T. (2000), Fundamentals of Physical Acoustics, Wiley–IEEE, p. 141, ISBN   0-471-31979-1
  10. 1 2 Halliday, Resnick & Walker 2005, p. 432.
  11. 1 2 3 Halliday, Resnick & Walker 2005, p. 434.
  12. Serway & Faughn 1992, p. 472.
  13. Serway & Faughn 1992, p. 475-476.
  14. String Resonance. Digital Sound & Music. May 21, 2014. YouTube Video ID: oZ38Y0K8e-Y. Retrieved August 22, 2020.
  15. Halliday, Resnick & Walker 2005, p. 450.
  16. Nave, C. R. (2016). "Standing Waves". HyperPhysics. Georgia State University. Retrieved August 23, 2020.
  17. Streets 2010, p. 6.
  18. 1 2 3 4 5 6 Halliday, Resnick & Walker 2005, p. 457.
  19. Streets 2010, p. 15.
  20. 1 2 Serway & Faughn 1992, p. 478.
  21. Halliday, Resnick & Walker 2005, p. 458.
  22. Halliday, Resnick & Walker 2005, p. 451.
  23. Serway & Faughn 1992, p. 477.
  24. Thomas-Palmer, Jonathan (October 16, 2019). Longitudinal Standing Waves Demonstration. Flipping Physics. Event occurs at 4:11. YouTube video ID: 3QbmvunlQR0. Retrieved August 23, 2020.
  25. Mould, Steve (April 13, 2017). A better description of resonance. YouTube. Event occurs at 6:04. YouTube video ID: dihQuwrf9yQ. Retrieved August 23, 2020.
  26. Weisstein, Eric W. "Helmholtz Differential Equation--Cartesian Coordinates". MathWorld--A Wolfram Web Resource. Retrieved January 2, 2021.
  27. Gallis, Michael R. (February 15, 2008). 2D Standing Wave Patterns (rectangular fixed boundaries). Animations for Physics and Astronomy. Pennsylvania State University. Also available as YouTube Video ID: NMlys8A0_4s. Retrieved December 28, 2020.
  28. R S Rao, Microwave Engineering, pp. 153–154, PHI Learning, 2015 ISBN   8120351592.
  29. K A Tsokos, Physics for the IB Diploma, p. 251, Cambridge University Press, 2010 ISBN   0521138213.
  30. 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)
  31. 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.
  32. Pedrotti, Frank L.; Pedrotti, Leno M. (2017). Introduction to Optics (3 ed.). Cambridge University Press. ISBN   978-1-108-42826-2.
  33. 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.
  34. 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.
  35. 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.
  36. 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.
  37. 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.
  38. "Precision Engineering and Manufacturing Solutions – IST Precision". Archived from the original on 31 July 2016. Retrieved 28 April 2018.
  39. Chen, Pu (2014). "Microscale Assembly Directed by Liquid-Based Template". Advanced Materials. 26 (34): 5936–5941. doi:10.1002/adma.201402079. PMC   4159433 . PMID   24956442.
  40. Lemmin, Ulrich (2012), "Surface Seiches", in Bengtsson, Lars; Herschy, Reginald W.; Fairbridge, Rhodes W. (eds.), Encyclopedia of Lakes and Reservoirs, Encyclopedia of Earth Sciences Series, Springer Netherlands, pp. 751–753, doi:10.1007/978-1-4020-4410-6_226, ISBN   978-1-4020-4410-6
  41. "Lake Huron Storm Surge July 13, 1995". NOAA. Archived from the original on 2008-09-16. Retrieved 2023-01-01.
  42. Korgen, Ben (February 2000). "Bonanza for Lake Superior: Seiches Do More Than Move Water". University of Minnesota Duluth. Archived from the original on 2007-12-27.
  43. "Seiche". Archived from the original on 2019-01-26. Retrieved 2023-01-01.
  44. Johnson, Scott K. (30 June 2013). "Japanese earthquake literally made waves in Norway". Ars Technica. Archived from the original on 30 July 2022. Retrieved 2023-01-01.

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In physics and mathematics, wavelength or spatial period of a wave or periodic function is the distance over which the wave's shape repeats. In other words, it is the distance between consecutive corresponding points of the same phase on the wave, such as two adjacent crests, troughs, or zero crossings. Wavelength is a characteristic of both traveling waves and standing waves, as well as other spatial wave patterns. The inverse of the wavelength is called the spatial frequency. 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.

<span class="mw-page-title-main">Wave</span> Repeated oscillation around equilibrium

In physics, mathematics, and related fields, a wave is a propagating dynamic disturbance of one or more quantities. Waves can be periodic, in which case those quantities oscillate repeatedly about an equilibrium (resting) value at some frequency. When the entire waveform moves in one direction, it is said to be a traveling wave; by contrast, a pair of superimposed periodic waves traveling in opposite directions makes a standing wave. In a standing wave, the amplitude of vibration has nulls at some positions where the wave amplitude appears smaller or even zero. Waves are often described by a wave equation or a one-way wave equation for single wave propagation in a defined direction.

The propagation constant of a sinusoidal electromagnetic wave is a measure of the change undergone by the amplitude and phase of the wave as it propagates in a given direction. The quantity being measured can be the voltage, the current in a circuit, or a field vector such as electric field strength or flux density. The propagation constant itself measures the change per unit length, but it is otherwise dimensionless. In the context of two-port networks and their cascades, propagation constant measures the change undergone by the source quantity as it propagates from one port to the next.

<span class="mw-page-title-main">Transverse wave</span> Moving wave whose oscillations are perpendicular to the direction of the wave

In physics, a transverse wave is a wave whose oscillations are perpendicular to the direction of the wave's advance. This is in contrast to a longitudinal wave which travels in the direction of its oscillations. Water waves are an example of transverse wave.

Fourier optics is the study of classical optics using Fourier transforms (FTs), in which the waveform being considered is regarded as made up of a combination, or superposition, of plane waves. It has some parallels to the Huygens–Fresnel principle, in which the wavefront is regarded as being made up of a combination of spherical wavefronts whose sum is the wavefront being studied. A key difference is that Fourier optics considers the plane waves to be natural modes of the propagation medium, as opposed to Huygens–Fresnel, where the spherical waves originate in the physical medium.

<span class="mw-page-title-main">Sine wave</span> Mathematical curve that describes a smooth repetitive oscillation; continuous wave

A sine wave, sinusoidal wave, or just sinusoid is a mathematical curve defined in terms of the sine trigonometric function, of which it is the graph. It is a type of continuous wave and also a smooth periodic function. It occurs often in mathematics, as well as in physics, engineering, signal processing and many other fields.

<span class="mw-page-title-main">Debye model</span> Method in physics

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 in a solid. It treats the vibrations of the atomic lattice (heat) as phonons in a box, in contrast to the Einstein photoelectron 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 of solids, which is proportional to – the Debye T 3 law. Similarly to the Einstein photoelectron model, it recovers the Dulong–Petit law at high temperatures. Due to simplifying assumptions, its accuracy suffers at intermediate temperatures.

<span class="mw-page-title-main">Dipole antenna</span> Antenna consisting of two rod shaped conductors

In radio and telecommunications a dipole antenna or doublet is the simplest and most widely used class of antenna. The dipole is any one of a class of antennas producing a radiation pattern approximating that of an elementary electric dipole with a radiating structure supporting a line current so energized that the current has only one node at each end. A dipole antenna commonly consists of two identical conductive elements such as metal wires or rods. The driving current from the transmitter is applied, or for receiving antennas the output signal to the receiver is taken, between the two halves of the antenna. Each side of the feedline to the transmitter or receiver is connected to one of the conductors. This contrasts with a monopole antenna, which consists of a single rod or conductor with one side of the feedline connected to it, and the other side connected to some type of ground. A common example of a dipole is the "rabbit ears" television antenna found on broadcast television sets.

In physics, a wave vector is a vector used in describing a wave, with a typical unit being cycle per metre. It has a magnitude and direction. Its magnitude is the wavenumber of the wave, and its direction is perpendicular to the wavefront. In isotropic media, this is also the direction of wave propagation.

In optics, the Fraunhofer diffraction equation is used to model the diffraction of waves when plane waves are incident on a diffracting object, and the diffraction pattern is viewed at a sufficiently long distance from the object, and also when it is viewed at the focal plane of an imaging lens. In contrast, the diffraction pattern created near the diffracting object and is given by the Fresnel diffraction equation.

<span class="mw-page-title-main">Dispersion relation</span> Relation of wavelength/wavenumber as a function of a waves frequency

In the physical sciences and electrical engineering, dispersion relations describe the effect of dispersion on the properties of waves in a medium. A dispersion relation relates the wavelength or wavenumber of a wave to its frequency. Given the dispersion relation, one can calculate the phase velocity and group velocity of waves in the medium, as a function of frequency. In addition to the geometry-dependent and material-dependent dispersion relations, the overarching Kramers–Kronig relations describe the frequency dependence of wave propagation and attenuation.

<span class="mw-page-title-main">String vibration</span>

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.

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.

<span class="mw-page-title-main">Acoustic resonance</span> Resonance phenomena in sound and musical devices

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

The theoretical and experimental justification for the Schrödinger equation motivates the discovery of the Schrödinger equation, the equation that describes the dynamics of nonrelativistic particles. The motivation uses photons, which are relativistic particles with dynamics described by Maxwell's equations, as an analogue for all types of particles.

<span class="mw-page-title-main">Diffraction from slits</span>

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.

<span class="mw-page-title-main">Stokes wave</span> Nonlinear and periodic surface wave on an inviscid fluid layer of constant mean depth

In fluid dynamics, a Stokes wave is a nonlinear and periodic surface wave on an inviscid fluid layer of constant mean depth. This type of modelling has its origins in the mid 19th century when Sir George Stokes – using a perturbation series approach, now known as the Stokes expansion – obtained approximate solutions for nonlinear wave motion.

<span class="mw-page-title-main">Envelope (waves)</span> Smooth curve outlining the extremes of an oscillating signal

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 into an instantaneous amplitude. The figure illustrates a modulated sine wave varying between an upper envelope 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.

<span class="mw-page-title-main">Trochoidal wave</span> Exact solution of the Euler equations for periodic surface gravity waves

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.