Animation of a standing wave(red) created by the superposition of a left traveling(blue) and right traveling(green) wave
Inphysics, astanding wave, also known as astationary wave, is awave that oscillates in time but whose peak amplitude profile does not move in space. The peakamplitude of the wave oscillations at any point in space is constant with respect to time, and the oscillations at different points throughout the wave arein phase. The locations at which the absolute value of the amplitude is minimum are callednodes, and the locations where the absolute value of the amplitude is maximum are called antinodes.
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 ofinterference between two waves traveling in opposite directions. The most common cause of standing waves is the phenomenon ofresonance, in which standing waves occur inside aresonator due to interference between waves reflected back and forth at the resonator'sresonant frequency.
Mouth of theCarmel River - Standing waves are formed where the Carmel River meets the Pacific Ocean.
As an example of the first type, under certain meteorological conditions standing waves form in the atmosphere in thelee of mountain ranges. Such waves are often exploited byglider pilots.
Standing waves andhydraulic jumps also form on fast flowingriver rapids and tidal currents such as theSaltstraumenmaelstrom. A requirement for this in river currents is a flowing water with shallow depth in which theinertia of the water overcomes itsgravity due to thesupercritical 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 popularriver surfing breaks.
As an example of the second type, astanding wave in atransmission line is a wave in which the distribution ofcurrent,voltage, orfield strength is formed by thesuperposition of two waves of the samefrequency propagating in opposite directions. The effect is a series ofnodes (zerodisplacement) andanti-nodes (maximumdisplacement) 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 isreflected from the other end by animpedancemismatch,i.e., discontinuity, such as anopen circuit or ashort.[8] The failure of the line to transfer power at the standing wave frequency will usually result inattenuation 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 apartial 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 thestanding wave ratio (SWR).[9]
Another example is standing waves in the openocean 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 ofmicrobaroms andmicroseisms.
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 differentboundary 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-dimensionalresonators. With standing waves on two-dimensional membranes such asdrumheads, 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 calledChladni figures. In three-dimensional resonators, such as musical instrumentsound boxes and microwavecavity 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.
Equation (1) does not describe a traveling wave. At any positionx,y(x,t) simply oscillates in time with an amplitude that varies in thex-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 thex-axis. As a result, a standing wave can form at any frequency.
At locations on thex-axis that areeven multiples of a quarter wavelength,
the amplitude is always zero. These locations are callednodes. At locations on thex-axis that areodd 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 calledanti-nodes. The distance between two consecutive nodes or anti-nodes is half the wavelength,λ/2.
Next, consider a string with fixed ends atx = 0 andx =L. The string will have some damping as it is stretched by traveling waves, but assume the damping is very small. Suppose that at thex = 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 frequencyf. In this situation, the driving force produces a right-traveling wave. That wavereflects 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 toboundary conditions wherey = 0 atx = 0 andx =L because the string is fixed atx =L and because we assume the driving force at the fixedx = 0 end has small amplitude. Checking the values ofy at the two ends,
This boundary condition is in the form ofthe 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]
2
Waves can only form standing waves on this string if they have a wavelength that satisfies this relationship withL. If waves travel with speedv along the string, then equivalently the frequency of the standing waves is restricted to[11][12]
The standing wave withn = 1 oscillates at thefundamental frequency and has a wavelength that is twice the length of the string. Higher integer values ofn correspond to modes of oscillation calledharmonics orovertones. Any standing wave on the string will haven + 1 nodes including the fixed ends andn anti-nodes.
To compare this example's nodes to the description of nodes for standing waves in the infinite length string, Equation (2) can be rewritten as
In this variation of the expression for the wavelength,n must be even. Cross multiplying we see that becauseL is a node, it is aneven multiple of a quarter wavelength,
This example demonstrates a type ofresonance and the frequencies that produce standing waves can be referred to asresonant frequencies.[11][13][14]
Next, consider the same string of lengthL, but this time it is only fixed atx = 0. Atx =L, the string is free to move in they direction. For example, the string might be tied atx =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 atx = 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 ofy = 0 atx = 0. However, atx =L where the string can move freely there should be an anti-node with maximal amplitude ofy. Equivalently, this boundary condition of the "free end" can be stated as∂y/∂x = 0 atx =L, which is in the form ofthe Sturm–Liouville formulation. The intuition for this boundary condition∂y/∂x = 0 atx =L is that the motion of the "free end" will follow that of the point to its left.
Reviewing Equation (1), forx =L the largest amplitude ofy 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
In this examplen only takes odd values. BecauseL is an anti-node, it is anodd multiple of a quarter wavelength. Thus the fundamental mode in this example only has one quarter of a complete sine cycle–zero atx = 0 and the first peak atx =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 calledresonant frequencies.
Consider a standing wave in a pipe of lengthL. The air inside the pipe serves as the medium forlongitudinalsound 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.
where
pmax is the pressure amplitude or the maximum increase or decrease in air pressure due to each wave,
If identical right- and left-traveling waves travel through the pipe, the resulting superposition is described by the sum
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 cannot 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 openorgan pipe or arecorder. 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]
Next, consider a pipe that is open atx = 0 (and therefore has a pressure node) and closed atx =L (and therefore has a pressure anti-node). The closed "free end" boundary condition for the pressure atx =L can be stated as∂(Δp)/∂x = 0, which is in the form ofthe Sturm–Liouville formulation. The intuition for this boundary condition∂(Δp)/∂x = 0 atx =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 aclarinet. 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]
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 withn = 2 for a pipe that is closed at both ends. Considering longitudinal displacement, 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, 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 positionx 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 thex-direction as the pressure varies and waves travel in either or both directions. The change in pressure Δp and longitudinal displacements are related as[22]
whereρ is thedensity 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 aπ⁄2 phase shift along thex-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
ARubens tube provides a way to visualize the pressure variations of the standing waves in a tube with two closed ends.[25]
Next, consider transverse waves that can move along a two dimensional surface within a rectangular boundary of lengthLx in thex-direction and lengthLy in they-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 thez-direction, withz = 0 defined as the height of the surface when it is still.
In two dimensions and Cartesian coordinates, thewave equation is
where
z(x,y,t) is the displacement of the surface,
c is the speed of the wave.
To solve this differential equation, let's first solve for itsFourier transform, with
Taking the Fourier transform of the wave equation,
This is aneigenvalue problem where the frequencies correspond to eigenvalues that then correspond to frequency-specific modes or eigenfunctions. Specifically, this is a form of theHelmholtz equation and it can be solved usingseparation of variables.[26] Assume
Dividing the Helmholtz equation byZ,
This leads to two coupled ordinary differential equations. Thex term equals a constant with respect tox that we can define as
Solving forX(x),
Thisx-dependence is sinusoidal–recallingEuler's formula–with constantsAkx andBkx determined by the boundary conditions. Likewise, they term equals a constant with respect toy that we can define as
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 forx,y, andt,
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 wherez(x,y,t) must be zero all around the rectangular boundary. For thex dependence,z(x,y,t) must vary in a way that it can be zero at bothx = 0 andx =Lx for all values ofy andt. As in the one dimensional example of the string fixed at both ends, the sinusoidal function that satisfies this boundary condition is
withkx restricted to
Likewise, they dependence ofz(x,y,t) must be zero at bothy = 0 andy =Ly, which is satisfied by
Restricting the wave numbers to these values also restricts the frequencies that resonate to
If the initial conditions forz(x,y,0) and its time derivativeż(x,y,0) are chosen so thet-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 integersn andm. As they oscillate in time, they do not travel and their spatial variation is sinusoidal in both thex- andy-directions such that they satisfy the boundary conditions. The fundamental mode,n = 1 andm = 1, has a single antinode in the middle of the rectangle. Varyingn andm gives complicated but predictable two-dimensional patterns of nodes and antinodes inside the rectangle.[27]
From the dispersion relation, in certain situations different modes–meaning different combinations ofn andm–may resonate at the same frequency even though they have different shapes for theirx- andy-dependence. For example, if the boundary is square,Lx =Ly, the modesn = 1 andm = 7,n = 7 andm = 1, andn = 5 andm = 5 all resonate at
Recalling thatω determines the eigenvalue in the Helmholtz equation above, the number of modes corresponding to each frequency relates to the frequency'smultiplicity as an eigenvalue.
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 constantphase 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 asuperposition 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 ajump 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).
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 avibrating string orair column'snatural frequency, a standing wave is created, allowingharmonics 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 soend 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 asoptical waveguides andoptical cavities.Lasers use optical cavities in the form of a pair of facing mirrors, which constitute aFabry–Pérot interferometer. Thegain medium in the cavity (such as acrystal) emits lightcoherently, exciting standing waves of light in the cavity.[32] The wavelength of light is very short (in the range ofnanometers, 10−9 m) so the standing waves are microscopic in size. One use for standing light waves is to measure small distances, usingoptical flats.
Interference betweenX-ray beams can form anX-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 materialsurfaces. The XSW is generated in the region where an X-ray beam interferes with adiffracted beam from a nearly perfectsingle crystal surface or a reflection from anX-ray mirror. By tuning the crystal geometry or X-ray wavelength, the XSW can be translated in space, causing a shift in theX-ray fluorescence orphotoelectron yield from the atoms near the surface. This shift can be analyzed to pinpoint the location of a particular atomic species relative to the underlyingcrystal structure or mirror surface. The XSW method has been used to clarify the atomic-scale details ofdopants in semiconductors,[34] atomic and molecularadsorption on surfaces,[35] and chemical transformations involved incatalysis.[36]
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 asensor to track changes infrequency orphase of the resonance of the fiber. One application is as a measurement device fordimensional metrology.[37][38]
TheFaraday 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]
Aseiche 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 simplestorm surge where no oscillation is present. In sizeable lakes, the period of such oscillations may be between minutes and hours, for exampleLake 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] SeeLake seiches.[42][43][44]
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^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.PMID20028144.
^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.
^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,ISBN978-1-4020-4410-6
