The wavelength of asine wave,λ, can be measured between any two points with the samephase, such as between crests (on top), or troughs (on bottom), or correspondingzero crossings as shown.
Inphysics andmathematics,wavelength orspatial period of awave orperiodic function is the distance over which the wave's shape repeats.[1][2] In other words, it is the distance between consecutive corresponding points of the samephase on the wave, such as two adjacent crests, troughs, orzero crossings. Wavelength is a characteristic of both traveling waves andstanding waves, as well as other spatial wave patterns.[3][4] Theinverse of the wavelength is called thespatial frequency. Wavelength is commonly designated by theGreek letterlambda (λ). For a modulated wave,wavelength may refer to thecarrier wavelength of the signal. The termwavelength may also apply to the repeatingenvelope of modulated waves or waves formed byinterference of several sinusoids.[5]
Assuming asinusoidal wave moving at a fixedwave speed, wavelength is inversely proportional to thefrequency of the wave: waves with higher frequencies have shorter wavelengths, and lower frequencies have longer wavelengths.[6]
Inlinear media, any wave pattern can be described in terms of the independent propagation of sinusoidal components. The wavelengthλ of a sinusoidal waveform traveling at constant speed is given by[7]where is called the phase speed (magnitude of thephase velocity) of the wave and is the wave'sfrequency. In adispersive medium, the phase speed itself depends upon the frequency of the wave, making therelationship between wavelength and frequency nonlinear.
In the case ofelectromagnetic radiation—such as light—infree space, the phase speed is thespeed of light, about3×108 m/s. Thus the wavelength of a 100 MHz electromagnetic (radio) wave is about:3×108 m/s divided by108 Hz = 3 m. The wavelength of visible light ranges from deepred, roughly 700 nm, toviolet, roughly 400 nm (for other examples, seeelectromagnetic spectrum).
Forsound waves in air, thespeed of sound is 343 m/s (atroom temperature and atmospheric pressure). The wavelengths of sound frequencies audible to the human ear (20 Hz–20 kHz) are thus between approximately 17 m and 17 mm, respectively. Somewhat higher frequencies are used bybats so they can resolve targets smaller than 17 mm. Wavelengths in audible sound are much longer than those in visible light.
Sinusoidal standing waves in a box that constrains the end points to be nodes will have an integer number of half wavelengths fitting in the box.A standing wave (black) depicted as the sum of two propagating waves traveling in opposite directions (red and blue)
Astanding wave is an undulatory motion that stays in one place. A sinusoidal standing wave includes stationary points of no motion, callednodes, and the wavelength is twice the distance between nodes.
The upper figure shows three standing waves in a box. The walls of the box are considered to require the wave to have nodes at the walls of the box (an example ofboundary conditions), thus determining the allowed wavelengths. For example, for an electromagnetic wave, if the box has ideal conductive walls, the condition for nodes at the walls results because the conductive walls cannot support a tangential electric field, forcing the wave to have zero amplitude at the wall.
The stationary wave can be viewed as the sum of two traveling sinusoidal waves of oppositely directed velocities.[8] Consequently, wavelength, period, and wave velocity are related just as for a traveling wave. For example, thespeed of light can be determined from observation of standing waves in a metal box containing an ideal vacuum.
Traveling sinusoidal waves are often represented mathematically in terms of their velocityv (in the x direction), frequencyf and wavelengthλ as:wherey is the value of the wave at any positionx and timet, andA is theamplitude of the wave. They are also commonly expressed in terms ofwavenumberk (2π times the reciprocal of wavelength) andangular frequencyω (2π times the frequency) as:in which wavelength and wavenumber are related to velocity and frequency as:or
In the second form given above, the phase(kx −ωt) is often generalized to(k ⋅r −ωt), by replacing the wavenumberk with awave vector that specifies the direction and wavenumber of aplane wave in3-space, parameterized by position vectorr. In that case, the wavenumberk, the magnitude ofk, is still in the same relationship with wavelength as shown above, withv being interpreted as scalar speed in the direction of the wave vector. The first form, using reciprocal wavelength in the phase, does not generalize as easily to a wave in an arbitrary direction.
Generalizations to sinusoids of other phases, and to complex exponentials, are also common; seeplane wave. The typical convention of using thecosine phase instead of thesine phase when describing a wave is based on the fact that the cosine is the real part of the complex exponential in the wave
Wavelength is decreased in a medium with slower propagation.Refraction: upon entering a medium where its speed is lower, the wave changes direction.Separation of colors by a prism (click for animation if it is not already playing)
The speed of a wave depends upon the medium in which it propagates. In particular, the speed of light in a medium is less than invacuum, which means that the same frequency will correspond to a shorter wavelength in the medium than in vacuum, as shown in the figure at right.
This change in speed upon entering a medium causesrefraction, or a change in direction of waves that encounter the interface between media at an angle.[9] Forelectromagnetic waves, this change in the angle of propagation is governed bySnell's law.
The wave velocity in one medium not only may differ from that in another, but the velocity typically varies with wavelength. As a result, the change in direction upon entering a different medium changes with the wavelength of the wave.
For electromagnetic waves the speed in a medium is governed by itsrefractive index according towherec is thespeed of light in vacuum andn(λ0) is the refractive index of the medium at wavelength λ0, where the latter is measured in vacuum rather than in the medium. The corresponding wavelength in the medium is
When wavelengths of electromagnetic radiation are quoted, the wavelength in vacuum usually is intended unless the wavelength is specifically identified as the wavelength in some other medium. In acoustics, where a medium is essential for the waves to exist, the wavelength value is given for a specified medium.
The variation in speed of light with wavelength is known asdispersion, and is also responsible for the familiar phenomenon in which light is separated into component colours by aprism. Separation occurs when the refractive index inside the prism varies with wavelength, so different wavelengths propagate at different speeds inside the prism, causing them torefract at different angles. The mathematical relationship that describes how the speed of light within a medium varies with wavelength is known as adispersion relation.
Various local wavelengths on a crest-to-crest basis in an ocean wave approaching shore[10]
Wavelength can be a useful concept even if the wave is notperiodic in space. For example, in an ocean wave approaching shore, shown in the figure, the incoming wave undulates with a varyinglocal wavelength that depends in part on the depth of the sea floor compared to the wave height. The analysis of the wave can be based upon comparison of the local wavelength with the local water depth.[10]
A sinusoidal wave travelling in a nonuniform medium, with loss
Waves that are sinusoidal in time but propagate through a medium whose properties vary with position (aninhomogeneous medium) may propagate at a velocity that varies with position, and as a result may not be sinusoidal in space. The figure at right shows an example. As the wave slows down, the wavelength gets shorter and the amplitude increases; after a place of maximum response, the short wavelength is associated with a high loss and the wave dies out.
The analysis ofdifferential equations of such systems is often done approximately, using theWKB method (also known as theLiouville–Green method). The method integrates phase through space using a localwavenumber, which can be interpreted as indicating a "local wavelength" of the solution as a function of time and space.[11][12]This method treats the system locally as if it were uniform with the local properties; in particular, the local wave velocity associated with a frequency is the only thing needed to estimate the corresponding local wavenumber or wavelength. In addition, the method computes a slowly changing amplitude to satisfy other constraints of the equations or of the physical system, such as forconservation of energy in the wave.
A wave on a line of atoms can be interpreted according to a variety of wavelengths.
Waves in crystalline solids are not continuous, because they are composed of vibrations of discrete particles arranged in a regular lattice. This producesaliasing because the same vibration can be considered to have a variety of different wavelengths, as shown in the figure.[13] Descriptions using more than one of these wavelengths are redundant; it is conventional to choose the longest wavelength that fits the phenomenon. The range of wavelengths sufficient to provide a description of all possible waves in a crystalline medium corresponds to the wave vectors confined to theBrillouin zone.[14]
This indeterminacy in wavelength in solids is important in the analysis of wave phenomena such asenergy bands andlattice vibrations. It is mathematically equivalent to thealiasing of a signal that issampled at discrete intervals.
The concept of wavelength is most often applied to sinusoidal, or nearly sinusoidal, waves, because in a linear system the sinusoid is the unique shape that propagates with no shape change – just a phase change and potentially an amplitude change.[15] The wavelength (or alternativelywavenumber orwave vector) is a characterization of the wave in space, that is functionally related to its frequency, as constrained by the physics of the system. Sinusoids are the simplesttraveling wave solutions, and more complex solutions can be built up bysuperposition.
In the special case of dispersion-free and uniform media, waves other than sinusoids propagate with unchanging shape and constant velocity. In certain circumstances, waves of unchanging shape also can occur in nonlinear media; for example, the figure shows ocean waves in shallow water that have sharper crests and flatter troughs than those of a sinusoid, typical of acnoidal wave,[16] a traveling wave so named because it is described by theJacobi elliptic function ofmth order, usually denoted ascn(x;m).[17] Large-amplitudeocean waves with certain shapes can propagate unchanged, because of properties of the nonlinear surface-wave medium.[18]
Wavelength of a periodic but non-sinusoidal waveform.
If a traveling wave has a fixed shape that repeats in space or in time, it is aperiodic wave.[19] Such waves are sometimes regarded as having a wavelength even though they are not sinusoidal.[20] As shown in the figure, wavelength is measured between consecutive corresponding points on the waveform.
Localizedwave packets, "bursts" of wave action where each wave packet travels as a unit, find application in many fields of physics. A wave packet has anenvelope that describes the overall amplitude of the wave; within the envelope, the distance between adjacent peaks or troughs is sometimes called alocal wavelength.[21][22] An example is shown in the figure. In general, theenvelope of the wave packet moves at a speed different from the constituent waves.[23]
UsingFourier analysis, wave packets can be analyzed into infinite sums (or integrals) of sinusoidal waves of differentwavenumbers or wavelengths.[24]
Louis de Broglie postulated that all particles with a specific value ofmomentump have a wavelengthλ =h/p, whereh is thePlanck constant. This hypothesis was at the basis ofquantum mechanics. Nowadays, this wavelength is called thede Broglie wavelength. For example, theelectrons in aCRT display have a De Broglie wavelength of about10−13 m. To prevent thewave function for such a particle being spread over all space, de Broglie proposed using wave packets to represent particles that are localized in space.[25] The spatial spread of the wave packet, and the spread of thewavenumbers of sinusoids that make up the packet, correspond to the uncertainties in the particle's position and momentum, the product of which is bounded byHeisenberg uncertainty principle.[24]
Pattern of light intensity on a screen for light passing through two slits. The labels on the right refer to the difference of the path lengths from the two slits, which are idealized here as point sources.
When sinusoidal waveforms add, they may reinforce each other (constructive interference) or cancel each other (destructive interference) depending upon their relative phase. This phenomenon is used in theinterferometer. A simple example is an experiment due toYoung where light is passed throughtwo slits.[26]As shown in the figure, light is passed through two slits and shines on a screen. The path of the light to a position on the screen is different for the two slits, and depends upon the angle θ the path makes with the screen. If we suppose the screen is far enough from the slits (that is,s is large compared to the slit separationd) then the paths are nearly parallel, and the path difference is simplyd sinθ. Accordingly, the condition for constructive interference is:[27]wherem is an integer, and for destructive interference is:
Thus, if the wavelength of the light is known, the slit separation can be determined from the interference pattern orfringes, andvice versa.
For multiple slits, the pattern is[28]whereq is the number of slits, andg is the grating constant. The first factor,I1, is the single-slit result, which modulates the more rapidly varying second factor that depends upon the number of slits and their spacing. In the figureI1 has been set to unity, a very rough approximation.
The effect of interference is toredistribute the light, so the energy contained in the light is not altered, just where it shows up.[29]
Diffraction pattern of a double slit has a single-slitenvelope.
The notion of path difference and constructive or destructive interference used above for the double-slit experiment applies as well to the display of a single slit of light intercepted on a screen. The main result of this interference is to spread out the light from the narrow slit into a broader image on the screen. This distribution of wave energy is calleddiffraction.
Two types of diffraction are distinguished, depending upon the separation between the source and the screen:Fraunhofer diffraction or far-field diffraction at large separations andFresnel diffraction or near-field diffraction at close separations.
In the analysis of the single slit, the non-zero width of the slit is taken into account, and each point in the aperture is taken as the source of one contribution to the beam of light (Huygens' wavelets). On the screen, the light arriving from each position within the slit has a different path length, albeit possibly a very small difference. Consequently, interference occurs.
In the Fraunhofer diffraction pattern sufficiently far from a single slit, within asmall-angle approximation, the intensity spreadS is related to positionx via a squaredsinc function:[30]withwhereL is the slit width,R is the distance of the pattern (on the screen) from the slit, and λ is the wavelength of light used. The functionS has zeros whereu is a non-zero integer, where are atx values at a separation proportion to wavelength.
Diffraction is the fundamental limitation on theresolving power of optical instruments, such astelescopes (includingradiotelescopes) andmicroscopes.[31]For a circular aperture, the diffraction-limited image spot is known as anAiry disk; the distancex in the single-slit diffraction formula is replaced by radial distancer and the sine is replaced by 2J1, whereJ1 is a first orderBessel function.[32]
The resolvablespatial size of objects viewed through a microscope is limited according to theRayleigh criterion, the radius to the first null of the Airy disk, to a size proportional to the wavelength of the light used, and depending on thenumerical aperture:[33]where the numerical aperture is defined as for θ being the half-angle of the cone of rays accepted by themicroscope objective.
Theangular size of the central bright portion (radius to first null of theAiry disk) of the image diffracted by a circular aperture, a measure most commonly used for telescopes and cameras, is:[34]whereλ is the wavelength of the waves that are focused for imaging,D theentrance pupil diameter of the imaging system, in the same units, and the angular resolutionδ is in radians.
As with other diffraction patterns, the pattern scales in proportion to wavelength, so shorter wavelengths can lead to higher resolution.
The termsubwavelength is used to describe an object having one or more dimensions smaller than the length of the wave with which the object interacts. For example, the termsubwavelength-diameter optical fibre means anoptical fibre whose diameter is less than the wavelength of light propagating through it.
Relationship between wavelength, angular wavelength, and other wave properties.
A quantity related to the wavelength is theangular wavelength (also known asreduced wavelength), usually symbolized byƛ ("lambda-bar" orbarred lambda). It is equal to the ordinary wavelength reduced by a factor of 2π (ƛ =λ/2π), with SI units of meter per radian. It is the inverse ofangular wavenumber (k = 2π/λ):ƛ=k-1. It is usually encountered in quantum mechanics, where it is used in combination with thereduced Planck constant (symbol ħ, h-bar) and theangular frequency (symbol ω = 2πf).
^To aid imagination, this bending of the wave often is compared to the analogy of a column of marching soldiers crossing from solid ground into mud. See, for example,Raymond T. Pierrehumbert (2010).Principles of Planetary Climate. Cambridge University Press. p. 327.ISBN978-0-521-86556-2.
^A. T. Fromhold (1991)."Wave packet solutions".Quantum Mechanics for Applied Physics and Engineering (Reprint of Academic Press 1981 ed.). Courier Dover Publications. pp. 59ff.ISBN0-486-66741-3.(p. 61) ... the individual waves move more slowly than the packet and therefore pass back through the packet as it advances
^Ming Chiang Li (1980)."Electron Interference". In L. Marton; Claire Marton (eds.).Advances in Electronics and Electron Physics. Vol. 53. Academic Press. p. 271.ISBN0-12-014653-3.
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