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Wavelength

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For other uses, see Wavelength (disambiguation).
Wavelength of a sine wave, λ, can be measured between any two points with the same phase, such as between crests, or troughs, or corresponding zero crossings as shown.

In physics, the wavelength of a sinusoidal wave is the spatial period of the wave—the distance over which the wave's shape repeats.[1] It 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. Wavelength is commonly designated by the Greek letter lambda (λ). The concept can also be applied to waves of other forms, as long as they are exactly periodic.[2] In some cases "wavelength" refers to a sinusoidal component of a wave, such as the carrier in a modulated transmission,[citation needed] or a sinusoidal "envelope" produced by interference between sinusoidal component waves.[citation needed]

Assuming a sinusoidal wave moving at a fixed wave speed, wavelength is inversely proportional to frequency: waves with higher frequencies have shorter wavelengths, and lower frequencies have longer wavelengths.[3]

Examples of wave-like phenomena are sound waves, light, and water waves. A sound wave is a periodic variation in air pressure, while in light and other electromagnetic radiation the strength of the electric and the magnetic field vary. Water waves are periodic variations in the height of a body of water. In a crystal lattice vibration, atomic positions vary periodically in both lattice position and time.

Wavelength is a measure of the distance between repetitions of a shape feature such as peaks, valleys, or zero-crossings, not a measure of how far any given particle moves. For example, in waves over deep water a particle in the water moves in a circle of the same diameter as the wave height, unrelated to wavelength.[4]

Contents

[edit] Sinusoidal waves

In linear media, any wave shape can be described in terms of the independent propagation of sinusoidal components.

The wavelength λ of a sinusoidal waveform traveling at constant speed v is given by:[5]

\lambda = \frac{v}{f},
Refraction: when a plane wave encounters a medium in which it has a slower speed, the wavelength decreases, and the direction adjusts accordingly.

where v is called the phase speed (magnitude of the phase velocity) of the wave and f is the wave's frequency.

In the case of electromagnetic radiation such as light in free space, the speed is the speed of light, about 3×108 m/s. For sound waves in air, the speed of sound is 343 m/s (1238 km/h) (at room temperature and atmospheric pressure). As an example, the wavelength of a 100 MHz electromagnetic (radio) wave is about: 3×108 m/s divided by 100×106 Hz = 3 metres.

Visible light ranges from deep red, roughly 700 nm, to violet, roughly 400 nm (430–750 THz) (for other examples, see electromagnetic spectrum). The wavelengths of sound frequencies audible to the human ear (20 Hz–20 kHz) are between approximately 17 m and 17 mm, respectively, assuming a typical speed of sound of about 343 m/s; the wavelengths in audible sound are much longer than those in visible light.

Frequency and wavelength can change independently, but only when the speed of the wave changes. For example, when light enters another medium, its speed and wavelength change while its frequency does not; this change of wavelength causes refraction, or a change in propagation direction of waves that encounter the interface between media at an angle.

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.

[edit] Standing waves

A standing wave (black) depicted as the sum of two propagating waves traveling in opposite directions (red and blue).

A standing wave is an undulatory motion that stays in one place. A sinusoidal standing wave includes stationary points of no motion, called nodes, and the wavelength is twice the distance between nodes. The wavelength, period, and wave velocity are related as before, if the stationary wave is viewed as the sum of two traveling sinusoidal waves of oppositely directed velocities.[6]

[edit] Mathematical representation

Traveling sinusoidal waves are often represented mathematically in terms of their velocity v (in the x direction), frequency f and wavelength λ as:

y (x, \ t) = A \cos \left( 2 \pi \left( \frac{x}{\lambda } - ft \right ) \right ) = A \cos \left( \frac{2 \pi}{\lambda} (x - vt) \right )

where y is the value of the wave at any position x and time t, and A is the amplitude of the wave. They are also commonly expressed in terms of (radian) wavenumber k (2π times the reciprocal of wavelength) and angular frequency ω (2π times the frequency) as:

y (x, \ t) = A \cos \left( kx - \omega t \right) = A \cos \left(k(x - v t) \right)

in which wavelength and wavenumber are related to velocity and frequency as:

k = \frac{2 \pi}{\lambda} = \frac{2 \pi f}{v} = \frac{\omega}{v}

or

\lambda = \frac{2 \pi}{k} = \frac{2 \pi v}{\omega} = \frac{v}{f}
Dispersion causes separation of colors when light is refracted by a prism.

The relationship between ω and λ (or k) is called a dispersion relation. This is not generally a simple inverse relation because the wave velocity itself typically varies with frequency.[7]

Wavelength is decreased in a medium with higher refractive index.

In the second form given above, the phase (kx − ωt) is often generalized to (k•r − ωt), by replacing the wavenumber k with a wave vector that specifies the direction and wavenumber of a plane wave in 3-space, parameterized by position vector r. In that case, the wavenumber k, the magnitude of k, is still in the same relationship with wavelength as shown above, with v 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; see plane wave.

[edit] General media

The speed of a wave depends upon the medium in which it propagates. In particular, the speed of light in most media is lower than in vacuum, which means that the same frequency will correspond to a shorter wavelength in the medium than in vacuum. The wavelength in the medium is

\lambda = \frac{\lambda_0}{n},
Various local wavelengths on a crest-to-crest basis in an ocean wave approaching shore.[8]

where λ0 is the wavelength in vacuum, and n is the refractive index of the medium. When wavelengths of electromagnetic radiation are quoted, the vacuum wavelength is usually 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.

In general, the refractive index is a function of wavelength. This variation of n with λ, called dispersion, causes different colors of light to be separated when light is refracted by a prism.

[edit] Nonuniform media

A sinusoidal wave in a nonuniform medium, with loss. 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.

Wavelength can be a useful concept even if the wave is not periodic in space. For example, in an ocean wave approaching shore, shown in the figure, the incoming wave undulates with a varying local 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.[8]

Waves that are sinusoidal in time but propagate through a medium whose properties vary with position (an inhomogeneous medium) may propagate at a velocity that varies with position, and as a result may not be sinusoidal in space. The analysis of differential equations of such systems is often done approximately, using the the WKB method (also known as the Liouville–Green method). The method integrates phase through space using a local wavenumber, which can be interpreted as indicating a "local wavelength" of the solution as a function of time and space.[9][10] 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 for conservation of energy in the wave.

[edit] Crystals

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 produces aliasing because the same vibration can be considered to have a variety of different wavelengths, as shown in the figure.[11] 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 medium corresponds to the wave vectors confined to the Brillouin zone.[12]

The aliasing of waves in crystals arises in the same fashion as when a signal is sampled in discrete intervals. This aliasing is important in the analysis of wave phenomena such as energy bands and lattice vibrations.

[edit] More general waveforms

 This section's factual accuracy is disputed. Please see the relevant discussion on the talk page. (July 2009)
Wavelength of an irregular periodic waveform at a particular moment in time. The same λ separates any two similarly situated points in the waveform.[13]

A wave moving in space is called a traveling wave. If the shape repeats itself, it is also a periodic wave.[13] At a fixed moment in time, a snapshot of the wave shows a repeating form in space, with characteristics such as peaks and troughs repeating at equal intervals. To an observer at a fixed location the amplitude appears to vary in time, and repeats itself with a certain period, for example T. If the spatial period of this wave is referred to as its wavelength, then during every period, one wavelength of the wave passes the observer. If the wave propagates with unchanging shape and the velocity in the medium is uniform, this period implies the wavelength is:

\lambda = vT \ .

This duality of space and time is expressed mathematically by the fact that the wave's behavior does not depend independently on position x and time t, but rather on the combination of position and time x − vt. A wave's amplitude u is then expressed as u(x − vt) and in the case of a periodic function u with period λ, that is, u(x + λ − vt) = u(x − vt), the periodicity of u in space means that a snapshot of the wave at a given time t finds the wave varying periodically in space with period λ. In a similar fashion, this periodicity of u implies a periodicity in time as well: u(x − v(t + T)) = u(x − vt) using the relation vT = λ described above, so an observation of the wave at a fixed location x finds the wave undulating periodically in time with period T = λ/v.[13]

Traveling waves with non-sinusoidal wave shapes can occur in linear dispersionless media such as free space, but also may arise in nonlinear media under certain circumstances. For example, large-amplitude ocean waves with certain shapes can propagate unchanged, because of properties of the nonlinear surface-wave medium.[14] An example is the cnoidal wave, a periodic traveling wave named because it is described by the Jacobian elliptic function of m-th order, usually denoted as cn (x; m).[15]

[edit] Wave packets

Main article: Wave packet
A propagating wave packet; in general, the envelope of the wave packet moves at a different speed than the constituent waves.[16]

Localized wave packets, "bursts" of wave action where each wave packet travels as a unit, find application in many fields of physics; the notion of a local wavelength also may be applied to these wave packets.[17] The wave packet has an envelope that describes the overall amplitude of the wave; within the envelope, the distance between adjacent peaks or troughs is a local wavelength.[18] Using Fourier analysis, wave packets can be analyzed into infinite sums (or integrals) of sinusoidal waves of different wavenumbers or wavelengths.[19]

Louis de Broglie postulated that all particles with a specific value of momentum have a wavelength

\lambda = \frac{h}{p},

where h is Planck's constant, and p is the momentum of the particle. This hypothesis was at the basis of quantum mechanics. Nowadays, this wavelength is called the de Broglie wavelength. For example, the electrons in a CRT display have a De Broglie wavelength of about 10−13 m. To prevent the wave function for such a particle being spread over all space, De Broglie proposed using wave packets to represent particles that are localized in space.[20] The spread of wavenumbers of sinusoids that add up to such a wave packet corresponds to an uncertainty in the particle's momentum, one aspect of the Heisenberg uncertainty principle.[19]

[edit] Subwavelength

The term subwavelength 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 term subwavelength-diameter optical fibre means an optical fibre whose diameter is less than the wavelength of light propagating through it.

A subwavelength particle is a particle smaller than the wavelength of light with which it interacts (see Rayleigh scattering). Subwavelength apertures are holes smaller than the wavelength of light propagating through them.

Subwavelength may also refer to a phenomenon involving subwavelength objects; for example, subwavelength imaging.

[edit] See also

  • Fraunhofer lines – dark lines in the solar spectrum, traditionally used as standard optical wavelength references

[edit] References

  1. ^ Hecht, Eugene (1987). Optics (2nd ed.). Addison Wesley. p. 15. ISBN 0-201-11609-X. 
  2. ^ Hecht, Eugene (1987). Optics (2nd ed.). Addison Wesley. p. 16. ISBN 0-201-11609-X. 
  3. ^ Theo Koupelis and Karl F. Kuhn (2007). In Quest of the Universe. Jones & Bartlett Publishers. ISBN 0763743879. http://books.google.com/books?id=WwKjznJ9Kq0C&pg=PA102&dq=wavelength+lambda+light+sound+frequency+wave+speed&lr=&as_brr=3&ei=nfIpSazAMIzukgSP04nODg. 
  4. ^ Paul R Pinet (2008). Invitation to Oceanography (5th ed.). Jones & Bartlett Publishers. p. 237. ISBN 0763759937. http://books.google.com/books?id=6TCm8Xy-sLUC&pg=PA237. 
  5. ^ David C. Cassidy, Gerald James Holton, Floyd James Rutherford (2002). Understanding physics. Birkhäuser. pp. 339 ff. ISBN 0387987568. http://books.google.com/books?id=rpQo7f9F1xUC&pg=PA340. 
  6. ^ John Avison (1999). The World of Physics. Nelson Thornes. p. 460. ISBN 9780174387336. http://books.google.com/books?id=DojwZzKAvN8C&pg=PA460&dq=%22standing+wave%22+wavelength&lr=&as_brr=3&ei=CDFMStC9JZDOlQTtzqgW. 
  7. ^ John A. Adam (2003). Mathematics in nature. Princeton University Press. p. 148. ISBN 0691114293. http://books.google.com/books?id=2gO2sBp4ipQC&pg=PA148. "The relation between the frequency of a wave and its wavelength λ … is referred to as a dispersion relation." 
  8. ^ a b Paul R Pinet. op. cit.. p. 242. ISBN 0763759937. http://books.google.com/books?id=6TCm8Xy-sLUC&pg=PA242. 
  9. ^ Bishwanath Chakraborty. Principles of Plasma Mechanics. New Age International. p. 454. ISBN 9788122414462. http://books.google.com/books?id=_MIdEiKqdawC&pg=PA454&dq=wkb+local-wavelength&lr=&as_brr=0&ei=ZHZASqOwLY7skwTuodyLDw. 
  10. ^ Jeffrey A. Hogan and Joseph D. Lakey (2005). Time-frequency and time-scale methods: adaptive decompositions, uncertainty principles, and sampling. Birkhäuser. p. 348. ISBN 9780817642761. http://books.google.com/books?id=YOf0SRzxz3gC&pg=PA348&dq=wkb+local-wavelength&lr=&as_brr=3&ei=oHhASubOH4SkkASI4ciEDw. 
  11. ^ See Figure 4.20 in A. Putnis (1992). Introduction to mineral sciences. Cambridge University Press. p. 97. ISBN 0521429471. http://books.google.com/books?id=yMGzmOqYescC&pg=PA97.  and Figure 2.3 in Martin T. Dove (1993). Introduction to lattice dynamics (4th ed.). Cambridge University Press. p. 22. ISBN 0521392934. http://books.google.com/books?id=vM50l2Vf7HgC&pg=PA22. 
  12. ^ Manijeh Razeghi (2006). Fundamentals of solid state engineering (2nd ed.). Birkhäuser. pp. 165 ff. ISBN 0387281525. http://books.google.com/books?id=6x07E9PSzr8C&pg=PA165. 
  13. ^ a b c Alexander McPherson (2009). "Waves and their properties". Introduction to Macromolecular Crystallography (2 ed.). Wiley. p. 77. ISBN 0470185902. http://books.google.com/books?id=o7sXm2GSr9IC&pg=PA77. 
  14. ^ Ken'ichi Okamoto (2001). Global environment remote sensing. IOS Press. p. 263. ISBN 9781586031015. http://books.google.com/books?id=tXQy5JdQyZoC&pg=PA263&dq=wave-length++non-sinusoidal&lr=&as_brr=3&ei=g1kwSsTyDYzqzATx06ydDg. 
  15. ^ Roger Grimshaw (2007). "Solitary waves propagating over variable topography". in Anjan Kundu. Tsunami and Nonlinear Waves. Springer. pp. 52 ff. ISBN 3540712550. http://books.google.com/books?id=2Dtgq-1CGWIC&pg=PA52. 
  16. ^ A. T. Fromhold (1991). "Wave packet solutions". Quantum Mechanics for Applied Physics and Engineering (Reprint of Academic Press 1981 ed.). Courier Dover Publications. pp. 59 ff. ISBN 0486667413. http://books.google.com/books?id=3SOwc6npkIwC&pg=PA59. "(p. 61) …the individual waves move more slowly than the packet and therefore pass back through the packet as it advances" 
  17. ^ Peter R. Holland (1995). The Quantum Theory of Motion: An Account of the de Broglie-Bohm Causal Interpretation of Quantum Mechanics. Cambridge University Press. p. 160. ISBN 9780521485432. http://books.google.com/books?id=BsEfVBzToRMC&pg=PA160&dq=wave-packet+local-wavelength&ei=4OhOSsbmNo32kASKyc31Ag. 
  18. ^ Jeffery Cooper (1998). Introduction to partial differential equations with MATLAB. Springer. p. 272. ISBN 0817639675. http://books.google.com/books?id=l0g2BcxOJVIC&pg=PA272. "The local wavelength λ of a dispersing wave is twice the distance between two successive zeros.…the local wave length and the local wave number k are related by k = 2π / λ." 
  19. ^ a b See, for example, Figs. 2.8 – 2.10 in Joy Manners (2000). "Heisenberg's uncertainty principle". Quantum Physics: An Introduction. CRC Press. pp. 53–56. ISBN 9780750307208. http://books.google.com/books?id=LkDQV7PNJOMC&pg=PA54&dq=wave-packet+wavelengths&lr=&as_drrb_is=q&as_minm_is=0&as_miny_is=&as_maxm_is=0&as_maxy_is=&as_brr=0&ei=UtpKSsvVH4WWkgTQ0om_Cw. 
  20. ^ Ming Chiang Li (1980). "Electron Interference". in L. Marton & Claire Marton. Advances in Electronics and Electron Physics. 53. Academic Press. p. 271. ISBN 0120146533. http://books.google.com/books?id=g5q6tZRwUu4C&pg=PA271. 

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