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{{cite book |title=Tsunami and Nonlinear Waves |author=Roger Grimshaw |editor=Anjan Kundu |url=http://books.google.com/books?id=2Dtgq-1CGWIC&pg=PA52 |pages=52 ''ff'' |chapter=Solitary waves propagating over variable topography |isbn=3540712550 |year=2007 |publisher=Springer}} </ref>
{{cite book |title=Tsunami and Nonlinear Waves |author=Roger Grimshaw |editor=Anjan Kundu |url=http://books.google.com/books?id=2Dtgq-1CGWIC&pg=PA52 |pages=52 ''ff'' |chapter=Solitary waves propagating over variable topography |isbn=3540712550 |year=2007 |publisher=Springer}} </ref>

==Local wavelength==

[[File:Three terms in sawtooth expansion.JPG|thumb|First three terms in Fourier expansion of sawtooth]]
[[File:Modulated wave.JPG|thumb|180px|Amplitude modulated waveform; the ''local wavelength'' varies from position to position inside the envelope]]
[[File:Wave packet (dispersion).gif|thumb|One wave packet in a possible train of wave packets]]
The definition of wavelength as the distance between similar positions in a periodic wave is shown in the two top figures at the right. This wavelength describes the true periodicity of the wave, but in the case of waves that contain a slowly varying amplitude and a much more rapid internal wavelike structure, the ''local wavelength'' is a useful idea. The local wavelength is twice the distance between two adjacent internal nodes of the waveform, or the distance between adjacent interior crests or between adjacent interior troughs.<ref name= Cooper>

{{cite book |title=Introduction to partial differential equations with MATLAB |author=Jeffery Cooper |url=http://books.google.com/books?id=l0g2BcxOJVIC&pg=PA272 |page=272 |isbn=0817639675 |year=1998 |publisher=Springer}}

</ref> In the case of carrier modulation (the ''carrier'' is the high frequency internal wave motion), the envelope of the wave form modulating the carrier wave introduces a distribution of local wave lengths in the vicinity of the wavelength of the carrier.<ref name=O'Reilly>

{{cite book |title=Telecommunications Principles |author= John J O'Reilly |url=http://books.google.com/books?id=8xdB0ttzAIIC&pg=PA58 |pages=58 ''ff'' |year=1989 |edition=2 |publisher=Springer}}</ref><ref name=Russ>

{{cite book |title=Sound synthesis and sampling |author=Martin Russ |url=http://books.google.com/books?id=Kv9RbnhQC2kC&pg=PA117 |page=117 |quote=For a sine carrier and non-sinusoidal wave modulator, there are actually the equivalent of several modulator frequencies: one for each harmonic in the modulator. |isbn=0240514297 |year=1996 |publisher=Focal Press }}

</ref>

A very common example of this situation occurs in the [[wave packet]], a waveform often used in [[quantum mechanics]] to describe the [[wave function]] of a particle. In the case of a single, isolated wave packet, the only wavelength is the local wavelength, because the envelope does not repeat itself periodically.

In representing the wave function of a particle, the wave packet is often taken to have a Gaussian shape and is called a ''[[Schrödinger_equation#Gaussian_wavepacket|Gaussian wave packet]]''. <ref name=wavepacket>

See for example {{cite book |url=http://books.google.com/books?id=7qCMUfwoQcAC&pg=PA60 |title=Quantum Mechanics |author=Walter Greiner, D. Allan Bromley |page=60 |isbn=3540674586 |edition=2 |year=2007 |publisher=Springer}} and {{cite book |title=Electronic basis of the strength of materials |author=John Joseph Gilman |url=http://books.google.com/books?id=YWd7zHU0U7UC&pg=PA57 |page=57 |year=2003 |isbn=0521620058 |publisher=Cambridge University Press}} and {{cite book |title=Principles of quantum mechanics |author= Donald D. Fitts |url=http://books.google.com/books?id=8t4DiXKIvRgC&pg=PA17 |page 17 |isbn=0521658411 |publisher=Cambridge University Press |year=1999}}

</ref> It is well known from the theory of Fourier analysis,<ref name= Brandt>

{{cite book |page=23 |url=http://books.google.com/books?id=VM4GFlzHg34C&pg=PA23 |title=The picture book of quantum mechanics |author=Siegmund Brandt, Hans Dieter Dahmen |isbn=0387951415 |year=2001 |edition =3rd |publisher=Springer}}

</ref> or from the [[Heisenberg uncertainty principle]] (in the case of quantum mechanics) that a narrow range of wavelengths is necessary to produce a localized wave packet, and the more localized the envelope, the larger the spread in required wavelengths. The Fourier transform of a Gaussian is itself a Gaussian.<ref name=Gaussian>

{{cite book |title=Modern mathematical methods for physicists and engineers |author= Cyrus D. Cantrell |page=677 |url=http://books.google.com/books?id=QKsiFdOvcwsC&pg=PA677 |isbn=0521598273 |publisher=Cambridge University Press |year=2000}}


</ref> Given the Gaussian:

:<math>f(x) = e^{-x^2 / (2\sigma^2)} \ , </math>

the Fourier transform is:

:<math>\tilde{ f} (k) = \sigma e^{-\sigma^2 k^2 / 2} \ . </math>

The Gaussian in space therefore is made up of waves:

:<math>f(x) = \frac{1}{\sqrt{2 \pi}} \int_{-\infty}^{\infty} \ \tilde{f} (k) e^{ikx} \ dk \ ; </math>

that is, a number of waves of wavelengths λ such that ''k''λ = 2 π.

The parameter σ decides the spatial spread of the Gaussian along the ''x''-axis, while the Fourier transform shows a spread in [[wave vector]] ''k'' determined by 1/σ. That is, the smaller the extent in space, the larger the extent in ''k'', and hence in λ = 2π/''k''.


==Subwavelength==
==Subwavelength==

Revision as of 06:14, 21 June 2009

For other uses, see Wavelength (disambiguation).
Wavelength of a sine wave

In physics, wavelength is the distance between repeating units of a waveform. It is commonly designated by the Greek letter lambda (λ). Examples of wave-like phenomena are light, water waves, and sound waves. Waves may have arbitrary shapes, but recur periodically in time or space or both. A wave that does not move in space but oscillates in time is called a standing wave.

Assuming the speed of the wave is fixed, wavelength is inversely proportional to frequency: waves with higher frequencies have shorter wavelengths, and lower frequencies have longer wavelengths.[1]

Different types of wave have different properties that may vary with position. For example, in a sound wave the air pressure oscillates, while in light and other electromagnetic radiation the strength of the electric and the magnetic field vary. In a crystal lattice vibration, atomic positions vary with location in the crystal lattice at a fixed time, and vary in time at a fixed lattice position. Wavelength is a measure of the distance between repetitions of the property, not how far any given particle moves during a given time interval.

Sinusoidal waves

Linear wave media are typically described in terms of the characteristics of sinusoidal waves, since sinusoids are the shapes that propagate unchanged in such media, and since any wave in such media can be described as a superposition of sinusoidal waves.

The wavelength λ of a sinusoidal waveform traveling at constant speed v is given by:[2]

where v is called the phase speed (magnitude of the phase velocity) of the wave and f is its 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) (in air at room temperature and atmospheric pressure).

For example, the wavelength for 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 (cf. refraction).

General media

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

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.

More general waveforms

Wavelength of an irregular periodic waveform

A wave with a fixed shape but moving in space is called a traveling wave. If the shape repeats itself in space, it is also a periodic wave.[3][4] To an observer at a fixed location, the amplitude of a traveling periodic wave will appear to vary in time, and if its velocity is constant, will repeat itself with a certain period, say T.

Every period, one wavelength of the wave passes the observer, showing the velocity of the wave, say v, to be related to the period by:

which implies the wavelength is related to period as:

In a physical system, such behavior can occur in linear dispersionless media, but also may arise in nonlinear media under certain circumstances. For example, in large-amplitude ocean waves, due to properties of the nonlinear surface-wave medium, wave shapes can propagate unchanged.[5] A related phenomenon 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).[6]

Local wavelength

First three terms in Fourier expansion of sawtooth
Amplitude modulated waveform; the local wavelength varies from position to position inside the envelope
One wave packet in a possible train of wave packets

The definition of wavelength as the distance between similar positions in a periodic wave is shown in the two top figures at the right. This wavelength describes the true periodicity of the wave, but in the case of waves that contain a slowly varying amplitude and a much more rapid internal wavelike structure, the local wavelength is a useful idea. The local wavelength is twice the distance between two adjacent internal nodes of the waveform, or the distance between adjacent interior crests or between adjacent interior troughs.[7] In the case of carrier modulation (the carrier is the high frequency internal wave motion), the envelope of the wave form modulating the carrier wave introduces a distribution of local wave lengths in the vicinity of the wavelength of the carrier.[8][9]

A very common example of this situation occurs in the wave packet, a waveform often used in quantum mechanics to describe the wave function of a particle. In the case of a single, isolated wave packet, the only wavelength is the local wavelength, because the envelope does not repeat itself periodically.

In representing the wave function of a particle, the wave packet is often taken to have a Gaussian shape and is called a Gaussian wave packet. [10] It is well known from the theory of Fourier analysis,[11] or from the Heisenberg uncertainty principle (in the case of quantum mechanics) that a narrow range of wavelengths is necessary to produce a localized wave packet, and the more localized the envelope, the larger the spread in required wavelengths. The Fourier transform of a Gaussian is itself a Gaussian.[12] Given the Gaussian:

the Fourier transform is:

The Gaussian in space therefore is made up of waves:

that is, a number of waves of wavelengths λ such that kλ = 2 π.

The parameter σ decides the spatial spread of the Gaussian along the x-axis, while the Fourier transform shows a spread in wave vector k determined by 1/σ. That is, the smaller the extent in space, the larger the extent in k, and hence in λ = 2π/k.

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.

De Broglie wavelength of particles

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

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.

See also

References

  1. ^ Theo Koupelis and Karl F. Kuhn (2007). In Quest of the Universe. Jones & Bartlett Publishers. ISBN 0763743879.
  2. ^ David C. Cassidy, Gerald James Holton, Floyd James Rutherford (2002). Understanding physics. Birkhäuser. pp. 339 ff. ISBN 0387987568.{{cite book}}: CS1 maint: multiple names: authors list (link)
  3. ^ Alexander McPherson (2009). "Waves and their properties". Introduction to Macromolecular Crystallography (2 ed.). Wiley. p. 77. ISBN 0470185902.
  4. ^ Robert Prigo (2007). Making physics fun. Corwin Press. pp. 80 ff. ISBN 1412926637.
  5. ^ Ken'ichi Okamoto (2001). Global environment remote sensing. IOS Press. p. 263. ISBN 9781586031015.
  6. ^ Roger Grimshaw (2007). "Solitary waves propagating over variable topography". In Anjan Kundu (ed.). Tsunami and Nonlinear Waves. Springer. pp. 52 ff. ISBN 3540712550.
  7. ^ Jeffery Cooper (1998). Introduction to partial differential equations with MATLAB. Springer. p. 272. ISBN 0817639675.
  8. ^ John J O'Reilly (1989). Telecommunications Principles (2 ed.). Springer. pp. 58 ff.
  9. ^ Martin Russ (1996). Sound synthesis and sampling. Focal Press. p. 117. ISBN 0240514297. For a sine carrier and non-sinusoidal wave modulator, there are actually the equivalent of several modulator frequencies: one for each harmonic in the modulator.
  10. ^ See for example Walter Greiner, D. Allan Bromley (2007). Quantum Mechanics (2 ed.). Springer. p. 60. ISBN 3540674586. and John Joseph Gilman (2003). Electronic basis of the strength of materials. Cambridge University Press. p. 57. ISBN 0521620058. and Donald D. Fitts (1999). Principles of quantum mechanics. Cambridge University Press. ISBN 0521658411. {{cite book}}: Text "page 17" ignored (help)
  11. ^ Siegmund Brandt, Hans Dieter Dahmen (2001). The picture book of quantum mechanics (3rd ed.). Springer. p. 23. ISBN 0387951415.
  12. ^ Cyrus D. Cantrell (2000). Modern mathematical methods for physicists and engineers. Cambridge University Press. p. 677. ISBN 0521598273.
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