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[[File:Periodic waves in shallow water.png|right|thumb|Near-periodic waves over shallow water]] |
[[File:Periodic waves in shallow water.png|right|thumb|Near-periodic waves over shallow water]] |
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In general, the propagation of a disturbance takes a different form in different media, for example, media where the velocity of propagation depends upon wavelength ([[dispersion|dispersive]] media) or upon the amplitude of the wave (non-linear media). The wavelength is functionally related to frequency by the physics of the medium. For example, sinusoids are simple solutions (eigensolutions or [[eigenfunction]]s) of [[linear partial differential equation]]s that describe linear physical media, and their frequencies and wavelengths are related by the [[dispersion relation]]s of those media. From these simple sinusoidal solutions, more complex solutions can be built up by [[superposition principle|superposition]]. Just which wavelengths can contribute to a waveform is decided by [[boundary conditions]]. (See the section on [[Wavelength#Standing_waves|standing waves]].) |
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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. The wavelength (or alternatively [[wavenumber]] or [[wave vector]]) is a characterization of the wave in space, that is functionally related to its frequency, as constrained by the physics of the system. That is, sinusoids are the simplest solutions (eigensolutions or [[eigenfunction]]s) of [[linear partial differential equation]]s that describe linear physical media, and their frequencies and wavelengths are related by the [[dispersion relation]]s of those equations or media. From these simple [[Wave#Travelling_waves|traveling wave]] solutions, more complex solutions can be built up by [[superposition principle|superposition]]. |
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In dispersive media, a wave that is periodic in time will not necessarily repeat in space, and most commonly will not have a well-defined wavelength. However, even in a dispersive medium, a sinusoidal, or nearly sinusoidal, wave propagates with no shape change – just a phase change and potentially an amplitude change (for example, if the medium is lossy, or if the wave is spreading out over a larger region). The same is true for a waveform that is composed as a combination of many sinusoids, all of the same wavelength but differing in amplitude, or phase, or both.<ref name=Rayleigh> |
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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 unchanghing shape can also 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. Large-amplitude [[ocean wave]]s with certain shapes can propagate unchanged, because of properties of the nonlinear surface-wave medium.<ref>{{cite book |
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See {{cite book |title=Encyclopedia Britannica |author=[[Lord Rayleigh]] |chapter=Wave theory |url=http://books.google.com/books?id=r54UAAAAYAAJ&pg=PA422&dq=%22only+kind+of+wave+which+can+be+propagated+without+a+change+of+form%22&hl=en&sa=X&ei=npKWT83pL-vJiQLb19zeCQ&ved=0CDAQ6AEwAA#v=onepage&q=%22only%20kind%20of%20wave%20which%20can%20be%20propagated%20without%20a%20change%20of%20form%22&f=false |year=1890 |publisher=The Henry G Allen Company |edition=9th |page=422}} |
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| title = Global environment remote sensing |
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| author = Ken'ichi Okamoto |
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</ref> Arbitrary waveforms can be analyzed as combinations of sinusoidal waves using a [[Fourier transform]], so that the different propagation speeds of the individual components with different wavelengths can be handled separately. |
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| publisher = IOS Press |
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| year = 2001 |
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Under special circumstances, waves other than sinusoids propagate with unchanging shape and constant velocity, called ''traveling waves''. For example, they occur in media that are both linear and non-dispersive, an example being the propagation of sound waves in air.<ref name=Shutilov> |
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| isbn = 9781586031015 |
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{{cite book |title=Fundamental Physics of Ultrasound |author=Vladimir Aleksandrovich Shutilov |chapter=§6. The velocity of sound |url=http://books.google.com/books?id=Bt19qfLnw4kC&pg=PA41 |pages=41 ''ff'' |isbn=2881246842 |publisher=CRC Press |year=1988}} |
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| page = 263 |
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</ref> 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 a [[cnoidal wave]],<ref name=Pilipchuk> |
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| url = http://books.google.com/books?id=tXQy5JdQyZoC&pg=PA263&dq=wave-length++non-sinusoidal |
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{{cite book |
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}}</ref> An example is the [[cnoidal wave]], a periodic traveling wave named because it is described by the [[Jacobi elliptic function]] of ''m''-th order, usually denoted as {{nowrap|''cn''(''x''; ''m'')}}.<ref name=Kundo> |
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|title=Nonlinear Dynamics: Between Linear and Impact Limits |
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|author=Valery N. Pilipchuk |
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|url=http://books.google.com/books?id=pqIlJNq-Ir8C&pg=PA127 |
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|page=127 |
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|chapter=Figure 4.4: Transition from quasi-harmonic to cnoidal wave |
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|isbn= 3642127983 |
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|year=2010 |
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|publisher=Springer |
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}} |
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</ref> a traveling wave so-named because it is described by the [[Jacobi elliptic function]] of ''m''-th order, usually denoted as {{nowrap|''cn''(''x''; ''m'')}}.<ref name=Ludu> |
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{{cite book |
{{cite book |
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|title= |
|title=Tsunami and Nonlinear Waves |
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|author= |
|author=Roger Grimshaw |
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|editor=Anjan Kundu |
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|url=http://books.google.com/books?id=HIu9_8QKo6UC&pg=PA469 |
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|url=http://books.google.com/books?id=2Dtgq-1CGWIC&pg=PA52 |
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|pages=469 ''ff'' |
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|pages=52 ''ff'' |
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|chapter=§18.3 Special functions |
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|chapter=Solitary waves propagating over variable topography |
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|isbn= 3642228941 |
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|isbn=364209032X |
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|year=2012 |
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| |
|year=2007 |
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|publisher=Springer |
|publisher=Springer |
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}}</ref> Large-amplitude [[ocean wave]]s with certain shapes can propagate unchanged, because of properties of the nonlinear surface-wave medium.<ref>{{cite book |
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| title = Nonlinear Ocean Waves and the Inverse Scattering Transform |
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| author = Alfred Osborne |
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| publisher = Academic Press |
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| year = 2010 |
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| chapter=Chapter 1: Brief history and overview of nonlinear water waves |
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| isbn = 0125286295 |
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| pages = 3 ''ff'' |
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| url = http://books.google.com/books?id=wdmsn9icd7YC&pg=PA3 |
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}}</ref> |
}}</ref> |
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[[File:Nonsinusoidal wavelength.JPG|thumb|Wavelength of a periodic but non-sinusoidal waveform.]] |
[[File:Nonsinusoidal wavelength.JPG|thumb|Wavelength of a periodic but non-sinusoidal waveform.]] |
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Mathematically, the amplitude ''f'' of an unchanging waveform moving with a velocity ''v'' can be expressed as {{nowrap|''f''(''x'' − ''vt'')}}, with ''x'' = position and ''t'' = time. The amplitude at location {{nowrap|''x'' + Δ''x''}} at time {{nowrap|''t'' + Δ''t''}} is the same as that at location ''x'' at time ''t'', if Δ''x'' and Δ''t'' are related by {{nowrap|1=Δ''x'' = ''v''Δ''t''}}. |
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If a traveling wave has a fixed shape that repeats, it is a ''periodic wave''.<ref name=McPherson> |
If a traveling wave has a fixed shape that repeats, it is a ''periodic wave''.<ref name=McPherson> |
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|edition=2 |
|edition=2 |
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|publisher=Wiley |
|publisher=Wiley |
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}}</ref> |
}}</ref> Such waves may have a well-defined wavelength even though they are not sinusoidal. As shown in the figure, wavelength is measured between consecutive corresponding points on the waveform. Mathematically, the amplitude ''f'' of an unchanging waveform moving with a velocity ''v'' can be expressed as {{nowrap|''f''(''x'' − ''vt'')}}, with ''x'' = position and ''t'' = time. The amplitude at location {{nowrap|''x'' + Δ''x''}} at time {{nowrap|''t'' + Δ''t''}} is the same as that at location ''x'' at time ''t'', if Δ''x'' and Δ''t'' are related by {{nowrap|1=Δ''x'' = ''v''Δ''t''}}. |
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In more general linear media (that is [[dispersion|dispersive]] media), a wave that is periodic in time will not necessarily repeat in space, so may not have a well-defined wavelength. Such waves are typically analyzed into sinusoidal waves via the [[Fourier series]], so that the different propagation speeds and wavelengths of their different frequency components can be separately handled. To an observer at a fixed location, the amplitude varies in time and repeats itself with a certain ''period'', ''T''. During every period, an integer number of each component wavelength of the wave passes the observer, but with different relative phases at different observer locations. |
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For a fixed-shape waveform that also is periodic, its wavelength λ is defined mathematically in one spatial dimension ''x'' by the definition of periodicity: |
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:<math>f(x-vt+\lambda)=f(x-vt)\ . </math> |
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If this periodicity requirement is satisfied by λ, it also is satisfied by 2λ, 3λ, and so on. The least value of λ that works corresponds to the wavelength of ''f''. This relation shows that if ''x'' increases by λ, ''f'' is not changed. Likewise, if ''t'' increases by λ/''v'', ''f'' is unchanged. This time interval is the ''period'' of the wave, ''T'': |
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:<math>T = \lambda/v \ . </math> |
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To an observer at a fixed location, the amplitude of such a periodic wave varies in time and repeats itself with the period, ''T''. During every period, a wavelength of the wave passes the observer. |
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The mathematical implications of the periodicity condition were thoroughly examined by [[Joseph Fourier]], who demonstrated that any such waveform could be expressed as a sum of sinusoidal waves, today called a [[Fourier series]]: |
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:"''Fourier's theorem'' states that a function ''f(x)'' of spatial period λ, can be synthesized as a sum of harmonic functions whose wavelengths are integral submultiples of λ (''i.e.'' λ, λ/2, λ/3, ''etc.'')."<ref name=Schaum> |
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{{cite book |title=Schaum's Outline of Theory and Problems of Optics |publisher=McGraw-Hill Professional |url=http://books.google.com/books?id=ZIZmyOG-DxwC&pg=PA205&dq=%22can+be+synthesized+as+a+sum+of+harmonic+functions+whose+wavelengths+are+integral+submultiples%22&hl=en&sa=X&ei=7rCVT4jJCYKpiQL2iuWICg&ved=0CDIQ6AEwAA#v=onepage&q=%22can%20be%20synthesized%20as%20a%20sum%20of%20harmonic%20functions%20whose%20wavelengths%20are%20integral%20submultiples%22&f=false |page=205 |author=Eugene Hecht |year=1975 |isbn=0070277303}} |
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</ref> |
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===Envelope waves=== |
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The term ''wavelength'' is also sometimes applied to the [[Envelope (waves)|envelopes]] of waves, such as the traveling sinusoidal envelope patterns that result from the [[Interference (wave propagation)|interference]] of two sinusoidal waves close in frequency; such envelope characterizations are used in illustrating the derivation of [[group velocity]], the speed at which slow envelope variations propagate.<ref> |
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{{cite book |
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| title = Air and Water: The Biology and Physics of Life's Media |
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| author = Mark W. Denny |
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| publisher = Princeton University Press |
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| year = 1995 |
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| isbn = 9780691025186 |
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| page = 289 |
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| url = http://books.google.com/books?id=XjNS6v7q130C&pg=PA289&dq=envelope+wavelength+group-velocity |
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}}</ref> |
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===Wave packets=== |
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[[File:Wave packet (dispersion).gif|thumb|A propagating wave packet]] |
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{{Main|Wave packet}} |
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Localized [[wave packet]]s, "bursts" of wave action where each wave packet travels as a unit, find application in many fields of physics; the notion of a wavelength also may be applied to these wave packets.<ref> |
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{{cite book |
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| title = Subquantum Kinetics: A Systems Approach to Physics and Cosmology |
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| author = Paul A. LaViolette |
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| publisher = Starlane Publications |
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| year = 2003 |
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| isbn = 9780964202559 |
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| page = 80 |
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| url = http://books.google.com/books?id=8HQJAvA1EqkC&pg=PA80&dq=wave-packet-wavelength+de-Broglie-wavelength |
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}}</ref> |
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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 sometimes called a ''local wavelength''.<ref> |
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{{cite book |
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| title = The Quantum Theory of Motion: An Account of the de Broglie–Bohm Causal Interpretation of Quantum Mechanics |
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| author = Peter R. Holland |
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| publisher = Cambridge University Press |
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| year = 1995 |
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| isbn = 9780521485432 |
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| page = 160 |
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| url = http://books.google.com/books?id=BsEfVBzToRMC&pg=PA160&dq=wave-packet+local-wavelength |
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}}</ref><ref name=Cooper> |
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{{cite book |
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|title=Introduction to partial differential equations with MATLAB |
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|author=Jeffery Cooper |
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|url=http://books.google.com/books?id=l0g2BcxOJVIC&pg=PA272 |
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|page=272 |
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|isbn=0817639675 |
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|publisher=Springer |
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|year=1998 |
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|quote=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. |
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}}</ref> An example is shown in the figure. In general, the ''envelope'' of the wave packet moves at a different speed than the constituent waves.<ref name= Fromhold>{{cite book |title=Quantum Mechanics for Applied Physics and Engineering |author=A. T. Fromhold |chapter=Wave packet solutions |pages=59 ''ff'' |quote=(p. 61) ... the individual waves move more slowly than the packet and therefore pass back through the packet as it advances |
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|url=http://books.google.com/books?id=3SOwc6npkIwC&pg=PA59 |
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|isbn=0486667413 |publisher=Courier Dover Publications |
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|year=1991 |
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|edition=Reprint of Academic Press 1981 |
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}}</ref> |
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Using [[Fourier analysis]], wave packets can be analyzed into infinite sums (or integrals) of sinusoidal waves of different [[wavenumber]]s or wavelengths.<ref name=Manners>See, for example, Figs. 2.8–2.10 in |
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{{cite book |
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| title = Quantum Physics: An Introduction |
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| author = Joy Manners |
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| publisher = CRC Press |
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| year = 2000 |
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| isbn = 9780750307208 |
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| chapter=Heisenberg's uncertainty principle |
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| pages = 53–56 |
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| url = http://books.google.com/books?id=LkDQV7PNJOMC&pg=PA54&dq=wave-packet+wavelengths |
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}}</ref> |
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[[Louis de Broglie]] postulated that all particles with a specific value of [[momentum]] have a wavelength |
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:<math>\lambda = \frac{h}{p},</math> |
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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 [[electron]]s in a [[cathode ray tube|CRT]] display have a De Broglie wavelength of about 10<sup>−13</sup> 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.<ref name=Marton> |
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{{cite book |
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|title=Advances in Electronics and Electron Physics |
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|page=271 |
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|url=http://books.google.com/books?id=g5q6tZRwUu4C&pg=PA271 |
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|isbn=0120146533 |year=1980 |publisher=Academic Press |
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|volume=53 |editor=L. Marton and Claire Marton |
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|author=Ming Chiang Li |
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|chapter=Electron Interference |
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}}</ref> The spread of [[wavenumber]]s 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]].<ref name=Manners/> |
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==Interference and diffraction== |
==Interference and diffraction== |
Revision as of 14:36, 25 April 2012
![](https://upload.wikimedia.org/wikipedia/commons/thumb/6/62/Sine_wavelength.svg/220px-Sine_wavelength.svg.png)
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, as well as other spatial wave patterns.[2][3] Wavelength is commonly designated by the Greek letter lambda (λ). The concept can also be applied to periodic waves of non-sinusoidal shape.[1][4] 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.[5] The SI unit of wavelength is the meter.
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.[6]
Examples of wave-like phenomena are sound waves, light, and water waves. A sound wave is a variation in air pressure, while in light and other electromagnetic radiation the strength of the electric and the magnetic field vary. Water waves are variations in the height of a body of water. In a crystal lattice vibration, atomic positions vary.
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 sinusoidal waves over deep water a particle in the water moves in a circle of the same diameter as the wave height, unrelated to wavelength.[7]
Sinusoidal waves
In linear media, any wave pattern 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:[8]
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 phase speed is the speed of light, about 3×108 m/s. Thus the wavelength of a 100 MHz electromagnetic (radio) wave is about: 3×108 m/s divided by 108 Hz = 3 metres. The wavelength of visible light ranges from deep red, roughly 700 nm, to violet, roughly 400 nm (for other examples, see electromagnetic spectrum).
For sound waves in air, the speed of sound is 343 m/s (at room 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. Note that the wavelengths in audible sound are much longer than those in visible light.
![](https://upload.wikimedia.org/wikipedia/commons/thumb/b/b1/Waves_in_Box.svg/220px-Waves_in_Box.svg.png)
Standing waves
![](https://upload.wikimedia.org/wikipedia/commons/thumb/7/7d/Standing_wave_2.gif/220px-Standing_wave_2.gif)
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.[9]
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:
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 ( times the reciprocal of wavelength) and angular frequency ω ( 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 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. The typical convention of using the cosine phase instead of the sine phase when describing a wave is based on the fact that the cosine is the real part of the complex exponential in the wave
General media
![](https://upload.wikimedia.org/wikipedia/commons/thumb/b/b7/Refraction_-_Huygens-Fresnel_principle.svg/220px-Refraction_-_Huygens-Fresnel_principle.svg.png)
![](https://upload.wikimedia.org/wikipedia/commons/thumb/f/f5/Light_dispersion_conceptual_waves.gif/220px-Light_dispersion_conceptual_waves.gif)
The speed of a wave depends upon the medium in which it propagates. In particular, the speed of light in a medium is lower than in vacuum, 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 causes refraction, or a change in direction of waves that encounter the interface between media at an angle.[10] For electromagnetic waves, this change in the angle of propagation is governed by Snell'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 its refractive index according to
where c is the speed of light in vacuum and n(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 vacuum wavelength is known as dispersion, and is also responsible for the familiar phenomenon in which light is separated into component colors by a prism. Separation occurs when the refractive index inside the prism varies with wavelength, so different wavelengths propagate at different speeds inside the prism, causing them to refract at different angles.
Nonuniform media
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.[11]
![](https://upload.wikimedia.org/wikipedia/commons/a/a5/Cochlea_wave_animated.gif)
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 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 of differential equations of such systems is often done approximately, using 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.[12][13] 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.
Crystals
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.[14] 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 the Brillouin zone.[15]
This indeterminacy in wavelength in solids is important in the analysis of wave phenomena such as energy bands and lattice vibrations. It is mathematically equivalent to the aliasing of a signal that is sampled at discrete intervals.
More general waveforms
![](https://upload.wikimedia.org/wikipedia/commons/thumb/d/d0/Periodic_waves_in_shallow_water.png/220px-Periodic_waves_in_shallow_water.png)
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. The wavelength (or alternatively wavenumber or wave vector) is a characterization of the wave in space, that is functionally related to its frequency, as constrained by the physics of the system. That is, sinusoids are the simplest solutions (eigensolutions or eigenfunctions) of linear partial differential equations that describe linear physical media, and their frequencies and wavelengths are related by the dispersion relations of those equations or media. From these simple traveling wave solutions, more complex solutions can be built up by superposition.
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 unchanghing shape can also 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. Large-amplitude ocean waves with certain shapes can propagate unchanged, because of properties of the nonlinear surface-wave medium.[16] An example is the cnoidal wave, a periodic traveling wave named because it is described by the Jacobi elliptic function of m-th order, usually denoted as cn(x; m).[17]
If a traveling wave has a fixed shape that repeats, it is a periodic wave.[18] Such waves may have a well-defined wavelength even though they are not sinusoidal. As shown in the figure, wavelength is measured between consecutive corresponding points on the waveform. Mathematically, the amplitude f of an unchanging waveform moving with a velocity v can be expressed as f(x − vt), with x = position and t = time. The amplitude at location x + Δx at time t + Δt is the same as that at location x at time t, if Δx and Δt are related by Δx = vΔt.
In more general linear media (that is dispersive media), a wave that is periodic in time will not necessarily repeat in space, so may not have a well-defined wavelength. Such waves are typically analyzed into sinusoidal waves via the Fourier series, so that the different propagation speeds and wavelengths of their different frequency components can be separately handled. To an observer at a fixed location, the amplitude varies in time and repeats itself with a certain period, T. During every period, an integer number of each component wavelength of the wave passes the observer, but with different relative phases at different observer locations.
Interference and diffraction
Double-slit interference
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 the interferometer. A simple example is an experiment due to Young where light is passed through two slits.[19] 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 separation d) then the paths are nearly parallel, and the path difference is simply d sin θ. Accordingly the condition for constructive interference is:[20]
where m 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 or fringes, and vice versa.
For multiple slits, the pattern is [21]
where q is the number of slits, and g 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 figure I1 has been set to unity, a very rough approximation.
It should be noted that the effect of interference is to redistribute the light, so the energy contained in the light is not altered, just where it shows up.[22]
Single-slit diffraction
![](https://upload.wikimedia.org/wikipedia/commons/thumb/8/8b/Double-slit_diffraction_pattern.png/200px-Double-slit_diffraction_pattern.png)
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 called diffraction.
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 and Fresnel 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 (Huygen's 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 a small-angle approximation, the intensity spread S is related to position x via a squared sinc function:[23]
- with
where L 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 function S has zeros where u is a non-zero integer, where are at x values at a separation proportion to wavelength.
Diffraction-limited resolution
Diffraction is the fundamental limitation on the resolving power of optical instruments, such as telescopes (including radiotelescopes) and microscopes.[24] For a circular aperture, the diffraction-limited image spot is known as an Airy disk; the distance x in the single-slit diffraction formula is replaced by radial distance r and the sine is replaced by 2J1, where J1 is a first order Bessel function.[25]
The resolvable spatial size of objects viewed through a microscope is limited according to the Rayleigh 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 the numerical aperture:[26]
where the numerical aperture is defined as for θ being the half-angle of the cone of rays accepted by the microscope objective.
The angular size of the central bright portion (radius to first null of the Airy disk) of the image diffracted by a circular aperture, a measure most commonly used for telescopes and cameras, is:[27]
where λ is the wavelength of the waves that are focused for imaging, D the entrance pupil diameter of the imaging system, in the same units, and the angular resolution δ is in radians.
As with other diffraction patterns, the pattern scales in proportion to wavelength, so shorter wavelengths can lead to higher resolution.
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. Such structures have applications in extraordinary optical transmission, and zero-mode waveguides, among other areas of photonics.
Subwavelength may also refer to a phenomenon involving subwavelength objects; for example, subwavelength imaging.
Angular wavelength
A quantity related to the wavelength is the angular wavelength (also known as reduced wavelength), usually symbolized by ƛ (lambda-bar). It is equal to the "regular" wavelength "reduced" by a factor of 2π (ƛ = λ/2π). It is usually encountered in quantum mechanics, where it is used in combination with the reduced Planck constant (symbol ħ, h-bar) and the angular frequency (symbol ω) or angular wavenumber (symbol k).
See also
- Emission spectrum
- Fraunhofer lines – dark lines in the solar spectrum, traditionally used as standard optical wavelength references
- Index of wave articles
- Length measurement
- Spectral line
- Spectrum
- Spectrum analysis
References
- ^ a b Hecht, Eugene (1987). Optics (2nd ed.). Addison Wesley. pp. 15–16. ISBN 0-201-11609-X.
- ^ Raymond A. Serway, John W. Jewett. Principles of physics (4th ed.). Cengage Learning. pp. 404, 440. ISBN 053449143X.
- ^ A. A. Sonin (1995). The surface physics of liquid crystals. Taylor & Francis. p. 17. ISBN 2881249957.
- ^ Brian Hilton Flowers (2000). "§21.2 Periodic functions". An introduction to numerical methods in C++ (2nd ed.). Cambridge University Press. p. 473. ISBN 0198506937.
- ^
Keqian Zhang and Dejie Li (2007). Electromagnetic Theory for Microwaves and Optoelectronics. Springer,. p. 533. ISBN 9783540742951.
{{cite book}}
: CS1 maint: extra punctuation (link) - ^ Theo Koupelis and Karl F. Kuhn (2007). In Quest of the Universe. Jones & Bartlett Publishers. ISBN 0763743879.
- ^ Paul R Pinet (2008). Invitation to Oceanography (5th ed.). Jones & Bartlett Publishers. p. 237. ISBN 0763759937.
- ^
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) - ^ John Avison (1999). The World of Physics. Nelson Thornes. p. 460. ISBN 9780174387336.
- ^ 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. ISBN 0521865565.
- ^ a b Paul R Pinet. op. cit. p. 242. ISBN 0763759937.
- ^ Bishwanath Chakraborty. Principles of Plasma Mechanics. New Age International. p. 454. ISBN 9788122414462.
- ^ 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.
- ^ See Figure 4.20 in A. Putnis (1992). Introduction to mineral sciences. Cambridge University Press. p. 97. ISBN 0521429471. and Figure 2.3 in Martin T. Dove (1993). Introduction to lattice dynamics (4th ed.). Cambridge University Press. p. 22. ISBN 0521392934.
- ^ Manijeh Razeghi (2006). Fundamentals of solid state engineering (2nd ed.). Birkhäuser. pp. 165 ff. ISBN 0387281525.
- ^ Ken'ichi Okamoto (2001). Global environment remote sensing. IOS Press. p. 263. ISBN 9781586031015.
- ^ Roger Grimshaw (2007). "Solitary waves propagating over variable topography". In Anjan Kundu (ed.). Tsunami and Nonlinear Waves. Springer. pp. 52 ff. ISBN 364209032X.
- ^ Alexander McPherson (2009). "Waves and their properties". Introduction to Macromolecular Crystallography (2 ed.). Wiley. p. 77. ISBN 0470185902.
- ^ Greenfield Sluder and David E. Wolf (2007). "IV. Young's Experiment: Two-Slit Interference". Digital microscopy (3rd ed.). Academic Press. p. 15. ISBN 0123740258.
- ^
Halliday, Resnick, Walker (2008). "§35-4 Young's interference experiment". Fundamentals of Physics (Extended 8th ed.). Wiley-India. p. 965. ISBN 8126514426.
{{cite book}}
: CS1 maint: multiple names: authors list (link) - ^
Kordt Griepenkerl (2002). "§9.8.2 Diffraction by a grating". In John W Harris, Walter Benenson, Horst Stöcker, Holger Lutz (ed.). Handbook of physics. Springer. pp. 307 ff. ISBN 0387952691.
{{cite book}}
: CS1 maint: multiple names: editors list (link) - ^ Douglas B. Murphy (2002). Fundamentals of light microscopy and electronic imaging. Wiley/IEEE. p. 64. ISBN 047123429X.
- ^ John C. Stover (1995). Optical scattering: measurement and analysis (2nd ed.). SPIE Press. p. 64. ISBN 9780819419347.
- ^ Graham Saxby (2002). "Diffraction limitation". The science of imaging. CRC Press. p. 57. ISBN 075030734X.
- ^ Grant R. Fowles (1989). Introduction to Modern Optics. Courier Dover Publications. pp. 117–120. ISBN 9780486659572.
- ^ James B. Pawley (1995). Handbook of biological confocal microscopy (2nd ed.). Springer. p. 112. ISBN 9780306448263.
- ^ Ray N. Wilson (2004). Reflecting Telescope Optics I: Basic Design Theory and Its Historical Development. Springer. p. 302. ISBN 9783540401063.