Editing Wavelength (section)

== More general waveforms ==
[[File:Periodic waves in shallow water.png|right|thumb|Near-periodic waves over shallow water]]

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.<ref name=Rayleigh>
See {{cite book |title=Encyclopædia Britannica |author=Lord Rayleigh |author-link=Lord Rayleigh |chapter=Wave theory |chapter-url=https://books.google.com/books?id=r54UAAAAYAAJ&q=%22only+kind+of+wave+which+can+be+propagated+without+a+change+of+form%22&pg=PA422 |year=1890 |publisher=The Henry G Allen Company |edition=9th |page=422}}
</ref> 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. Sinusoids are the simplest [[Wave|traveling wave]] solutions, and more complex solutions can be built up by [[superposition principle|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 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>
{{cite book
 |title=Nonlinear Dynamics: Between Linear and Impact Limits
 |author=Valery N. Pilipchuk
 |chapter-url=https://books.google.com/books?id=pqIlJNq-Ir8C&pg=PA127
 |page=127
 |chapter=Figure 4.4: Transition from quasi-harmonic to cnoidal wave
 |isbn= 978-3642127984
 |year=2010
 |publisher=Springer
 }}</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>
{{cite book
 |title=Nonlinear Waves and Solitons on Contours and Closed Surfaces
 |author=Andrei Ludu
 |chapter-url=https://books.google.com/books?id=HIu9_8QKo6UC&pg=PA469
 |pages=469 ''ff''
 |chapter=§18.3 Special functions
 |isbn= 978-3642228940
 |year=2012
 |edition=2nd
 |publisher=Springer
 }}</ref> Large-amplitude [[ocean wave]]s with certain shapes can propagate unchanged, because of properties of the nonlinear surface-wave medium.<ref>{{cite book
 | title = Nonlinear Ocean Waves and the Inverse Scattering Transform
 | author = Alfred Osborne
 | publisher = Academic Press
 | year = 2010
 | chapter=Chapter 1: Brief history and overview of nonlinear water waves
 | isbn = 978-0-12-528629-9
 | pages = 3 ''ff''
 | chapter-url = https://books.google.com/books?id=wdmsn9icd7YC&pg=PA3
 }}</ref>

[[File:Nonsinusoidal wavelength.svg|thumb|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 a ''periodic wave''.<ref name=McPherson>
{{cite book
 |title=Introduction to Macromolecular Crystallography
 |author=Alexander McPherson |chapter-url=https://books.google.com/books?id=o7sXm2GSr9IC&pg=PA77
 |page=77
 |chapter=Waves and their properties
 |isbn=978-0-470-18590-2
 |year=2009
 |edition=2
 |publisher=Wiley
 }}</ref> Such waves are sometimes regarded as having a wavelength even though they are not sinusoidal.<ref name=may>
{{cite book
 |title=Fourier Analysis
 |url=https://books.google.com/books?id=gMPVFRHfgGYC&pg=PA1
 |page=1
 |isbn=978-1-118-16551-5
 |year=2011
 |publisher=John Wiley & Sons
 |author=Eric Stade
 }}</ref> As shown in the figure, wavelength is measured between consecutive corresponding points on the waveform.

=== Wave packets ===
[[File:Wave packet (dispersion).gif|thumb|A propagating wave packet]]

{{Main|Wave packet}}

Localized [[wave packet]]s, "bursts" of wave action where each wave packet travels as a unit, find application in many fields of physics. A 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>
{{cite book
 | title = The Quantum Theory of Motion: An Account of the de Broglie–Bohm Causal Interpretation of Quantum Mechanics
 | author = Peter R. Holland
 | publisher = Cambridge University Press
 | year = 1995
 | isbn = 978-0-521-48543-2
 | page = 160
 | url = https://books.google.com/books?id=BsEfVBzToRMC&q=wave-packet+local-wavelength&pg=PA160
 }}</ref><ref name=Cooper>
{{cite book
 |title=Introduction to partial differential equations with MATLAB
 |author=Jeffery Cooper
 |url=https://books.google.com/books?id=l0g2BcxOJVIC&pg=PA272
 |page=272
 |isbn=0-8176-3967-5
 |publisher=Springer
 |year=1998
 |quote=The local wavelength ''λ'' of a dispersing wave is twice the distance between two successive zeros. ... the local wavelength and the local wave number ''k'' are related by ''k'' = 2π / ''λ''.
 }}</ref> An example is shown in the figure. In general, the ''envelope'' of the wave packet moves at a speed different from 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
 |chapter-url=https://books.google.com/books?id=3SOwc6npkIwC&pg=PA59
 |isbn=0-486-66741-3 |publisher=Courier Dover Publications
 |year=1991
 |edition=Reprint of Academic Press 1981
 }}</ref>

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
{{cite book
 | title = Quantum Physics: An Introduction
 | author = Joy Manners
 | publisher = CRC Press
 | year = 2000
 | isbn = 978-0-7503-0720-8
 | chapter=Heisenberg's uncertainty principle
 | pages = 53–56
 | chapter-url = https://books.google.com/books?id=LkDQV7PNJOMC&q=wave-packet+wavelengths&pg=PA54
 }}</ref>

[[Louis de Broglie]] postulated that all particles with a specific value of [[momentum]] ''p'' have a wavelength ''λ'' = ''h''/''p'', where ''h'' is the [[Planck constant]]. 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 {{val|e=−13|u=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>
{{cite book
 |title=Advances in Electronics and Electron Physics
 |page=271
 |chapter-url=https://books.google.com/books?id=g5q6tZRwUu4C&pg=PA271
 |isbn=0-12-014653-3 |year=1980 |publisher=Academic Press
 |volume=53 |editor1=L. Marton |editor2=Claire Marton |author=Ming Chiang Li
 |chapter=Electron Interference
 }}</ref> The spatial spread of the wave packet, and the spread of the [[wavenumber]]s of sinusoids that make up the packet, correspond to the uncertainties in the particle's position and momentum, the product of which is bounded by [[Heisenberg uncertainty principle]].<ref name=Manners/>