Editing Wave (section)

== Wave in elastic medium ==
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{{Main|Wave equation|d'Alembert's formula}}

Consider a traveling [[transverse wave]] (which may be a [[pulse (physics)|pulse]]) on a string (the medium). Consider the string to have a single spatial dimension. Consider this wave as traveling

[[File:Nonsinusoidal wavelength.svg|class=skin-invert-image|thumb|right|200 px|Wavelength ''λ'' can be measured between any two corresponding points on a waveform.]]
[[File:Superpositionprinciple.gif|class=skin-invert-image|thumb|Animation of two waves, the green wave moves to the right while blue wave moves to the left, the net red wave amplitude at each point is the sum of the amplitudes of the individual waves. Note that {{nowrap|1=''f''(''x'', ''t'') + ''g''(''x'', ''t'') = ''u''(''x'', ''t'')}}. |left]]
* in the <math>x</math> direction in space. For example, let the positive <math>x</math> direction be to the right, and the negative <math>x</math> direction be to the left.
* with constant [[amplitude]] <math>u</math>
* with constant velocity <math>v</math>, where <math>v</math> is
** independent of [[wavelength]] (no [[dispersion relation|dispersion]])
** independent of amplitude ([[linear]] media, not [[Nonlinearity|nonlinear]]).<ref name=Helbig>{{cite book |title = Seismic waves and rays in elastic media |chapter-url = https://books.google.com/books?id=s7bp6ezoRhcC&pg=PA134 |pages = 131 ''ff'' |author = Michael A. Slawinski |chapter = Wave equations |isbn = 978-0-08-043930-3 |year = 2003 |publisher = Elsevier }}</ref><ref name=Ostrovsky>{{cite book |title = Modulated waves: theory and application |url = https://www.amazon.com/gp/product/0801873258 |author1=Lev A. Ostrovsky |author2=Alexander I. Potapov |name-list-style=amp |publisher = Johns Hopkins University Press |isbn = 978-0-8018-7325-6 |year = 2001 }}</ref>
* with constant [[waveform]], or shape

This wave can then be described by the two-dimensional functions
{{unbulleted list | style = padding-left: 1.5em
| <math>u(x,t) = F(x - v t)</math> (waveform <math>F</math> traveling to the right)
| <math>u(x,t) = G(x + v t)</math> (waveform <math>G</math> traveling to the left)
}}
or, more generally, by [[d'Alembert's formula]]:<ref name=Graaf>{{cite book |title = Wave motion in elastic solids |last = Graaf |first= Karl F |edition = Reprint of Oxford 1975 |publisher = Dover |year = 1991 |url = https://books.google.com/books?id=5cZFRwLuhdQC |pages = 13–14 |isbn = 978-0-486-66745-4 }}</ref>
<math display="block">u(x,t) = F(x - vt) + G(x + vt). </math>
representing two component waveforms <math>F</math> and <math>G</math> traveling through the medium in opposite directions. A generalized representation of this wave can be obtained<ref>For an example derivation, see the steps leading up to eq. (17) in {{cite web |url = http://prism.texarkanacollege.edu/physicsjournal/wave-eq.html |title = Kinematic Derivation of the Wave Equation |last = Redfern |first= Francis |work = Physics Journal |access-date = 2012-12-11 |archive-date = 2013-07-24 |archive-url = https://web.archive.org/web/20130724011045/http://prism.texarkanacollege.edu/physicsjournal/wave-eq.html |url-status = dead }}</ref> as the [[partial differential equation]]
<math display="block">\frac{1}{v^2}\frac{\partial^2 u}{\partial t^2}=\frac{\partial^2 u}{\partial x^2}.</math>

General solutions are based upon [[Duhamel's principle]].<ref name=Struwe>{{cite book |title = Geometric wave equations |author1=Jalal M. Ihsan Shatah |author2=Michael Struwe |chapter-url = https://books.google.com/books?id=zsasG2axbSoC&pg=PA37 |chapter = The linear wave equation |pages = 37''ff'' |isbn = 978-0-8218-2749-9 |year = 2000 |publisher = American Mathematical Society Bookstore }}</ref>

=== Wave forms ===
{{main|Waveform}}
[[File:Waveforms.svg|class=skin-invert-image|thumb|right|280 px|[[Sine wave|Sine]], [[Square wave (waveform)|square]], [[Triangle wave|triangle]] and [[Sawtooth wave|sawtooth]] waveforms]]

The form or shape of ''F'' in [[d'Alembert's formula]] involves the argument ''x'' − ''vt''. Constant values of this argument correspond to constant values of ''F'', and these constant values occur if ''x'' increases at the same rate that ''vt'' increases. That is, the wave shaped like the function ''F'' will move in the positive ''x''-direction at velocity ''v'' (and ''G'' will propagate at the same speed in the negative ''x''-direction).<ref name=Lyons>{{cite book |url = https://books.google.com/books?id=WdPGzHG3DN0C&pg=PA128 |pages = 128 ''ff'' |title = All you wanted to know about mathematics but were afraid to ask |author = Louis Lyons |isbn = 978-0-521-43601-4 |publisher = Cambridge University Press |year = 1998 }}</ref>

In the case of a periodic function ''F'' with period ''λ'', that is, ''F''(''x'' + ''λ'' − ''vt'') = ''F''(''x'' − ''vt''), the periodicity of ''F'' in space means that a snapshot of the wave at a given time ''t'' finds the wave varying periodically in space with period ''λ'' (the [[wavelength]] of the wave). In a similar fashion, this periodicity of ''F'' implies a periodicity in time as well: ''F''(''x'' − ''v''(''t'' + ''T'')) = ''F''(''x'' − ''vt'') provided ''vT'' = ''λ'', so an observation of the wave at a fixed location ''x'' finds the wave undulating periodically in time with period ''T'' = ''λ''/''v''.<ref name="McPherson0">{{cite book |title = Introduction to Macromolecular Crystallography |last = McPherson |first= Alexander |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>

=== Amplitude and modulation ===
{{Main|Amplitude modulation}}
{{See also|Frequency modulation|Phase modulation}}

[[File:Amplitudemodulation.gif|class=skin-invert-image|thumb|Amplitude modulation can be achieved through ''f''(''x'',''t'') = 1.00×sin(2π/0.10×(''x''−1.00×''t'')) and ''g''(''x'',''t'') = 1.00×sin(2π/0.11×(''x''−1.00×''t'')) only the resultant is visible to improve clarity of waveform.|left]]
[[File:Wave packet.svg|class=skin-invert-image|right|thumb|Illustration of the ''envelope'' (the slowly varying red curve) of an amplitude-modulated wave. The fast varying blue curve is the ''carrier'' wave, which is being modulated.]]

The amplitude of a wave may be constant (in which case the wave is a ''c.w.'' or ''[[continuous wave]]''), or may be ''modulated'' so as to vary with time and/or position. The outline of the variation in amplitude is called the ''envelope'' of the wave. Mathematically, the [[Amplitude modulation|modulated wave]] can be written in the form:<ref name=Jirauschek>{{cite book |url = https://books.google.com/books?id=6kOoT_AX2CwC&pg=PA9 |page = 9 |title = FEW-cycle Laser Dynamics and Carrier-envelope Phase Detection |author = Christian Jirauschek |isbn = 978-3-86537-419-6 |year = 2005 |publisher = Cuvillier Verlag }}</ref><ref name="Kneubühl">{{cite book |title = Oscillations and waves |author = Fritz Kurt Kneubühl |url = https://books.google.com/books?id=geYKPFoLgoMC&pg=PA365 |page = 365 |year = 1997 |isbn = 978-3-540-62001-3 |publisher = Springer }}</ref><ref name=Lundstrom>{{cite book |url = https://books.google.com/books?id=FTdDMtpkSkIC&pg=PA33 |page = 33 |author = Mark Lundstrom |isbn = 978-0-521-63134-1 |year = 2000 |title = Fundamentals of carrier transport |publisher = Cambridge University Press }}</ref>
<math display="block">u(x,t) = A(x,t) \sin \left(kx - \omega t + \phi \right) , </math>
where <math>A(x,\ t)</math> is the amplitude envelope of the wave, <math>k</math> is the ''[[wavenumber]]'' and <math>\phi</math> is the ''[[phase (waves)|phase]]''. If the [[group velocity]] <math>v_g</math> (see below) is wavelength-independent, this equation can be simplified as:<ref name=Chen>{{cite book |chapter-url = https://books.google.com/books?id=LxzWPskhns0C&pg=PA363 |author = Chin-Lin Chen |title = Foundations for guided-wave optics |page = 363 |chapter = §13.7.3 Pulse envelope in nondispersive media |isbn = 978-0-471-75687-3 |year = 2006 |publisher = Wiley }}</ref>
<math display="block">u(x,t) = A(x - v_g t) \sin \left(kx - \omega t + \phi \right) , </math>
showing that the envelope moves with the group velocity and retains its shape. Otherwise, in cases where the group velocity varies with wavelength, the pulse shape changes in a manner often described using an ''envelope equation''.<ref name=Chen/><ref name="Recami">{{cite book |last1=Longhi |first1=Stefano |title=Localized Waves |last2=Janner |first2=Davide |publisher=Wiley-Interscience |year=2008 |isbn=978-0-470-10885-7 |editor1=Hugo E. Hernández-Figueroa |page=329 |chapter=Localization and Wannier wave packets in photonic crystals |editor2=Michel Zamboni-Rached |editor3=Erasmo Recami |chapter-url=https://books.google.com/books?id=xxbXgL967PwC&pg=PA329}}</ref>

=== Phase velocity and group velocity ===
{{Main|Phase velocity|Group velocity}}
{{See also|Envelope (waves)#Phase and group velocity}}
[[Image:Wave group.gif|class=skin-invert-image|thumb|The red square moves with the [[phase velocity]], while the green circles propagate with the [[group velocity]].|left]]

There are two velocities that are associated with waves, the [[phase velocity]] and the [[group velocity]].

Phase velocity is the rate at which the [[phase (waves)|phase]] of the wave [[Wave propagation|propagates in space]]: any given phase of the wave (for example, the [[crest (physics)|crest]]) will appear to travel at the phase velocity. The phase velocity is given in terms of the [[wavelength]] {{mvar|λ}} (lambda) and [[Wave period|period]] {{mvar|T}} as
<math display="block">v_\mathrm{p} = \frac{\lambda}{T}.</math>

[[Image:Wave opposite-group-phase-velocity.gif|class=skin-invert-image|thumb|right|A wave with the group and phase velocities going in different directions]]

Group velocity is a property of waves that have a defined envelope, measuring propagation through space (that is, phase velocity) of the overall shape of the waves' amplitudes—modulation or envelope of the wave.