Editing Wave equation (section)

===General solution===

==== Algebraic approach ====

For the one-dimensional wave equation a relatively simple general solution may be found. Defining new variables<ref>{{cite web | url = http://mathworld.wolfram.com/dAlembertsSolution.html | title = d'Alembert's Solution | author = Eric W. Weisstein| publisher = [[MathWorld]] | access-date = 2009-01-21 | author-link = Eric W. Weisstein }}</ref>
<math display="block">\begin{align}
\xi  &= x - c t, \\
\eta &= x + c t
\end{align}</math>
changes the wave equation into
<math display="block">\frac{\partial^2 u}{\partial \xi \partial \eta}(x, t) = 0,</math>
which leads to the general solution
<math display="block">u(x, t) = F(\xi) + G(\eta) = F(x - c t) + G(x + c t).</math>

In other words, the solution is the sum of a right-traveling function {{mvar|F}} and a left-traveling function {{mvar|G}}. "Traveling" means that the shape of these individual arbitrary functions with respect to {{mvar|x}} stays constant, however, the functions are translated left and right with time at the speed {{mvar|c}}. This was derived by [[Jean le Rond d'Alembert]].<ref>D'Alembert (1747) [https://books.google.com/books?id=lJQDAAAAMAAJ&pg=PA214 "Recherches sur la courbe que forme une corde tenduë mise en vibration"] (Researches on the curve that a tense cord forms [when] set into vibration), ''Histoire de l'académie royale des sciences et belles lettres de Berlin'', vol.&nbsp;3, p.&nbsp;214–219.
* See also: D'Alembert (1747) [https://books.google.com/books?id=lJQDAAAAMAAJ&pg=PA220 "Suite des recherches sur la courbe que forme une corde tenduë mise en vibration"] (Further researches on the curve that a tense cord forms [when] set into vibration), ''Histoire de l'académie royale des sciences et belles lettres de Berlin'', vol.&nbsp;3, p.&nbsp;220–249. 
* See also: D'Alembert (1750) [https://books.google.com/books?id=m5UDAAAAMAAJ&pg=PA355 "Addition au mémoire sur la courbe que forme une corde tenduë mise en vibration,"] ''Histoire de l'académie royale des sciences et belles lettres de Berlin'', vol.&nbsp;6, p.&nbsp;355–360.</ref>

Another way to arrive at this result is to factor the wave equation using two first-order [[Differential operator|differential operators:]] 
<math display="block">\left[\frac{\partial}{\partial t} - c\frac{\partial}{\partial x}\right] \left[\frac{\partial}{\partial t} + c\frac{\partial}{\partial x}\right] u = 0.</math>
Then, for our original equation, we can define
<math display="block">v \equiv \frac{\partial u}{\partial t} + c\frac{\partial u}{\partial x},</math>
and find that we must have
<math display="block">\frac{\partial v}{\partial t} - c\frac{\partial v}{\partial x} = 0.</math>

This [[advection equation]] can be solved by interpreting it as telling us that the directional derivative of {{mvar|v}} in the {{math|(1, ''-c'')}} direction is 0. This means that the value of {{mvar|v}} is constant on [[Method of characteristics|characteristic]] lines of the form {{math|1=''x'' + ''ct'' = ''x''<sub>0</sub>}}, and thus that {{mvar|v}} must depend only on {{math|''x'' + ''ct''}}, that is, have the form {{math|''H''(''x'' + ''ct'')}}. Then, to solve the first (inhomogenous) equation relating {{mvar|v}} to {{mvar|u}}, we can note that its homogenous solution must be a function of the form {{math|''F''(''x'' - ''ct'')}}, by logic similar to the above. Guessing a particular solution of the form {{math|''G''(''x'' + ''ct'')}}, we find that

<math display="block"> \left[\frac{\partial}{\partial t} + c\frac{\partial}{\partial x}\right] G(x + ct) = H(x + ct).</math>

Expanding out the left side, rearranging terms, then using the change of variables {{math|1=''s'' = ''x'' + ''ct''}} simplifies the equation to

<math display="block"> G'(s) = \frac{H(s)}{2c}.</math>

This means we can find a particular solution {{math|''G''}}  of the desired form by integration. Thus, we have again shown that {{mvar|u}} obeys {{math|1=''u''(''x'', ''t'') = ''F''(''x'' - ''ct'') + ''G''(''x'' + ''ct'')}}.<ref>{{cite web |url=http://math.arizona.edu/~kglasner/math456/linearwave.pdf |title=First and second order linear wave equations |website=math.arizona.edu |archive-url=https://web.archive.org/web/20171215022442/http://math.arizona.edu/~kglasner/math456/linearwave.pdf |archive-date=2017-12-15}}</ref>

For an [[initial-value problem]], the arbitrary functions {{mvar|F}} and {{mvar|G}} can be determined to satisfy initial conditions:
<math display="block">u(x, 0) = f(x),</math><math display="block">u_t(x, 0) = g(x).</math>

The result is [[d'Alembert's formula]]:
<math display="block">u(x, t) = \frac{f(x - ct) + f(x + ct)}{2} + \frac{1}{2c} \int_{x-ct}^{x+ct} g(s) \, ds.</math>

In the classical sense, if {{math|''f''(''x'') ∈ ''C<sup>k</sup>''}}, and {{math|''g''(''x'') ∈ ''C''<sup>''k''−1</sup>}}, then {{math|''u''(''t'', ''x'') ∈ ''C<sup>k</sup>''}}. However, the waveforms {{mvar|F}} and {{mvar|G}} may also be [[generalized functions]], such as the delta-function. In that case, the solution may be interpreted as an impulse that travels to the right or the left.

The basic wave equation is a [[linear differential equation]], and so it will adhere to the [[superposition principle]]. This means that the net displacement caused by two or more waves is the sum of the displacements which would have been caused by each wave individually. In addition, the behavior of a wave can be analyzed by breaking up the wave into components, e.g. the [[Fourier transform]] breaks up a wave into sinusoidal components.

==== Plane-wave eigenmodes ====
{{Main|Helmholtz equation}} 

Another way to solve the one-dimensional wave equation is to first analyze its frequency [[eigenmodes]]. A so-called eigenmode is a solution that oscillates in time with a well-defined ''constant'' angular frequency {{mvar|ω}}, so that the temporal part of the wave function takes the form {{math|1=''e''<sup>−''iωt''</sup> = cos(''ωt'') − ''i'' sin(''ωt'')}}, and the amplitude is a function {{math|''f''(''x'')}} of the spatial variable {{mvar|x}}, giving a [[separation of variables]] for the wave function:
<math display="block">u_\omega(x, t) = e^{-i\omega t} f(x).</math>

This produces an [[ordinary differential equation]] for the spatial part {{math|''f''(''x'')}}:
<math display="block">\frac{\partial^2 u_\omega }{\partial t^2} = \frac{\partial^2}{\partial t^2} \left(e^{-i\omega t} f(x)\right) = -\omega^2 e^{-i\omega t} f(x) = c^2 \frac{\partial^2}{\partial x^2} \left(e^{-i\omega t} f(x)\right).</math>

Therefore,
<math display="block">\frac{d^2}{dx^2}f(x) = -\left(\frac{\omega}{c}\right)^2 f(x),</math>
which is precisely an [[eigenvalue equation]] for {{math|''f''(''x'')}}, hence the name eigenmode. Known as the [[Helmholtz equation]], it has the well-known [[plane-wave]] solutions
<math display="block">f(x) = A e^{\pm ikx},</math>
with [[wave number]] {{math|1= ''k'' = ''ω''/''c''}}.

The total wave function for this eigenmode is then the linear combination
<math display="block">u_\omega(x, t) = e^{-i\omega t} \left(A e^{-ikx} + B e^{ikx}\right) = A e^{-i (kx + \omega t)} + B e^{i (kx - \omega t)},</math>
where complex numbers {{mvar|A}}, {{mvar|B}} depend in general on any initial and boundary conditions of the problem.

Eigenmodes are useful in constructing a full solution to the wave equation, because each of them evolves in time trivially with the phase factor <math>e^{-i\omega t},</math> so that a full solution can be decomposed into an [[eigenmode expansion]]:
<math display="block">u(x, t) = \int_{-\infty}^\infty s(\omega) u_\omega(x, t) \, d\omega,</math>
or in terms of the plane waves,
<math display="block">\begin{align}
u(x, t) &= \int_{-\infty}^\infty s_+(\omega) e^{-i(kx+\omega t)} \, d\omega + \int_{-\infty}^\infty s_-(\omega) e^{i(kx-\omega t)} \, d\omega \\
 &= \int_{-\infty}^\infty s_+(\omega) e^{-ik(x+ct)} \, d\omega + \int_{-\infty}^\infty s_-(\omega) e^{ik (x-ct)} \, d\omega \\
 &= F(x - ct) + G(x + ct),
\end{align}</math>
which is exactly in the same form as in the algebraic approach. Functions {{math|''s''<sub>±</sub>(''ω'')}} are known as the [[Fourier component]] and are determined by initial and boundary conditions. This is a so-called [[frequency-domain]] method, alternative to direct [[time-domain]] propagations, such as [[FDTD]] method, of the [[wave packet]] {{math|''u''(''x'',&nbsp;''t'')}}, which is complete for representing waves in absence of time dilations. Completeness of the Fourier expansion for representing waves in the presence of time dilations has been challenged by [[chirp]] wave solutions allowing for time variation of {{mvar|ω}}.<ref>{{Cite journal |author1=V. Guruprasad |title=Observational evidence for travelling wave modes bearing distance proportional shifts |doi=10.1209/0295-5075/110/54001 |journal=[[Europhysics Letters|EPL]] |volume=110 |issue=5 | date=2015 |pages=54001 |arxiv=1507.08222 |bibcode=2015EL....11054001G |s2cid=42285652 }}</ref> The chirp wave solutions seem particularly implied by very large but previously inexplicable radar residuals in the [[flyby anomaly]] and differ from the sinusoidal solutions in being receivable at any distance only at proportionally shifted frequencies and time dilations, corresponding to past chirp states of the source.