Editing Wave equation (section)

==== 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.