Editing Wavelength (section)

=== 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:
<math display="block">  y (x, \ t) = A \cos \left( 2 \pi \left( \frac{x}{\lambda } - ft \right ) \right )  = A \cos \left( \frac{2 \pi}{\lambda} (x - vt) \right )</math>
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 [[wavenumber]] ''k'' (2π times the reciprocal of wavelength) and [[angular frequency]] ''ω'' (2π times the frequency) as:
<math display="block">  y (x, \ t) = A \cos \left( kx - \omega t \right)  = A \cos \left(k(x - v t) \right) </math>
in which wavelength and wavenumber are related to velocity and frequency as:
<math display="block"> k = \frac{2 \pi}{\lambda} = \frac{2 \pi f}{v} = \frac{\omega}{v},</math>
or
<math display="block"> \lambda = \frac{2 \pi}{k} = \frac{2 \pi v}{\omega} = \frac{v}{f}.</math>

In the second form given above, the phase {{nowrap|(''kx'' − ''ωt'')}} is often generalized to {{nowrap|('''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
<math display="block">A e^{ i \left( kx - \omega t \right)}. </math>