Editing Wavenumber (section)

=== Plane waves in linear media ===
The propagation factor of a [[sinusoidal plane wave]] propagating in the positive x direction in a linear material is given by<ref name="Harrington_1961">{{Citation |last=Harrington |first= Roger F. |year= 1961 |title= Time-Harmonic Electromagnetic Fields |edition= 1st |publisher= McGraw-Hill |isbn=0-07-026745-6  }}</ref>{{rp|p=51}}
: <math>  P = e^{-jkx}  </math>
where
* <math>k = k' - jk'' = \sqrt{-\left(\omega \mu '' + j \omega \mu' \right) \left(\sigma + \omega \varepsilon '' + j \omega \varepsilon  ' \right) }\;</math>
* <math>k' =</math> [[phase constant]] in the units of [[radian]]s/meter
* <math>k'' =</math> [[attenuation constant]] in the units of [[neper]]s/meter
* <math>\omega =</math> angular frequency
* <math>x =</math> distance traveled in the ''x'' direction
* <math>\sigma =</math> [[Electrical resistivity and conductivity|conductivity]] in [[Siemens (unit)|Siemens]]/meter
* <math>\varepsilon = \varepsilon' - j\varepsilon'' =</math> [[Permittivity#Complex permittivity|complex permittivity]]
* <math>\mu = \mu' - j\mu'' =</math> [[Permeability (electromagnetism)#Complex permeability|complex permeability]]
* <math>j=\sqrt{-1}</math>

The [[sign convention]] is chosen for consistency with propagation in lossy media. If the attenuation constant is positive, then the wave amplitude decreases as the wave propagates in the x-direction.

[[Wavelength]], [[phase velocity]], and [[skin effect|skin depth]] have simple relationships to the components of the wavenumber:
: <math> \lambda = \frac {2 \pi} {k'} \qquad  v_p = \frac {\omega} {k'}   \qquad  \delta = \frac 1 {k''}  </math>