Editing Propagation constant (section)

==Phase constant==
In [[electromagnetic theory]], the '''phase constant''', also called '''phase change constant''', '''parameter''' or '''coefficient''' is the imaginary component of the propagation constant for a plane wave.  It represents the change in phase per unit length along the path traveled by the wave at any instant and is equal to the [[real part]] of the [[wavenumber#In wave equations|angular wavenumber]] of the wave.  It is represented by the symbol ''β'' and is measured in units of radians per unit length.

From the definition of (angular) wavenumber for [[Transverse mode|transverse electromagnetic]] (TEM) waves in lossless media,
:<math>k = \frac{2\pi}{\lambda} = \beta</math>

For a [[transmission line]], the [[telegrapher's equations]] tells us that the wavenumber must be proportional to frequency for the transmission of the wave to be undistorted in the [[time domain]].  This includes, but is not limited to, the ideal case of a lossless line.  The reason for this condition can be seen by considering that a useful signal is composed of many different wavelengths in the frequency domain.  For there to be no distortion of the [[waveform]], all these waves must travel at the same velocity so that they arrive at the far end of the line at the same time as a [[group velocity|group]].  Since wave [[phase velocity]] is given by

:<math>v_p = \frac{\lambda}{T} = \frac{f}{\tilde{\nu}} = \frac{\omega}{\beta},</math>

it is proved that ''β'' is required to be proportional to ''ω''. In terms of primary coefficients of the line, this yields from the telegrapher's equation for a distortionless line the condition

:<math>\beta = \omega \sqrt{LC},</math>

where ''L'' and ''C'' are, respectively, the inductance and capacitance per unit length of the line. However, practical lines can only be expected to approximately meet this condition over a limited frequency band.

In particular, the phase constant <math> \beta </math> is not always equivalent to the [[wavenumber]] <math>k</math>. The relation
:<math> \beta = k </math>
applies to the TEM wave, which travels in free space or TEM-devices such as the [[coaxial cable]] and [[Twin-lead|two parallel wires transmission lines]]. Nevertheless, it does not apply to the [[Transverse mode|TE]] wave (transverse electric wave) and [[Transverse mode|TM]] wave (transverse magnetic wave). For example,<ref>{{cite book |first=David |last=Pozar |author-link=David M. Pozar |year=2012 |title=Microwave Engineering |edition=4th |publisher=John Wiley &Sons |isbn=978-0-470-63155-3 |pages=62–164 }}</ref> in a hollow [[waveguide]] where the TEM wave cannot exist but TE and TM waves can propagate,

:<math>k=\frac{\omega}{c} </math>

:<math>\beta=k\sqrt{1-\frac{\omega_c^2}{\omega^2}}</math>

Here <math> \omega_{c} </math>  is the [[cutoff frequency]]. In a rectangular waveguide, the cutoff frequency is

:<math> \omega_{c} = c \sqrt{\left(\frac{n \pi}{a}\right)^2 + \left(\frac{m \pi}{b}\right) ^2}, </math>

where <math>m,n \ge 0</math> are the mode numbers for the rectangle's sides of length <math>a</math> and <math>b</math> respectively. For TE modes, <math> m,n \ge 0</math> (but <math> m = n = 0</math> is not allowed), while for TM modes <math> m,n \ge 1 </math>.

The phase velocity equals

:<math>v_p=\frac{\omega}{\beta}=\frac{c}{\sqrt{1-\frac{\omega_\mathrm{c}^2}{\omega^2}}}>c </math>