Waveguide Propagation Constant
The propagation constant (γ) for waveguides defines the amplitude and phase of every element of the wave when it gets transmitted along the waveguide. The factor for each element can be expressed as:
exp[j ω t - γm,n z]
where ω is the angular (2 π x) frequency and z is the direction of propagation.
For example, if γm,n is real,then the phase of each element is constant. So, the amplitude decreases exponentially as the value of z increases. In such a case, there is no significant propagation and the frequency is also below the cut-off frequency of the waveguide.
Let us consider the case where γm,n is imaginary: then the amplitude of each element remains constant whereas the phase changes with z implying that propagation takes place within the waveguide.
γm,n is purely imaginary if there is a total lossless system. In fact, there are always some losses involved; so γm,n will contain a real part (αm,n) as well as an imaginary part (βm,n).
γm,n = αm,n + j βm,n
This theory of waveguides and the related equations are true for all kinds of waveguides, be they circular or rectangular.