Wave Number or Phase Constant?

There have been many confused by the two terms - wave number and phase constant. Are they related? They do not seem to be interchangeable, or are they?

This short article may be of help in answering those questions.

A wave solution in free-space leads to wave number k, which is for TEM mode of propagation. Here k is a vector along the wave direction that makes it the phase constant (as relates to phase velocity). If k has real and imaginary terms, the imaginary part refers to attenuation due to losses.

In transmission lines, the term beta is used. If the transmission mode is TEM, then both are identical. If the transmission line has loss, we have attenuation constant alpha also. Alpha and beta together are referred to as gamma, the propagation constant.

In a waveguide there is no TEM mode, and kz is used to denote the direction of propagation (z-axis is normally used propagation direction; x-axis and y-axis are bounded by the structure walls). Note that it is only one component of k, the others being kx, ky. In that scenario, the phase velocity is determined by kz and not k. Many books use the term Kz or beta interchangeably, to refer to the phase constant and related phase velocity, and adding to the confusion.

Since kz is expected to be real for propagation, under some conditions, it may cease to be real. This is the evanescent wave scenario and reason for the cutoff frequency limits in waveguides.

Best way to understand and appreciate above is to solve the equations in free-space, transmission lines and waveguides.