Q:

Propagating modes in a waveguide, what do they represent?

In a hollow rectangular wave guide of dimensions $a \times b$ for example, I know how to apply the boundary conditions to find the solutions. In particular, for TE (or TM) modes we have the expression $$k=\sqrt{\left(\frac{\omega}{c}\right)^2-\pi^2\left[\left(\frac{m}{a}\right)^2+\left(\frac{n}{b}\right)^2\right]} $$ as our dispersion relation. I understand that in order to excite a certain TE$_{mn}$ mode the driving frequency must exceed the cutoff frequency for such mode. My question is what does it mean to excite a mode? Is it that we may only find waves propagating with distinct frequencies that correspond to the excited modes? 

A:

For some given $\omega$, one can only excite specific traveling modes corresponding to different values of $k$ as shown by the dispersion relation. This means that a traveling mode can be excited at any frequency as long as it exceeds the cutoff frequency $\omega_{mn}\equiv c\pi \sqrt{\big(\frac{m}{a}\big)^2+\big(\frac{n}{b}\big)^2}$ for that mode.

