Confirm that the energy in the TEmn mode travels at the group velocity. [Hint: Find the time averaged Poynting vector and the energy density (use Prob. 9.12 if you wish). Integrate over the cross section of the wave guide to get the energy per unit time and per unit length carried by the wave, and take their ratio.]
Reference prob 9.12
In the complex notation there is a clever device for finding the time average of a product. Suppose f (r, t) = A cos (k · r – ω + δa) and g(r, t) = B cos (k · r − ωt + δb). Show that where the star denotes complex conjugation. [Note that this only works if the two waves have the same k and ω, but they need not have the same amplitude or phase.] For example,
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In the question, our aim is to show that the energy in the mode travels at the group velocity.
We have to start with the time averaged Poynting vector and the energy density and then integrate them over the cross section of the waveguide to get the energy per unit time and energy per unit length and take their ratio to prove that it will be same as the group velocity.