Confirm that the energy in the TEmn mode travels at the group velocity. [Hint: Find the...

Confirm that the energy in the TE_mn 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 + δ_a) and g(r, t) = B cos (k · r − ωt + δ_b). 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,