Question

Derive an expression for group velocity as a function of phase velocity and frequency?

Answer 1

Consider some basic cosine waves of the form Ei=E0cos(ωit−kiz)Ei=E0cos⁡(ωit−kiz) with different amplitudes, frequencies and phases. We know a combination of such waves could result in a wave which has an envelope that is traveling at a group velocity and also has a phase velocity.

When there are some basic waves with equal amplitudes propagating in the same direction of propagation, deriving the expression for the resultant wave is easily possible using some simple trigonometric identities. say when there are two waves of the form: E1=E0cos(ω1t−k1z)E1=E0cos⁡(ω1t−k1z)and E2=E0cos(ω2t−k2z)E2=E0cos⁡(ω2t−k2z) we'll have:

E1+E2=E0(cos(ω1t−k1z)+cos(ω2t−k2z))=2E0cos(ω1−ω22t−k1−k22z)cos(ω1+ω22t−k1+k22z)={2E0cos(ω1−ω22t−k1−k22z)}cos(ω1+ω22t−k1+k22z)E1+E2=E0(cos⁡(ω1t−k1z)+cos⁡(ω2t−k2z))=2E0cos⁡(ω1−ω22t−k1−k22z)cos⁡(ω1+ω22t−k1+k22z)={2E0cos⁡(ω1−ω22t−k1−k22z)}cos⁡(ω1+ω22t−k1+k22z)

First part refers to envelope of the resultant wave and group velocity will be derived from this part and the other part deals with the phase velocity.