# Group - and phase velocity in communication via electromagnetic waves (radio)

I'd like to present the way I see group and phase velocities in the context of radio waves, and see whether or not my understanding is in some way flawed or lacking. I will summarize with a specific question in the end.

Consider a radio wave. This wave is a combination ( a sum ) of individual waves, each with a constant wavenumber $$k$$, a constant frequency $$\omega$$. These individual waves tend to have the form $$\psi_i=A\cos(\omega_it-k_ix)$$

and actually, it turns out that a sum of such waves can be rewritten as a single product of waves. This is a product of trigonometric functions, with different values for the angular frequency $$\omega_{avg}$$, which is derived from the average/differences of angular frequencies in the wave group.

Consider the trig function with the lowest frequency $$\omega_{avg,min}$$. Due to mathematical reasons, this wave acts as an "envelope" for the whole product wave, the radio wave in question. Now the group velocity is the velocity of this envelope, but the phase velocity refers to the velocity of these individual waves which form the resulting radiowave. Also, because $$\omega_{avg,min}$$ is an average of all other $$\omega_i$$'s, it must mean the group velocity cannot exceed any phase velocity.

Now, if I want to communicate with someone via radio, I need to do something to one of these individual waves ( with constant frequency etc ). So the speed of communication is determined by some specific phase velocity. But what role does the group velocity $$v_g$$ have in information transfer - if any? Moreover, if we have a system with dispersion, then $$v_p$$ and $$v_g$$ are related by the equation $$v_g = \frac{c^2}{v_p}$$

which seems to add further complexity ( $$c$$ is light speed ).