Solution:

phase velocity $\left(v_p\right)$ and greer velocity $\left(v_g\right)$:

$u_p=\frac{\omega}{\beta}=\frac{v}{\sqrt{1-(f_c / f)^2}}$

Group velocity $\left(\nu_g\right)$.

It is the velocity with which the energy flows down the waveguide.It is also defined as the velocity associated with the propagation of now band-modulated signal

$v_g=\frac{d \omega}{d \beta}=v \sqrt{1-(f_c/ f)^2}$