In general, a wave propagating in +z direction can be described by a wave function:
\[\Psi(t,z) = g(\omega t-\beta z)\]
When we look at a wave, we would observe the constant phase and sense it moving at the phase velocity.
\[\omega t-\beta z = \text{const.}\]
\[\omega -\beta \frac{dz}{dt} = 0\Rightarrow v_p = \frac{dz}{dt} = \frac{\omega}{\beta}\]
2. Group Velocity
In the figure below, when the two waves align, we would sense them moving at the group velocity.
\[\begin{cases}
\lambda _{gA} + \Delta z = v_{pzA}\Delta t\\
\lambda _{gB} + \Delta z = v_{pzB}\Delta t
\end{cases}\Rightarrow\Delta t = \frac{\lambda_{gA} - \lambda_{gB}}{v_{pzA} - v_{pzB}},~\Delta z = \frac{\lambda_{gA} v_{pzB} - \lambda_{gB} v_{pzA}}{v_{pzA} - v_{pzB}}\]
\[\Rightarrow v_g = \frac{\Delta z}{\Delta t} = \frac{\lambda_{gA} v_{pzB} - \lambda_{gB} v_{pzA}}{\lambda_{gA} - \lambda_{gB}} = \frac{\omega_B - \omega_A}{\beta_{zB} - \beta_{zA}}\]
In the narrow-band case,
\[v_g = \frac{d\omega}{d\beta_z}\]
3. Example 1: Parallel-Plate Metallic Waveguide
Graph \(\omega\)-\(\beta_z\) for a parallel-plate metallic waveguide.
The slope of the graph at a given \(\beta_z\) is the group velocity \(v_g=d\omega/d\beta_z\) and the slope of the line passing origin at a given \(\beta_z\) is the phase velocity \(v_p=\omega/\beta_z\)
4. Example 2: Narrow-band Amplitude Modulation
The wave shown in the video has the form:
\[E(z,t)=E_0[1+m\cos{(\Delta\omega\cdot t-\Delta\beta_z\cdot z)}]\cos{(\omega t-\beta_zz)}\]
where the parameters are:
\[E_0=1,~m=0.1,~\omega=10\pi,~\Delta\omega=\pi,~\beta_z=1,~\Delta\beta_z=0.15\]
The velocity at which the high-frequency wave moves is the phase velocity \(v_p=\omega/\beta_z\), while the envelope moves at group velocity \(v_g=\Delta\omega/\Delta\beta_z\) .
\[\begin{cases}
\lambda _{gA} + \Delta z = v_{pzA}\Delta t\\
\lambda _{gB} + \Delta z = v_{pzB}\Delta t
\end{cases}\Rightarrow\Delta t = \frac{\lambda_{gA} - \lambda_{gB}}{v_{pzA} - v_{pzB}},~\Delta z = \frac{\lambda_{gA} v_{pzB} - \lambda_{gB} v_{pzA}}{v_{pzA} - v_{pzB}}\]
\[\Rightarrow v_g = \frac{\Delta z}{\Delta t} = \frac{\lambda_{gA} v_{pzB} - \lambda_{gB} v_{pzA}}{\lambda_{gA} - \lambda_{gB}} = \frac{\omega_B - \omega_A}{\beta_{zB} - \beta_{zA}}\]
In the narrow-band case,
\[v_g = \frac{d\omega}{d\beta_z}\]
