Group Velocity
Understanding Group Velocity
Group velocity is the speed that matters for information transfer. While individual wave crests may travel at the phase velocity (which can exceed c in waveguide), the signal envelope, which carries the information, travels at the group velocity, which is always less than c in waveguide.
Group Velocity in Waveguide
v_g = c x sqrt(1 - (fc/f)^2). Near cutoff, group velocity approaches zero (the signal barely propagates). Far above cutoff, group velocity approaches c. The product of group velocity and phase velocity always equals c^2: v_g x v_p = c^2.
Implications
- Signal delay: Longer waveguide or slower group velocity means more delay.
- Dispersion: Different frequencies travel at different group velocities, spreading pulses.
- Group delay: tau = L / v_g. Group delay varies with frequency in waveguide.
v_g = c x sqrt(1 - (fc/f)^2)
Phase velocity in waveguide:
v_p = c / sqrt(1 - (fc/f)^2)
Relationship: v_g x v_p = c^2
In dielectric: v_g = c / sqrt(er) (approximately)
WR-90 at 10 GHz:
v_g = c x sqrt(1-(6.557/10)^2) = 0.755c
Frequently Asked Questions
What is group velocity?
Group velocity is the speed at which the signal envelope (modulation/information) travels through a medium. In free space it equals c. In waveguide, it is always less than c and varies with frequency.
How does group velocity relate to group delay?
Group delay = path length / group velocity. In waveguide, group velocity varies with frequency, so group delay also varies. This variation is the source of dispersion that spreads pulsed signals.
Can group velocity exceed the speed of light?
In some exotic media (anomalous dispersion), the group velocity can appear to exceed c. However, information still cannot travel faster than light. The front velocity (the speed of the first arrival of a signal) never exceeds c.