$\hbox{\rm\bf Group}\bigl(\theta\bigr)$ vs $\hbox{\rm\bf phase}\bigl(\theta\bigr)$ vs $\hbox{\rm\bf group}\bigl(\phi(\theta)\bigr)$

For another irony, consider the relation between group and phase velocity, usually written

$$V_{\hbox{\rm group}}^2\bigl(\phi_{\hbox{\rm group}}(\theta_{... ...ox{\rm phase}}) \over d \theta_{\hbox{\rm phase}} } \biggr)^2 .$$ (1)

This equation is just the scalar part of the statement ``the group velocity vector is the sum of the phase velocity vector (perpendicular to the wavefront) and the sideslip velocity vector (parallel to the wavefront)''. This equation immediately implies the inequality  

$$V_{\hbox{\rm group}}(\phi_{\hbox{\rm group}}) \geq V_{\hbox{\rm phase}}(\theta_{\hbox{\rm phase}}) .$$ (2)

or ``group velocities are greater than (or equal to) associated phase velocities''. Earlier in this paper, however, we saw that ``vertical phase velocities are greater than (or equal to) vertical group velocities'', or mathematically  

$$V_{\hbox{\rm group}}(\psi) \leq V_{\hbox{\rm phase}}(\psi) .$$ (3)

These inequalities seem contradictory at first glance, but they are not because they refer to different things. Equation (2) compares the phase and group velocities associated with the same point on a wavefront. Equation (3) compares the phase and group velocities for the same propagation direction ($\psi$). The solid vertical lines in Figure  show how the phase and group velocities for the same (vertical) propagation direction are related. The tilted dotted line in the same figure indicates how the vertical phase velocity also has an associated (non-vertical) group velocity. The non-vertical group velocity associated with the vertical phase is faster than the vertical phase, which is in turn faster than the vertical group!

 

Measurement velocity

$${\hbox{\rm V}}_{\hbox{\scriptsize\rm P}}(0^\circ)$$ 3.46

$${\hbox{\rm V}}_{\hbox{\scriptsize\rm P}}(45^\circ)$$ 3.86

$${\hbox{\rm V}}_{\hbox{\scriptsize\rm P}}(90^\circ)$$ 4.49

$${\hbox{\rm V}}_{\hbox{\scriptsize\rm SV}}(0^\circ)$$ 1.99

$${\hbox{\rm V}}_{\hbox{\scriptsize\rm SV}}(45^\circ)$$ 2.13

$${\hbox{\rm V}}_{\hbox{\scriptsize\rm SV}}(90^\circ)$$ 2.02

$${\hbox{\rm V}}_{\hbox{\scriptsize\rm SH}}(0^\circ)$$ 2.00

$${\hbox{\rm V}}_{\hbox{\scriptsize\rm SH}}(45^\circ)$$ 2.33

$${\hbox{\rm V}}_{\hbox{\scriptsize\rm SH}}(90^\circ)$$ 2.62