An interesting application of the interplay between phase velocity and group velocity is given by the pattern of waves generated by a moving ship. (This analysis is due to Lord Kelvin.)
Consider first a source of disturbance moving supersonically in a non-dispersive medium (such as the sound generated by a supersonic aeroplane). As Fig. 1 illustrates, all radiation is confined to a cone behind the aeroplane: sound radiated when the aeroplane was at an earlier position will have spread on the surface of a sphere, and the radii of the spheres grow linearly as you look further back in the aeroplane’s travel.
For example, by the time the aeroplane has reached position B, the sound that it made when it was at position A has expanded to the middle one of the three red circles in the figure. If the aeroplane is moving at speed $V$ and the speed of sound is $c$, it is easy to see that the angle $\theta$ of the cone satisfies
$$\sin \theta = \dfrac{ct}{Vt} = \dfrac{c}{V} \tag{1} .$$
This angle is called the Mach angle. By far the biggest signals occur on the boundary of the cone: for any given frequency component of the excitation, the waves from different points of radiation tend to interfere constructively there, whereas within the cone cancellation occurs. The result is the “sonic boom”.
For our dispersive water waves, things are more complicated. Consider a disturbance emitted when the ship was at point A in Fig. 2. By the time the ship has reached point B, $t$ seconds later, we can easily calculate where a wave of any given phase velocity would have reached in a non-dispersive medium. The energy in the wave would have gone predominantly in the direction given by eq. (1), a distance $c t$ where $c$ is the phase velocity. But because the waves are dispersive, the different frequency components generated by the moving ship would have travelled to different points, such as C, D and E. The “ray” direction, such as AC, is always perpendicular to the line BC (or AD to BD, or AE to BE): look back at the right angle marked in Fig. 1. This means that all such points must lie on a circle with AB as diameter (by a familiar geometrical theorem).
Now we know that in fact, each frequency component has not travelled at the speed $c$ but at the group velocity, $c_g = c/2$. The predominant directions of propagation, however, will still be governed by eq. (1) because wave interference effects are governed by the positions of individual peaks and troughs in the pattern, and hence by the phase velocity rather than the group velocity. As a result, the frequency component associated with the direction AC, for example, will in fact have propagated to point $\mathrm{C} ^\prime$ shown in Fig. 3. As a result the actual positions for different frequency components all lie on a scaled-down circle, of half the diameter: the red circle shown in Fig. 3. It follows from the geometry that the pattern of waves generated by the moving ship lie within the “Kelvin wedge”, shown by dashed red lines, with an angle
$$\psi = \sin^{-1} 1/3 \approx 19.5^\circ . \tag{2}$$
Finally, it is of interest to look at the pattern of waves at the boundary of the Kelvin wedge. Figure 4 shows a version of Fig. 3 that highlights the relevant details. The particular frequency component of the wave pattern that passes through the boundary of the wedge is shown in blue. The wave crests will be aligned perpendicular to this blue line at the point where it crosses the wedge boundary, on the line marked in green. This line is inclined at an angle $\phi$ to the direction of the ship’s travel. Noting that triangle PRB is right-angled and that triangle PRQ is isosceles, a little thought about geometry reveals that
$$\phi = \dfrac{1}{2}\left[ \dfrac{\pi}{2} – \psi \right] \approx 55^\circ . \tag{3}$$
Notice that both angles $\psi$ and $\phi$ are governed purely by the geometrical construction of Figs. 3 and 4, based on the fact that the group velocity is one-half of the phase velocity. Nothing else comes into the calculation: not the speed of the ship, the properties of water, nor the strength of gravity.
For fuller mathematical details of the Kelvin wave pattern, see section 3.10 of Lighthill [1].
[1] James Lighthill. “Waves in fluids”, Cambridge University Press (1978)