along, following a particular crest, he finds that it moves forward in the group and gradually gets weaker and dies out in the front, and mystically and mysteriously a weak one in the back works its way forward and gets stronger. In short, the waves are moving through the group while the group is only moving at half the speed that the waves are moving. Fig. 51-10. The wake of a boat Because the group velocities and phase velocities are not equal, then the waves are produced by an object moving through are no longer simply a cone, but it is much more interesting. We can see that in Fig. 51-10, which shows the waves produced by an object moving through the water. Note that it is quite different than what we would have for sound, in which the velocity is independent of wave length, where we would have wavefronts only along the cone, travelling outward Instead of that, we have waves in the back with fronts moving parallel to the motion of the boat, and then we have little waves on the sides at other angles. this entire pattern of waves can, with ing o the square root of the wavelength. The trick is enuity, be analyzed by knowing only this: that the hat the pattern of waves is stationary relative to the(constant-velocity )boat; any other pattern would get lost from the boat The water waves that we have been considering so far were long waves in which le force of restoration is due to gravitation. But when waves get very short in 9s6> the water, the main restoring force is capillary attraction, i. e, the energy of the surface, the surface tension. For surface tension waves, it turns out that the phase velocity Is v phase v2T/Mp(for ripples), where T is the surface tension and p the density. It is the exact opposite: the phase elocity is higher, the shorter the wavelength, when the wavelength gets very small λ,cm When we have both gravity and capillary action, as we always do, we get the com- bination of these two together Fig. 51-11. Phase velocity vs wave- length for water. he=√Tk/p+g/k, where k= 2r/A is the wave number. So the velocity of the waves of water is really quite complicated. The phase velocity as a function of the wavelength is shown in Fig. 51-11; for very short waves it is fast, for very long waves it is fast, and there is a minimum speed at which the waves can go. The group velocity can be calculated from the formula: it goes to 2 the phase velocity for ripples and 2 the phase velocity for gravity waves. To the left of the minimum the group velocity is higher than the pha ase velocit ity; to the right, the group velocity is less than the