Light

In the case of water waves, and perhaps of electric waves of very long period, we can observe the crests and so can see directly the wave-velocity and measure the amplitude, but in general it is impossible to do this. Usually we only observe some dynamical effect, such as that which occurs when the rays of light impinge on the retina of the eye. The exact form of this effect is special to each type of wave, but it is a general principle that it depends on the energy of the waves and this is proportional to the average value of the square of the amplitude of the wave-variable O. This quantity is the intensity ; it is often quite unnecessary to know the factor of proportionality associated with it. It is important to observe that the amplitude can be deduced from the intensity, but that nothing can be said about the phase ; this is a consequence of the dynamical fact that only by altering the phase can the wave medium do work on the observing instrument, so that only by spoiling the phase can we observe the wave.

The case of waves in three dimensions is of course more com plicated than that of one. In thinking about them the most im portant idea is that of the wave-front which is any continuous surface over which the phases are all the same. For mathematical treatment such waves are sometimes resolved into sets of plane waves going in all directions, but it is more often convenient to make use of spherical waves. It can be shown that, if a three dimensional medium propagates plane waves according to the rule then it can propagate a spherical wave of the form where r is the distance from the origin. The wave-fronts are spheres round the origin as centre, and the wave represents an emission from a point source at the origin. The amplitude de creases as the wave spreads out, and is always proportional to the inverse distance. The intensity is therefore proportional to the inverse square of the distance and this is the fundamental law of photometry (see PHOTOMETRY). This type of wave suffices for the discussion of a great many phenomena in optics, but it may be mentioned that it will require some unessential modifications when we come to discuss polarization and electromagnetic waves.

The Velocity of a Group of Waves.

Much of optical theory can be developed by the consideration of pure monochromatic waves, and indeed it may be said that experimental methods have tended to move in directions for which these suffice. The mono chromatic wave, however, gives a very incomplete account of events, because it ranges over an infinite extent of time and space, whereas we usually want to know about events in some limited region. We saw, for instance, in Huygens construction for double refraction, that the extraordinary wave has its front parallel to the ordinary, but yet that the rays go in a different direction. Even

in the simpler case of plane waves of unlimited breadth there is a similar complication when the wave-velocity depends on the wave length, as is the case in refracting media. Let us take a group of approximately monochromatic waves of limited length, and find the group-velocity with which it travels as a whole. To construct such a group we take a sine-curve multiplied by a factor such that the waves are not of uniform height, but fall away gradually to zero on both sides of the centre. The solution then shows that, though the crests travel forward with the wave-velocity appro priate to the wave-length, yet they alter in height as they go, so that after a time the crest which was highest at first will have be come quite inconspicuous, while another wave originally quite small will have grown up and taken its place as highest crest. Thus the group as a whole moves with a different velocity from its component waves. The group-velocity U is derived from the wave-velocity V by the formula d(kV)/dk, where k is the recip rocal of the wave-length. The phenomenon of group-velocity is easily observed from the deck of a ship, for after a very short time a large wave under observation becomes quite small while another behind it has grown at its expense. For water the wave velocity is proportional to the square root of the wave-length, and our formula then shows that the group-velocity is half the wave-velocity. For other types of waves it may be greater than the wave-velocity, and there is nothing to prevent it even being in the opposite direction, though no case of this is known. Only in the case where the wave-velocity is independent of the wave length is it equal to the group-velocity; in this case any wave can be propagated without change of form.

The most important application of the idea of group-velocity to optics arises in the measurement of the velocity of light. Every type of measure depends in some way on interrupting the light and thus gives the group-velocity. In free space the wave-velocity of light does not depend on the wave-length and so is the same as the group-velocity, but this is not so in other cases. The superior ity of the wave theory of light over the corpuscular theory was held proved when Foucault showed directly that light goes slower through water than air; but he did not in fact prove it, for the refractive index depends on wave-velocity, and his work dealt with group-velocity. However, since either can be deduced from the other, it is easy to verify the correctness of his result indirectly. In much of optics these considerations do not arise, and so they are frequently forgotten. Nevertheless they are indispensable for a full understanding of waves, and many difficulties have been caused in the past through neglecting them.