Light

The most familiar form of waves is of course the surface waves of water, but the term is extended to cover any vibratory effect propagated through a medium. For example if we take a long stretched string and strike it near one end, a small hump will form and will travel without change of shape down the string. Though the portion of the string humped is changing all the time, the geometrical form is unchanging and so we can endow this form with individuality and say that the wave goes at a certain speed. If x denotes the distance of any point on the string from the point of reference, or the origin, and 4) the displacement of the string at that point, the motion is described by the differential equation the solution of which is 4)=f (x—ct). This expresses the fact that, whatever the initial form, the form at time t is the same with the origin shifted a distance ct.

General Characteristics of Waves.

The vibration of a string is specially simple, because the wave undergoes no change of form, but in most cases, such as water-waves, that is not so, for as the wave travels it alters shape, and so it is no longer possible to speak of a definite velocity. It is therefore necessary to analyse the waves into definite types which can be studied separately, and the type which is incomparably the most useful is the sine-curve. That such an analysis is useful is a consequence of the principle of superposition enunciated by Young; mathematically this prin ciple follows from the fact that the differential equations of wave motion are linear, so that if we have two solutions of a wave prob lem we can derive a third by taking their sum. Thus any problem can be solved by expressing the initial state as a sum, or more usually an integral, of sine-curves of various wave-lengths, then solving the motion for each of these separately and finally recom bining them. This is the only method available for water-waves; and though plane waves of light are propagated in free space with out change of form, any obstacle or refractive medium destroys this simplicity and makes the analysis necessary.

In discussing the general characteristics of waves we may con veniently think of a stretched string, but when the appropriate name is given to the dependent variable (/) everything we shall say applies equally well to water-waves, sound, or light. Thus, cl) in the string means the displacement of the string sideways, in water waves the elevation of the surface above its average height, and for sound it might mean the variable part of the air pressure; for light we shall later specialize it rather more and identify it with the electric force. In all these cases the typical solution is

First considering the motion of a particle of the string, we keep x constant and see how 4 depends on t. It is a pure harmonic vi bration of amplitude A and frequency v, frequency meaning the total number of vibrations described in a second. The expression 27r(vt—x/X) --C, regarded as an angle, is the phase at the time t at the distance x from the origin. It can be interpreted very sim ply by the consideration that, when a point describes a circle at uniform speed, its projection on a diameter describes a harmonic motion; then the angle between this diameter and the radius through the point is the phase-angle. The term phase, though quite precise, is often used in a looser sense, because its absolute value does not matter, whereas differences in phase are of the very greatest importance in deciding the character of a wave. We next examine the shape of the string at any instant of time and see that it is a sine curve of wave-length X. The amplitude, frequency, wave-length and phase are the four quantities characteristic of any wave.

The wave is progressive because of the phase relations between the various points of the string. If at any instant we compare the phases at x and x+a, we see that the latter is an angle 27ra/X be hind, and that at a time a/Xv later it will have arrived at the phase which was at x initially. We therefore take Xv = V as the wave-velocity or phase-velocity; it is the rate at which the crests move. Though phase velocity plays a most important part in wave theory it is misleading to think of it as a real velocity. For ex ample, it is a general principle that no effect of any kind can be propagatet1 with a speed greater than c, the velocity of light in vacuo, but it is not very rare to find substances in which the phase velocity of light is considerably higher. When we say that no ef fect can travel quicker than c, we are thinking of the rate at which waves advance into a region previously undisturbed, but phase velocity only has a meaning in connection with a sine-curve, and a sine-curve extends indefinitely in both directions, so that there is no undisturbed region to which the principle can apply.