12.7 GROUP VELOCITY It will be readily seen that if all the component simple waves making up a group travel with the same velocity, the group will move with this velocity and maintain its form unchanged. If, however, the velocities vary with wavelength, this is no longer true, and the group will change its form as it progresses. This situation exists for water waves, and if one watches the individual waves in the group sent out by dropping a stone in still water, they will be found to be moving faster than the group as a whole, dying out at the front of the group and reappearing at the back. Hence in this case the group velocity is less than the wave velocity, a relation which always holds when the velocity of longer waves is greater than that of shorter ones. It is important to establish a relation between the group velocity and wave velocity, and this can easily be done by considering the groups formed by superimposing two waves of slightly different wavelength, such as those already discussed and illustrated in Fig. 12F (f). We shall suppose that the two waves have equal amplitudes but slightly different wavelengths, 21 and '1', and slightly di'erent velocities, v and v'. The primed quantities in each case will be taken as the larger. Then the propagation numbers and angular frequencies will also differ, such that k > k' and co > w'. The resultant wave will be given by the sum J' = asin (a): kx) + a 5310\"" k'x) Again applying the trigonometric relation of Eq. (12k), this equation becomes y=Zasin(w;mtk:kx)cos(w;mtkhkx) (12n) 2 Equations (120) and (1213}, although derived for an especially simple type of groUp, are quite general and can be shown to hold for any group whatever. e.g., the three illustrated in Fig. 12H(n). (c), and (e). The relation between wave and group velocities can also be derived in a less mathematical way by considering the motions of the two component wave trains in Fig. 121(o} and (b). At the instant shown, the crests A and A' of the two trains coincide to produce a maximum for the group. A little later the faster waves will have gained a distance A' H .1 on the slower ones, so that E' coincides with B and the maximum of the group will have moved back a distance L Since the difference in velocity of the two trains is do, the time required for this is dude. But in this time both wave trains have been moving to the right, the upper one moving a distance a dildo. The net displacement of the maximum of the group is thus vtdtfdu) .4 A in the time dildo. so that we obtain, for the group velocity, u=v{dA/du)l=v_ld_v dildo d1 in agreement with Eq. (1213}. A picture of the groups formed by two waves of slightly different frequency may easily be produced with the apparatus described in Sec. 12.5. It is merely neces- sary to adjust the two vibrating strips until the frequencies differ by only a few vibra tions per second. See Fig. 126. The group velocity is the important one for light, since it is the only velocity which we can observe experimentally. We know of no means of following the progress of an individual wave in a group of light waves; instead, We are obliged to measure the rate at which a wave train of nite length conveys the energy, a quantity which can be observed. The wave and group Velocities become the same in a medium having no dispersion, i.e., in which tie/til = i}, so that waves of all lengths travel with the same speed. This is accurately true for light traveling in a vacuum, so that there is no di'ereuce between group and wave velocities in this case. B. AAAAAAAAAAAAAAAAAAAAAAAA( ) (c) AX X-X FIGURE 121 Groups and group velocity of two waves of slightly different wavelength and frequency. In Figs. 121(a) and (b) the two waves are plotted separately, while (c) gives their sum, represented by this equation with / = 0. The resultant waves have the average wave- length of the two, but the amplitude is modulated to form groups. The individual waves, having the average of the two k's, correspond to variations of the sine factor in Eq. (12n), and according to Eq. (1 1z), their phase velocity is the quotient of the multipliers of f and x D = w+ w' k + k' That is, the velocity is essentially that of either of the component waves, since these velocities are very nearly the same. The envelope of modulation, indicated by the broken curves shown in Fig. 121, is given by the cosine factor. This has a much smaller propagation number, equal to the difference of the separate ones, and a correspondingly greater wavelength. The velocity of the groups is 1 = dw (120) K k' dk Since no limit has been set on the smallness of the differences, they may be treated as infinitesimals and the approximate equality becomes exact. Then, since ( = uk, we find for the relation between the group velocity u and the wave velocity v u=utk du dk If the variable is changed to 2, through k = 2x/1, one obtains the useful form 1 = 0 - 1. (12p) It should be emphasized that 1 here represents the actual wavelength in the medium. For light, this will not in most problems be the ordinary wavelength in air (see Sec. 23.7)