just means that the phase shift which is produced by the scattered light can be either positive or negative. It can be shown, however, that the speed at which you an send a signal is not determined by the index at one frequency, but depends on what the index is at many frequencies. What the index tells us is the speed at which he nodes(or crests)of the wave travel. The node of a wave is not a signal by itself. In a perfect wave, which has no modulations of any kind, i. e,, which is a steady oscillation, you cannot really say when it"starts, " so you cannot use it for a timing signal. In order to send a signal you have to change the wave somehow, make a notch in it, make it a little bit fatter or thinner. That means that you have to have more than one frequency in the wave, and it can be shown that the speed at which signals travel is not dependent upon the index alone, but upon the way that the index changes with the frequency. This subject we must also delay (until Chapter 48). Then we will calculate for you the actual speed of signals through such a piece of glass, and you will see that it will not be faster than the speed of light, although the nodes, which are mathematical points, do travel faster than the speed of light. Just to give a slight hint as to how that happens, you will not rea difficulty has to do with the fact that the responses of the charges are opposite to the field, i.e., the sign has gotten reversed. Thus in our expression for x(eq. 31. 16) the displacement of the charge is in the direction opposite to the driving field, because(wb -w2)is negative for small wo. The formula says that when the electric field is pulling in one direction, the charge is moving in the opposite direc How does the charge happen to be going in the opposite direction? It certainly does not start off in the opposite direction when the field is first turned on.When the motion first starts there is a transient, which settles down after awhile, a only then is the phase of the oscillation of the charge opposite to the driving field And it is then that the phase of the transmitted field can appear to be advanced with respect to the source wave. It is this advance in phase which is meant when we say that the"phase velocity "or velocity of the nodes is greater than c. In Fig 31-4 we give a schematic idea of how the waves might look for a case where the wave is suddenly turned on(to make a signal). You will see from the diagram that the signal (i. e. the start of the wave)is not earlier for the wave which ends up with an advance in phase Fig. 31-4. Wave"signals deloy of phose Let us now look again at our dispersion equation. We should remark that our analysis of the refractive index gives a result that is somewhat simpler than you would actually find in nature. To be completely accurate we must add some refinements. First, we should expect that our model of the atomic oscillator should have some damping force(otherwise once started it would oscillate forever, and we do not expect that to happen). We have worked out before(Eq. 23. 8)the motion of a damped oscillator and the result is that the denominator in Eq (31.16), and therefore in (31. 19), is changed from(ab3-w2)to(ao-w2+ ira), where n is the damping coefficient. We need a second modification to take into account the fact that there are several resonant frequencies for a particular kind of atom. It is easy to fix up