per meter in the direction of motion, and wo is the frequency, then the observed irequency for a moving man is 2/c2 (3413) For the case of light, we know that ko wo/c. So, in this particular probler he equation would read o(1 (34.14) which looks completely unlike formula(34.12)! Is the frequency that we would observe if we move toward a source different than the frequency that we would see if the source moved toward us? Of course not! The theory of relativity says that these two must be exactly equal. If we were expert enough mathematicians we would probably recognize that these two mathematical expressions are exactly equal! In fact, the necessary equality of the two expressions is one of the ways by which some people like to demonstrate that relativity requires a time dilation because if we did not put those square-root factors in, they would no longer be Since we know about relativity, let us analyze it in still a third way, which may appear a little more general. (It is really the same thing, since it makes no difference how we do it! )According to the relativity theory there is a relationship between position and time as observed by one man and position and time as seen by another who is moving relative to him. We wrote down those relationships long ago( Chapter 16). This is the Lorentz transformation and its inverse √1-v2/c2 t+ux/c (34.15) If we were standing still on the ground, the form of a wave would be cos (ot kx); all the nodes and maxima and minima would follow this form But what would man in motion, observing the same physical wave, see? Where the field is zero, the positions of all the nodes are the same(when the field is zero, everyone measures the field as zero); that is a relativistic invariant. So the form is the same for the other man too, except that we must transform it into his frame of reference os(at -kx) t'-ux/c2 1-u2/c If we regroup the terms inside the brackets, we get (at-kx) t ku k This is again a wave, a cosine wave, in which there is a certain frequency w,a constant multiplying t', and some other constant, k', multiplying x'. We call k' the wave number, or the number of waves per meter, for the other man. Therefore the other man will see a new frequency and a new wave number given by w+ kw (3417) k wv/ 1-2/c2 If we look at (34. 17), we see that it is the same formula(34 13), that we obtained by a more physical argument