Interference Phenomena. 529 



But there is a point of view sufficiently instructive in 

 my opinion to be mentioned. In the case of a grating it is 

 easy to see how a solitary impulse is spread out into a dis- 

 turbance lasting a finite time — it is not so clear that the same 

 holds for a prism. The easiest way to assure ourselves that 

 a prism acts in the same way as a grating, is to cease to 

 consider only homogeneous waves and to follow through the 

 prism a group of waves involving, as it always does, oscilla- 

 tions of different periods. 



Let us imagine a train of waves made up of a finite number 

 of oscillations, each following the law of a simple pendulum. 

 The analytical representation of such a group will introduce 

 wave-lengths which do not differ much. The front or rear 

 of such a train of waves may be taken to pass through a 

 medium like glass with a definite velocity, which is the group 

 velocity corresponding to the given range of 

 wave-lengths. In fig. 4 let A B be the Fig. 4. 



direction of the front of the incident wave, 

 C D that of the emergent wave, then, as the 

 group velocity in the prism is smaller than 

 the wave velocity, the front of the group 

 will after emergence be parallel to some 

 such direction as D K. If the light is 

 concentrated by means of a lens, the position 

 of the focus is determined by the condition 

 of agreement of phase ; that is to say, the 

 optical length from all points CD to the focus must be the 

 same, but in that case the optical length from the different 

 points of K D will not be the same, — in other words, the 

 front of the wave will take some time to pass through the 

 focus, just as it would if the spectrum were produced by a 

 grating. The same holds for the rear of the group and for 

 any disturbance, account being taken of the different group 

 velocities for different regions of the spectrum. To make the 

 analogy with the grating more complete, let us calculate the 

 distance C K through which the group has fallen behind the 

 front. The group velocity U is connected with the wave- 

 velocity Y by 



where k is inversely proportional to the wave-length. The 

 difference in the two velocities is therefore 



v u dx 



The wave-front will pass through a thickness t of the prism 



