404 Lord Rayleigh on the 



relative motion of the waves and of the group still continues, 

 and thus we have to introduce the factor Y/U, obtaining for 

 the number N of waves outside 



Y—U t' 

 N=V^ (4) - 



If X be the distance through which the pressure travels, 

 X = Yt'; and if Y be the (constant) velocity outside and X the 

 wave-length outside, \ = Y Q r. Thus 



To introduce optical notation, let /n = V /V, so that fi is the 

 refractive index. In terms of /jl 



Y ^ d P (a\ 



u =IM ~ x dx (6) > 



so that finally 



N =- X S w> 



in close correspondence with (1). A very simple formula 

 thus expresses the number of waves (after emergence) gene- 

 rated by the travel of the pressure over a distance X of a 

 dispersive medium. 



The above calculation has the advantage of being* clear of 

 the complication due to obliquity; but a very little modifi- 

 cation will adapt it to the case of a prism, especially if we 

 suppose that the waves considered are emergent at the second 

 face of the prism without refraction. In the figure, AC repre- 

 sents an incident plane pulse 

 whose trace runs along the first 

 face of the prism from A to B. 

 AF, BE is the direction of pro- 

 pagation of the refracted waves 

 under consideration, to which the 

 second face of the prism is per- 

 pendicular. As before, if t be — 

 the period, V the wave-velocity 

 ■of the waves propagated in di- 

 rection BE, U the corresponding 

 group-velocity, t' the time of 

 travel of the pulse from A to B, 

 the number of waves within the \F 



medium is 



V-U *' 

 V t' 



