﻿Fan-shaped Group of Waves in a Dispersing Medium. 1151 



made equal to AB of fig. 1, and if in tig. 1 the difference in \ 

 on the top and on the bottom is just \ 2 — ^-i> then these lines 

 represent the waves of fig. 1. Thus at P the direction o£ 

 the elementary wave is always vertical, and therefore 

 constant. If the difference in X in fig. 1 is X' 2 — X' l -=fc\ i — \ l , 



the argument applies if we make — = — f ry. In the 



A,2 — A/j A, 2 — A. ] 



general case the argument may be stated as follows : — 



Proof in General Case. 



The motion of a point of the group is such as to keep the 

 phase-difference between two nearly equal wave-lengths con- 

 stant to within quantities of the first order in dX. Therefore if 

 the angle between consecutive waves is so small that dispersion 

 effects may be treated as small quantities of the first order 

 in dX, then the motion of a point of the group may also be 

 said to be such as to keep constant the phase-difference 

 between the wave-lengths X\, X' 2 at the top and at the 

 bottom of an elementary wave. 



Let now two points be considered both moving with the 

 group velocity corresponding, say, to the bottom, and both 

 situated in the surface of an elementary wave at a given 

 instant — one at the top and the other at the bottom of the 

 wave. The wave moves slower at the top than at the 

 bottom, but both top and bottom move faster than the two 

 points. Thus the two points are overtaken by waves coming 

 from behind, and each of these waves is turning during the 

 motion. 



Since now the points move so as to keep the phase- difference 

 between the wave-lengths at the top and at the bottom con- 

 stant, and since the points have been once in the surface of a 

 wave (so that the constant phase-difference is zero) it is 

 apparent that if a wave reaches the top point it simulta- 

 neously reaches the bottom point, and thus the orientation of 

 a wave is unchanged if it is picked out by a point moving 

 with the group velocity*. 



Briefly: both motion with group velocity and motion with 



* It is essential to the argument here given that the words "group 

 velocity " should have a definite meaning. r l his implies that the Fourier 

 analysis of the group is confined to a sharp band on the scale of variations 

 in wave-length along the aperture. A closer examination shows that 

 this condition is fulfilled if the angle between the first and last waves 

 is large in comparison with the least angle distinguishable through 

 diffraction. 



