Jlfaiichestcr Memoirs, Vol. xliv. (1900), No. 0. 



VI. Geometrical Representation of the Relation 

 between Wave-Velocity and Group-Velocity. 



By Prof. Horace Lamb, M.A., LL.D., F.R.S. 



Received Decemhcr r2th, iSgg, read [an nary gth, igoo. 



The fact that in a group of progressive waves on 

 water tlic velocity of advance of the group as a whole is 

 much less than that of the individual waves composing it 

 seems to have been first distinctly recognised by Scott 

 Russell* The theoretical explanation has been given, 



* Brit. Ass. Rt'p., 1S44, p. 369. As Russell's share in the matter appears 

 to have been somewhat overlooked, the following quotations may be of 

 interest : 



"(Jne observation which I have made is curious. It is, that in the case 

 of oscillating waves of the second order, I have found that the motion of 

 propagation of the whole group is different from the apparent motion of wave- 

 transmission along the surface ; that in the group whose velocity of oscillation 

 is as observed 3 '57 feet per second, each wave having a seeming velocity of 

 3"57, the whole group moves forward in the direction of transmission with a 

 much slower velocity. The consequence of this is a difficulty in observing 

 these waves (especially such as are raised by the wind at sea), namely, that as 

 the eye follows the crest of the wave, this crest appears to run out of sight, 



and is lost in the small waves in which the group termiuates There 



is to be observed, therefore, this distinction in a group of waves of the second 

 order, between the velocity of individual wave-transmission and the velocity 

 of aggregate wave- propagation. 



" I have not found it possible to measure this velocity of aggregate pro- 

 pagation of a group of waves, from want of a point to observe. If I lix my 

 eye upon a single wave, I follow it along the group, and it gradually diminishes 

 and then disappears ; I take another and follow it, and it also disappears. 

 My eye, in following a wave-crest, follows the visible velocity of transmission 

 merely. After one or two such observations, I find that the whole group of 

 oscillations has been transferred along in the direction of transmission with a 

 velocity comparatively slower ; but I have not been able to measure this 



velocity of propagation of the wave-motion from one place to another 



" On Observations of the Waves of the Sea Besides the coexistence 



of different series of waves, we have the difficulty arising from the fact already 



May 4t/i, igoo. 



