446 Proceedings of the Royal Society of Edinburgh. [Sess. 
Reynolds for the case of a regular procession of water-waves could be ex- 
tended to apply to all kinds of waves. His result may be briefly stated as 
follows. When a large regular group of waves is advancing in any dis- 
persive medium, the energy propagated in one wave-period across a plane 
at right angles to the direction of the wave-motion is equal to the energy 
contained in one wave-length of the group multiplied by the ratio of 
group-velocity to wave-velocity. 
§ 4. It may be here remarked that the above results all refer to a 
regular group of waves, or rather to the regular part of a group which is 
irregular at the front or rear, or to a regular procession of mutually sup- 
porting groups, as in the explanation of group-velocity according to inter- 
ference principles given by Stokes and Lord Rayleigh. The difficulty of 
the subject lies in the application of the theory to the case of a finite group 
of waves, or to any irregular disturbance, and doubt has arisen as to whether 
the theory of group-velocity can be useful in determining the circumstances 
existing at any time in the actual front of a finite or infinite group of 
initially regular waves invading undisturbed space. Lord Kelvin, after 
examining mathematically the front of a large procession of initially 
regular waves advancing into smooth water, remarks : * “ The whole in- 
vestigation shows how very far from finding any definite ‘ group-velocity ’ 
we are, in any initially given group of two, three, four, or any number, 
however great, of waves.” In his last paper on “ Deep Sea Waves,” Proc. 
R.S.E., 1906, he examined more particularly the case of a finite group of 
initially regular waves, and came to the same conclusion (on finding that 
the fronts of the groups extend forward indefinitely to greater and greater 
distances as time goes on). With reference to diagrams which he gives of 
the water surface at different stages (reproduced on page 466 below), he 
says : “ The perceptible fronts of these two groups extend rightwards and 
leftwards from the end of the initial single static group, far beyond the 
4 hypothetical fronts,’ supposed to travel at half the wave-velocity, which 
(according to the dynamics of Osborne Reynolds and Rayleigh in 
their important and interesting consideration of the work required to feed 
a uniform procession of water-waves) would be the actual fronts if the free 
groups remained uniform. How far this if is from being realised is illus- 
trated by the diagrams of fig. 35, which show a great extension outwards 
in each direction far beyond distances travelled at half the wave- velocity.” 
§ 5. From these investigations it seems clear that the dynamical inter- 
pretation of group-velocity cannot be effectively applied to the circum- 
stances of a finite or irregular group of waves. The kinematical interpre- 
* Lord Kelvin, Proc. Roy. Soc. Edin ., vol. xxv., 1904. 
