432 SCIENCE PROGRESS 



It is unnecessary to do more than indicate here the wide 

 field of applications of the principle of group-velocity by stating 

 one or two recent investigations depending on it. It has been 

 applied in the difficult problem of Ship Resistance to determine 

 the part of the total resistance arising from wave-production in 

 experiments with models. The theory has been extended by 

 Lord Rayleigh to deal with the case of media in which there 

 is minimum wave-velocity such as water, when the influence of 

 gravity and surface tension combined is to be considered ; and 

 the same writer has discussed its application in the case of 

 Aberration in a Dispersive Medium. 



The questions which formed the basis of Lord Kelvin's 

 investigations on Water Waves, as to the cause of the formation 

 of the front and rear of groups of waves travelling in a dis- 

 persive medium, and as to the manner in which irregularity 

 invades a group of waves originally regular, from the mere 

 kinematical point of view, have been satisfactorily answered. 

 In view of the smallness of light waves, the applications of 

 principles primarily derived for the case of infinitely extended 

 media to groups of waves in lenses and prisms is fairly direct ; 

 nevertheless, a consistent development of many parts of Optical 

 Theory from the point of view of Group-velocity would still be 

 a useful undertaking. 



The questions raised by Lord Kelvin, however, have a 

 physical as well as a geometrical aspect. The problem regarding 

 the falling off from regularity of a group is simply, How is the 

 distribution of energy to be determined when a regular group 

 of waves enters a dispersive medium ? The kinematical investi- 

 gations in Lord Kelvin's work and its extensions are thus 

 intimately connected with and are the necessary preliminaries 

 to the study of the passage of energy by means of wave motion 

 through a dispersive medium, and have thus a very important 

 bearing on the modern Theory of Radiation. Some rather 

 important results in this connection can be very simply derived 

 from the solution given by Lord Kelvin in 1887 for the case of 

 the waves produced by a single impulse in a dispersive 

 medium. His evaluation of the integral in equation (1) is as 

 follows : 



cos[k {x - tf(k)} + ?] 

 ^ = V + 27rt{2f (k) + kf"(k)j 



