220-221] EFFECT OF ARBITRARY INITIAL CONDITIONS. 381 



221. A remarkable result of the dependence of the velocity of 

 propagation on the wave-length is furnished by the theory of 

 group-velocity. It has often been noticed that when an isolated 

 group of waves, of sensibly the same length, is advancing over 

 relatively deep water, the velocity of the group as a whole is less 

 than that of the waves composing it. If attention be fixed on a 

 particular wave, it is seen to advance through the group, gradually 

 dying out as it approaches the front, whilst its former position in 

 the group is occupied in succession by other waves which have 

 come forward from the rear. 



The simplest analytical representation of such a group is 

 obtained by the superposition of two systems of waves of the same 

 amplitude, and of nearly but not quite the same wave-length. 

 The corresponding equation of the free surface will be of the form 



f] = a sin k (x ct) + a sin k' (x c'i) 



/I* V l*f* J/J '// \ / _L Z*' Zv _l V /f 



/ A/ ft/ tv\j fv \s i \ I fv ~T fv A/v \^ IV L/ 



= 2a cos 



If k, k' be very nearly equal, the cosine in this expression varies 

 very slowly with x ; so that the wave-profile at any instant has 

 the form of a curve of sines in which the amplitude alternates 

 gradually between the values and 2a. The surface therefore 

 presents the appearance of a series of groups of waves, separated 

 at equal intervals by bands of nearly smooth water. The motion 

 of each group is then sensibly independent of the presence of the 

 others. Since the interval between the centres of two successive 

 groups is %7r/(k k'), and the time occupied by the system in 

 shifting through this space is 27r/(kc k'c'), the group -velocity is 

 (kc k'c')l(k k'), or d (kc)/dk, ultimately. In terms of the wave- 

 length X (= 2-7T/&), the group- velocity is 



dc 



This result holds for any case of waves travelling through a 

 uniform medium. In the present application we have 



c= f tanhM (3), 



