159.] WAVES IN DEEP WATER. 191 



These formulae, in which c is a function of k given by (28), may 

 be readily verified by means of Fourier s expression for an arbitrary 

 function as a definite integral, viz. 



/(*) = Ifcdk {/ &quot;_ d\/(X) cos k (X - *)} . 



When the initial conditions are arbitrary, the subsequent 

 motion is made up of systems of waves, of all possible lengths, 

 travelling in either direction, each with the velocity proper to 

 its own wave-length, as given by (28). Hence, in general, the 

 form of the free surface is continually altering, the only exception 

 being when the wave-length of every component system which 

 is present in sensible amplitude is large compared with the depth 

 of the fluid. In this case the velocity of propagation Jgh is 

 independent of the wave-length, so that if we have waves 

 travelling in one direction only, the wave-profile remains un 

 changed in form as it advances. Compare Art. 148. 



A curious result of the dependence of the velocity of propaga 

 tion on the wave-length occurs when we have two systems of 

 waves of the same amplitude, and of nearly but not quite equal 

 wave-lengths, travelling in the same direction. The equation 

 of the free surface is then of the form 



y = a sin k (x ct) + a sin k (.? c t) 



k - k kc - k c \ , ik + k kc 4- k c , 

 = 2a cos x --- - t sm x -- t 



- c - c \ , 



g x --- - t) s 



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

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

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

 slowly 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 interval 



between the centres of two successive groups is ,- p, arid the 



. kc k c d . kc , . , , 



velocity of advance of the groups is ~j~j, ~ &amp;gt; or ^/. ultimately. 



