880 SURFACE WAVES. [CHAP. IX 



and the equation of the free surface is 



,= rnr dx/<x)*(x-*)VcoB** 



(iv). 



These formulae, in which c is a function of k given by Art. 218 (4), may be 

 readily verified by means of Fourier's expression for an arbitrary function as 

 a definite integral, viz. 



- 

 "" J 



dk \\ d\f(\)co$k(\-x)\ (v). 



\J - J 



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. 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 (gtifi 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, as in Art. 167. 



In the case of infinite depth, the formulae (iii), (iv) take the simpler forms 



dk*r L ( f" ] . 1,1 



~ T~ (gk) \ I <*X/ (X) cos * (X - X) f sin g*K*t 



n % L (J -oo J 



/ /"oo 1 . . ~~l 



(vi), 



(vii). 



The problem of tracing out the consequences of a limited initial disturbance, 

 in this case, received great attention at the hands of the earlier investigators 

 in the subject, to the neglect of the more important and fundamental pro- 

 perties of simple-harmonic trains. Unfortunately, the results, even on the 

 simplest suppositions which we may make as to the nature and extent of the 

 original disturbance, are complicated and difficult of interpretation. We shall 

 therefore content ourselves with the subjoined references, which will enable 

 the reader to make himself acquainted with what has been achieved in this 

 branch of the subject*. 



* Poisson, " Memoire sur la Theorie des Ondes," Mem. de VAcad. roy. des 

 Sciences, 1816. 



Cauchy, 1. c. ante p. 18. 



Sir W. Thomson, "On the Waves produced by a Single Impulse in Water of 

 any Depth, or in a Dispersive Medium," Proc. Roy. Soc., Feb. 3, 18 7. 



W. Burnside, " On Deep- Water Waves resulting from a Limited Original 

 Disturbance," Proc. Lond. Math. Soc., t. xx., p. 22 (1888). 



