114 Sir W. Thomson on the Front and Rear of 



The most general upper-surface dynamical condition which 

 can be imposed is 



iVo)=/^) ( 7 )> 



where / denotes an arbitrary function of the two independent 

 variables. 



Suppose now the water to be at rest at time 0. It is clear 

 from dynamical considerations that the solution of (4), subject 

 to the conditions (5), (7), (3), is fully determinate : and when 

 it is found, (1) gives the position at time t of the fluid-particle 

 which at time was in any position [x, y) ; and so completes 

 the solution of the problem. 



The particular solution which we are now going to work 

 out to represent a uniform procession of waves commencing 

 at time 0, and produced and maintained by the application of 

 changing pressure to the surface in the neighbourhood of the 

 zero of x, must, as its appropriate form of (7), fulfil the 

 condition 



p ( 0) = $(x) sin cot + F (x) cos cot . . (8), 



where $(x) and ¥(x) denote functions which vanish for all 

 large positive or negative values of x. 



If we wish to make only a single procession, in the direc- 

 tion of x positive for example, we may take 



8(*)=F(«-fewy) (9). 



A perfectly general formula is easily (by the Fourier- 

 Poisson-Cauchy method) written down to express the value 

 of P; and so, by (1) and (6), the complete solution of the 

 problem : for § and F any given arbitrary functions. 



It is obvious that, so far as ^ is concerned, the general 

 solution for x any considerable multiple of ±1, and exceeding 

 ±1 by not less than two or three times the wave-length, 

 27rg/co 2 , must, for values of t great enough to have let the 

 front of the procession pass the place x y be 



l} = 31 sin [*>*--— O-/)] +Acos[©*-— (*-f)] 

 i for x positive, 



[2 - 1 r" 2 — i 



cot- —(—x+f) — Acos cot— — ( — ,s + f) I 



for x negative, 



where 21 and/ denote quantities calculable from the form of §; 

 and A and f similarly from F. Further, it is obvious that 

 the front of each procession will, for any value of t not less 



K10); 



