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XIII. On the Front and Rear of a Free Procession of Waves 

 in Deep Water. By Sir William Thomson, F.R.S* 



PRELIMINARY. 



General Problem of Deep-Sea Wave-Motion, in two dimensions. 

 (Infinitesimal Motion.) 



npAKINGr x horizontal, and y vertically downwards ; let 

 J- (# + ?, 3Z + 7 ?) De > a t time t, the position of the particle 

 whose position at time is (x, y) ; let <J> denote the velocity- 

 potential at (x, y, t) ; and let P denote its time-integral, 



We have 





t C*d<f> d? , f* <z$ dP n . 



H,**"* 5 and v= ) d % = W ' ' W- 



Let p be the pressure at (x + g, y + y)> (The motion being 

 infinitesimal,) we have 



P = C+g(y+ V )-^ (2), 



or, in virtue of (1), 



~ dV d 2 P 



PrQ+a+t^-^ (3). 



The kinematical conditions are, the equation of continuity, 



d? + W=° (4) ' 



and the boundary equaion, in two parts — one relating to the 

 upper surface, the other to the bottom. The latter, for our 

 present case of infinitely deep water, is simply 



P = when y=zco (5). 



To find the former, or upper-surface kinematical equation, at 

 time t, let it be y = at time 0, and let I) be the height at 

 time t above the level y = 0, of the upper-surface particle 

 whose coordinates at time are (x, 0). Remembering that 

 y positive was taken as downwards, we have, by (1), 



f) (6). 



M 



* Communicated by the Author ; having been read before the Royal 

 Society of Edinburgh, Friday, January 7th, 1887. 



