Growth of a Train of Waves. 15 



downwards : and let II denote constant surface-pressure, and 



rake — as the value of the arbitrary constant. C. Thus ('ISO) 



9 " 



gives, at the disturbed surface 



0=-^F(x, l + 5„ t) +^=-|F(^ 1, *)+<7?i (151). 



/r v, - ■ ^' V - .'M (/ , 



The equality between the second and third members of 

 this formula is due to the disturbance being infinitely small, 



which makes - F(a?, 1 + ?i, r F(a', 1, f) an infinitely small 



at at 



quantity of the second order, negligible in comparison with 

 g£ v which is an infinitely small quantity of the first order. 



§ 121. For a sinusoidal wave-disturbance of wave-length 

 2w/m, travelling #-wards with velocity v, we have, as in {66) 

 above, 



F(«, .:, 0=-^~ m(z - 1) sin m{x-vt) . . (152). 



For surface-equation (151) becomes 



= kmvcos m(x — vt) — g£ x . . . (153). 



This gives as the equation of the free surface 



Jj=7iC0s m[.v— vt) .... (151 ), 



where /t= ■ (loo). 



9 



Xow by (149) and (152) with 2 = 1, we find 



£i==-cosm(.i; — vt) .... (156). 



Comparison of this with (151) gives 



r- = r //m = A r //27r .... (157). 



§ 122. Let us now rind A (activity), the rate of doing 

 work by the pressure of the water on one side upon the 

 water on the other side of a vertical plane (,r). We have 



A= (**</./>£=[ rf:| [-'J +0(*_1 + C)] (158). 



«/ J. %) 1 



Eliminating from this f and F by (149) and (152), we find 

 A= —tocos mix— rt) I dze- m (~- l >r— kmve~ m ^ z -^ cos m(sp— vt) 



+g(z- l + C)] . (1.39). 



