07i Waves in water supposed frictionless. 369 



pressure in an irrotationally moving fluid, 



C-p^^+^-ffx) (6) 



Using (4) and (5) in this and putting C = ^o-V 2 , we find 



— p = ak— nh(V— ct) 2 e nx sin n(y — at) — gx\ . . . (7) 



Similarly if p 1 denote the pressure at any point (x } y) of the 

 water, since in this case q* is infinitesimal, we have 



—p , = nha Q 6~ nx smn(y — at)—gx i ... (8) 



the density of the water being taken as unity. 



Now let T be the cohesive tension of the separating surface of 



air and water. The curvature of the surface at any point (a?, y) 



d 2 x 

 given by equation (1), being y^, is equal to 



— n*hmin(x — *f) (9) 



Hence at any point (x, y) fulfilling (1), 



p-p' = Wh&inn(x-at); .... (10) 



and by (7) and (8), with for x its value by (1) (which, as h is 

 infinitesimal, only affects their last terms), we have 



^-y = A{n[<r(V— «) 9 + « 9 ]-^(l-o-)}sinn(af-«/). . (11) 



This, compared with (10), gives 



rc[o-(V-a) 2 + a 2 ]-#(l-0-)=W. . . (12) 

 Let 



-i/«$2^ m 



which (being the value of a for V = 0) is the velocity of propa- 



2tt 



gation of waves with no wind, when the wave-length is — , 



Then (12) becomes 



««+<r(V-«)« 



=wr 



I+, (14) 



which determines a, the velocity of the same waves when there 

 is wind, of velocity V, in the direction of propagation of the 

 waves. Solving the quadratic, we have 



•=i+7 ± t^-(i+5j»J- • • (15) 



This result leads to the following conclusions : — 

 Phil Mag. S. 4. Vol. 42. No. 281. Nov. 1871. 2 B 



