368 Sir W. Thomson on the Influence of Wind 



Part III. The Influence of Wind on Waves in water supposed 

 frictionless. {Letter to Professor Tait, of date August 16, 1871.) 



Taking OX vertically downwards and OY horizontal, let 



x=hsmn(y — ctt) (1) 



be the equation of the section of the water by a plane perpendi- 

 cular to the wave-ridges \ and let h (the half wave-height) be 



o 



infinitely small in comparison with — (the wave-length). The 



^-component of the velocity of the water at the surface is then 



— nah cos n(y—ut) ; (2) 



and this (because h is infinitesimal) must be the value of -j- for 



the point (0, y) , if <j> denote the velocity-potential at any point 

 (x, y) of the water. Now because 



dx*~*~ dy*~ V ' 



and (j> is a periodic function of y, and a function of x which 

 becomes zero when x = oo, it must be of the form 



P cos [ny — e)e~ nx , 



where P and e are independent of x and y. Hence, taking ~, 



putting x=0 in it, and equating it to (2), we have 



— Yn cos (ny — e) = —nuh cos (ny—nctt) ; 



and therefore Y = uh, and e^nut ; so that we have 



<j) = uh€- nx cos n(y— at). . ... (3) 



This, it is to be remarked, results simply from the assumptions 

 that the water is frictionless, that it has been at rest, and that 

 its surface is moving in the manner specified by (1). 



If the air were a frictionless liquid moving irrotationally, 

 with a constant velocity V at heights above the water (that is to 

 say, values of —x) considerably exceeding the wave-length, its 

 velocity potential ifr, found on the same principle, would be 



(V— u)he nx cosn (y — ett)+Vy. ... (4) 



Let now q denote the resultant velocity at any point (x, y) of 

 the air. Neglecting infinitesimals of the order (nh) 2 , we have 



±q <2 = ±Y*-Y(Y-ct)nhe n *smn(y-at). . . (5) 



Now, if p denote the pressure at any point (x } y) in the air, and 

 a the density of the air, we have by the general equation for 



