384 SURFACE WAVES. [CHAP. IX 



222. The theory of progressive waves may be investigated, in 

 a very compact manner, by the method of Art. 172*. 



Thus if <j>, >|r be the velocity- and stream-functions when the 

 problem has been reduced to one of steady motion, we assume 



= - (x + iy) + 



whence <f>/c = # (ae~ ky jSe^) sin kx,\ , 



Tjr/c = - ?/ + (ae-ty + $<&) cos Tex j 



This represents a motion which is periodic in respect to x, super- 

 posed on a uniform current of velocity c. We shall suppose that 

 ka. and k{3 are small quantities ; in other words that the amplitude 

 of the disturbance is small compared with the wave-length. 



The profile of the free surface must be a stream-line ; we will 

 take it to be the line ty = 0. Its form is then given by (1), viz. to 

 a first approximation we have 



y = (* + &) COB fas ........................ (2), 



shewing that the origin is at the mean level of the surface. 

 Again, at the bottom (y = h) we must also have i/r = const. ; this 

 requires 



ae kh _|_ fi e -kh _ Q 



The equations (1) may therefore be put in the forms 



<f>/c = {K+C cosh k (y 4- h) sin kx t \ 

 ^Ir/c = y+C sinh k(y + h) cos kx } ' 



The formula for the pressure is 



p . (7dd>\ 2 fd<b\*\ 



= const. ay * < I -^- + I -=*- r 

 p 2 [\<&c/ \ctyx J 



c 2 

 = const. #2/ - {1 2A:Ocosh k(y + h) cos &#}, 



neglecting A; 2 (7 2 . Since the equation to the stream-line -fy = is 



y = C sinh M cos kx (4), 



approximately, we have, along this line, 



- = const. + (Arc 2 coth kh g) y. 

 * Lord Kayleigh, I. c. ante p. 279. 



