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 (/&amp;gt;, -\/r be the velocity- and stream-functions when the 

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



(&amp;lt;/&amp;gt; -I- i^)/c = - (x + iy} + me ik( *+ iy) + i/3e- ik(x+iy} , 



whence &amp;lt;/c = - x - (oLe~ ky - /3e ky ) sin kx,\ , . 



^/ c = -y + (ae~ ky + /3e ky ) cos kx } 



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 ^r = 0. Its form is then given by (1), viz. to 

 a first approximation we have 



y = (a + /3) cos kx (2), 



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

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

 requires 



Pe- kh =0. 

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



(j)/c = x + C cosh k (y + h) sin kx, ) , , 



Tfr/c = y + C sinh A; (y + A) cos &# j 



The formula for the pressure is 



= const. gy i- \ { -2- I + ( -^~ 

 p \\CUB / \ay 



C 2 



= const. gy -= {1 2kC cosh k(y + h) cos &#}, 



neglecting A: 2 2 . Since the equation to the stream-line ty = is 



y = (7 sinh kh cos &# (4), 



approximately, we have, along this line, 



- = const. + (kc z coth M a) y. 

 P 



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



