III-76 



2tt 

 ITie wavelength of the surface is 5 = — , hence the direction cosines , a^, 



are given by 



a = a + -^ 

 n k 



(III- 68) 



Since this wave is travelling upward and the time dependence is given by e 

 we must have y - 0. The boundary condition which must be satisfied is 



•iujt 



p. (x,S(x)) + p_(x,S(x)) = 

 mc sc 



(III- 69) 



An expression giving any particular coefficient Aj. in terms of the others can be 

 obtained by first factoring out the term e^^(°'^"'"^tz) from the sum p^^^^ + Pg^,. 

 Then, on the surface z = h cos px, the boundary condition gives 



OD 



^ik(Y - Yj.)h cos px _^ V^ ^ ^ 



npx+k (y - Y ) h cos px 



- (III-70) 



Using the expansion 



i 8 cos px V^ .m impx ^ ,., 

 e = / 1 e ^ J (8) 



(III-71) 



it can be shown that 



\ - - Jo[(^ - y)^^'Y> '" \ Vt[<% - \' H <"'-"> 



n^^-t 



If kh is so small that all the terms involving a Bessel function of order greater 

 than zero may be ignored, we obtain the so-called Rayleigh approximation 



\ = - (-i)^Jo [<\-^)^^] 



(III- 73) 



artbur Il.littlcJnir. 



S-7001-0307 



