65 



waves at order J is: 



d>{x,y,z,t) = UrX + Uyy + 



'"^^ "" cosh A'[](/2 + z) 



.E E, ■- E A>^^^::iif^s..„ (4.0, 



[] = («1,«2,--- ,«iv) 



o-[] = {iiSi + i2S2 -{ 1- inSn) 



Sn = {kx,nX + ky^nV " UjJ + On) 



where Ux and Uy are the components of the depth uniform Eulerian current, k^.n and 

 ky^n are the vector wave number components of the nth wave in the x and y directions. 

 There is a separate summation for each wave considered. 



As the notation in Eq. 4.10 is quite confusing, examples with two and three waves 

 are given below. With two waves (A'^ = 2): 



0(x, J/, z, t) = UxX + Uyy + 



, E ■^■-» cosh A^., A ' -■M..g,+..g.) (4.11) 



Sn = {kx.nX + ky,ny - UJnt + On) H = [l, 2] 



•''^•21,22 ^^ y \'^lf^x,ii i i2'^x,i2) 'T {ilKy,ii -r 'i2^y,i2) 



And with three waves (A'^ = -3): 



(f>{x, y, z,t) = UxX + Uyy + 

 J J-\i,\ ^-In|-h2l , ^ ,, , , 



!l=-J ,2 = -J+Kl| i3 = -^+hl| + N2| ''^'^ 



(4.12) 



