- 42 - 



The analysis procedure follows from Longuet-Higgins , 

 Cartwright, and Smith (1963). The water surface displacement is 

 expressed in terms of the complex directional amplitude spectrum, 

 F(6,B), as: 



J-co Jq 



where 6 = frequency 



Q = direction (assuming Q measured counter-clockwise 



from the positive x-axis) 

 kx = kcosE), wave-number in the x-direction 

 ky = ksinB, wave-number in the y-direction, 

 and \l i = -I. 



The water surface slope and curvature follow from (4.1) as: 



~ (x,v,t) = f f ""-ik no^ fl Ffa.fl^.^^^t-k x-k y),- . (A. 2) 



-^ (x,y,t) = 1 I -ik cos e F(a,e)e"'"'- x^'V^Se do 

 



r. f" /•2TT 



-^ (x,y,t) = - ik sin 9 F (a, 6) e"" ^'^^"'^x''"^^ 96 3ct (4.3) 



9x^ J-oc Jo 



