The water surface displacement consistent with the assumption of one 

 direction per frequency is 



N-l 

 n(x, y, j) = I F(n) exp (Mnu^t - k^n) x - k y (n) y]} 

 n=o 



(14) 



N-l 



V /i2imj\ 



= I [a(n) - ib(n)] exp ) 



n= V N / 



where uk is the primary analysis frequency (= 2ir/record length = 2ir/T = 

 2iTAf), and 0(n) the direction of wave propagation at frequency o)(n) = nu . 

 The wave number components, k^C 11 ) an d k(n) , are expressed in terms of the 

 wave number, k(n) , and wave direction, 9(n) , as 



k x (n) = k(n) cos 0(n) (15) 



k y (n) = k(n) sin 0(n) (16) 



The cross spectrum, S (n) , of the two measured water surface displacements 

 (or dynamic pressures) is given by 



S 12 (n) = |F(n)| 2 {exp - i [k(n) cos 0(n)(x 2 - x : ) 



(17) 

 + k(n) sin 0(n)(y 2 - y^]} 



Denoting the separation distance and angle as % and g, respectively, the 

 cross spectrum can be expressed as (see Fig. 1) 



S 12 (n) = |F(n)| 2 {cos [k(n) £cos (0(n) - g) ] - i sin [k(n) Jlcos (0(n) - g)]} 

 = cospectrum (n) - i quad-spectrum (n) (18) 



= Ci 2 (n) - iQi 2 (n) 



Thus, from equation (18), the wave direction 0(n) associated with each wave 

 frequency can be expressed as 



( 1 _ [ Q 12 (n) 

 0(n) = g + cos 1 ( tan 1 



k(n) % |_ C !2 (n) 



(19) 



The above relationship has two roots, one of which must be selected based 

 on physical considerations of the most likely direction of wave propagation. 

 In the present case, assuming no wave reflection from the beach, the ambiguity 

 in wave direction is ruled out; for wave sensors nearly parallel to the beach, 

 the minus sign in equation (19) is appropriate. 



11 



