frequency bandwidth can be determined from Equation 40, and wave amplitude 

 C^ can be found by solving Equation 41c to form 



C,^ = (2 Af S,,)^'2 (42) 



or, if Equation 40 is used to define Af , 



[ NAt J 



[2 S.^V/2 



(43) 



The frequency of element k is the (k - l)'^'^ increment of the resolution 

 frequency bandwidth Af in the form 



fk = (k - 1) Af (44) 



or, if Equation 40 is used to define Af , 



'^ = ^^1^ ^^5) 



The value of the independent variable time at the n*^^ time step of both time 

 series x^ and y^^ is 



t„ = (n - 1) At (46) 



Finally, the initial phase is a uniform random deviate in the range from to 

 In radians. This definition conforms to the assumption of a Gaussian random 

 process for natural ocean waves. Notation for random phase of the k*^*^ wave 

 component is 



^,k = 27r U,k[0,l] (47) 



where Uj,i^[0,l] represents a uniform random deviate in the range to 1 , and, 

 in this example, one associated with time series x^^ . A variable of this 

 type can be obtained routinely from a computer-based random number generator. 



33. With the definitions of Equations 43, 45, 46, and 47, the contribu- 

 tion to a time series at the n*^^ time step from the k*^^ frequency component is 



18 



