fo contribute to the whole range of w . Hence, to avoid developing quasi -dependent 

 or redundant equation pairs, it is necessary to pick the <t : in such a way that no one 



'^ -interval will contribute the majority of the energy in two successive <" -inter- 

 vals. 



The equations that finally result from a triangular array of dimension 30 and, 

 because of the finite difference approximation, represent the original integral equa- 

 tion plus a slight perturbation term e (w). The corresponding form of (7a) would be 



E (w) + € ('^) = / F (o-,>|/) \d_±_ 

 c. 9 a) 



do- 



(12) 



A measure of the degree of accuracy of the approximation, and hence «(<*'), was 

 obtained in the following way. A known H (o- ) was supplied to {7a) and (8a) and 

 E (cj) directly evaluated to four decimal places by Simpson's rule integration. Using 

 this E (to), the equations 



were solved for H' on an IBM 7094 computer. The ratio H' (^)/H (o") then 

 gives the required accuracy measure as a function of frequency. 



The input H (o") that was used was the latest Pierson-Moskowitz (1963) empirical 

 wave spectrum. 



H(cr)=_V 

 0-5 



exp 



^-3 



i o- W j 



where d^ =8.1 x 10 ^ and d2 = 0.74. The directionality is assumed proportional to 

 cos and normalized as before (K2). For the comparison, wind speeds of 18 and 30 

 knots (9.3 and 15.4 m/sec) were used. The given H {'^ ), estimated H' [^ ) and ratio 

 of H'/H are shown in Figure (11). 



For the 18 knot test case, the agreement between H and H' is quite good with 

 an error of about 1 ,5% near the spectral peak. For the 30 knot case, the error is 

 approximately 3% near the peak but almost 10% for the last (lowest) value of cr . 

 This is due to a combination of rapid changes of H and the grossness of the finite 

 difference approximation to the original integral in the low frequency range. Although 

 an error of 10% is acceptable, in practice no significant energy will be found at these 

 lowest frequencies, and so the accuracy is somewhat better than 10%. 



It should be mentioned in passing that since there is an upper limit (o" max) °" ^^^ 

 o" -range that is being considered, it should be necessary to subtract from all of the 

 E (t*'), the contribution due to o" 2^ *^ max* ^'^'^ would be done by evaluating the 

 integral via Simpson's rule, as before, from o" max ^° °^ equal, say Av . Waves with 

 frequencies greater than 4 Tt contribute essentially nothing to the frequency range 



30 



