III-115 



In the analytical evaluation of I, y was taken to be zero. Careful numerical 

 evaluation^"^' of I shows that for large a (a > > 1), y can have an appreciable 

 effect on the value of I . Table III-4 shows the value of I obtained by numerical 

 methods, and the analytical approximation for y = over a broad range of a. 

 The generally close agreement illustrates the accuracy of the analytical approxi- 

 mations . 



TABLE III-4 



COMPARISON OF NUMERICAL AND ANALYTICAL INTEGRATION OF I 



AT Y 







I 



Numerical 



I 



Analytical 

 Approximation 



0.1 



5 



1.0 



2 



5.0 



10.0 



50.0 



100.0 



0.0002 

 0.302 

 2.01 

 10.43 

 76.73 

 321.54 

 8395 

 33742 



-0.152 

 -0.087 

 1.52 

 9.8 

 75.2 

 319.40 

 8357 

 33614 



Figure III -42 shows the effect of a nonzero value of v in the evaluation 

 of I. Let I (a) denote the value of I for Y = 0. Then, for nonzero Y , 1(a) can 

 be given by 



1(a) = 6(a, Y)ya) 



(111-161) 



For certain Y , the values of o(a, y) are plotted in Figure III-42. As an example 

 of the effect of a nonzero value of Y . suppose y = 0.2. This corresponds to an 

 angle of incidence of 11.5° . At a = 10 (typical sea conditions), the true value of 

 I is 483, which is 50% more than the value obtained assuming y = when 

 evaluating the integral . - 



(2) Unpublished work by R. F. Meyer and L. Lawrence. 



Arthur m.llittlcJnir. 



S-7001-0307 



