Recent Progress in the Calculation of Potential Flows 



Fig. 15 - The approximation treatment 



cosine function that must be approximated by a polynomial. The step length 

 must be chosen so that d is kept reasonably small. 



Now by means of Eq. (23), Eq. (19) can be written as 



f(x) !'" (Xx+S) dx . i- ' --■ (24) 



Next, we expand the trigonometric expression and designate the sine form of in- 

 tegral by Sn and the cosine form by C^, where the index n corresponds to that 

 specifying the range of integration. We obtain 



f(x) [sin \x cos S + cos \x sin S] dx 



and a similar form for C^. These expressions can be written as 



[G(x) sin \x + H(x) cos kx] dx , 



and 



pn h 



'^- nh 



[G(x) cos kx - H(x) sin X.x] dx 



(25) 



(26a) 



(26b) 



where G(x) = f(x) cos o(x), andH(x) = f(x) sin &(x). 



If the variation of h is such that d is always less than about one radian, and if f 

 is a not too rapidly varying function, then G and H are both sufficiently smooth to 

 be approximated by low-order polynomials. It should be emphasized that it is 



353 



