Joosen 



•*i(-ij""-i-r 



X e cos {q2( ''^i" "^^^ ^-'-" J 



cos (9 \/l - 47 cos 





X e e 4 1 ^^g {q4(T]i-f)q sin 61} — . W-'i) 



cos 6 \J 1 - 4'y cos 6 



In this derivation the bow region BB^ and the stern region SS^ are assumed to 

 be of order e and therefore ^^- i/e is of order unity if ^j is in the neighbour- 

 hood of bow or stern. 



If these regions are of order xmity the expression (3.2) becomes zero in the 

 limiting case. This fact follows from the application of the method of stationary 

 phase, which will be discussed somewhat later in this section. 



From these results it must be concluded that the series expansion is not 

 viniformly convergent in the neighbourhood of |^j | = 1. In the following it is 

 assumed that the ship is sharply pointed. 



With this restriction the final results for cpj is only produced by the last 

 two terms of (3.1). These integrals are of the form: 



1 tt/ 2 i 



3= [ d^ f F(^,Od^ f l^f B(q,5) e^°^^''^de (3.5) 



where 



D(^,0) = qqk{(^i-a cos 5 ± e(77j- f) sin ^}. (3.6) 



For this integral the method of stationary phase can be applied in the ^, 

 plane, e' ^ being the large parameter. The general theory of the method of sta- 

 tionary phase applied on multiple integrals can be fovind, e.g., in [12]. 



Here only the first order term in e will be derived using the above men- 

 tioned theory for the case of a double integral with the point of stationary phase 

 Inside the integration domain. 



Let this point be denoted by (^g, 0^^); then ^^(^^,0^) = 0, ^0(^^,0^) = 0. 



In the neighbourhood of ( ^^ , ^^ ) , D( ^, 0) can be written as 



174 



