Non-Linear Shtp Wave Theory 
obtained subsequently from (8) and (12) with x 
II .2 - Solution for two Sources. 
oF 
We derive now i!" and £5 , the potentials related to the 
flow past two sources of strengh ¢€; ,e, at z) =x, - ih, z, = xy - ih 
respectively. The advantage of the body discretization is the easy 
extension of the solution for n=2 toany other n. The first order 
solution is given (Wehausen and Laitone 1965) by 
Meet Bast hats 
fi agbs ae In (z z;) aay In (z z,) = In (z - = - 2ih) + 
€k : € j z- Zz; - 2ih 
+ 3, (z - z, - 2ih) a )- 
F 
€k Z- Z, - 2ih 
SE 2 a) 
F 
The function w({¢) is defined as 
7 it Se 
w(f)=at+iB =e EI (it) =e f(e /r) ad (17) 
where the ) plane is cut along the real negative axis and the integra- 
tion is carried out below the cut, such that (9) is satified. 
The equation of the far downstream wave is obtained from 
the w terms of (16), by the residue in (17), as follows 
Zz 
aiht/ F 
By =A gees PP) Flats : 
x, x. 
k 
A é-cos ——a7 1 4Ccos’——. eos + 
K 5 k 2 ) 2 
F F F 
x x x 
sin —— (x 00) (18) 
SF 
i Bowikinsse - 
The second order potential is now split into two parts 
fee age + £ i where ay ” is the body correction 
with the free surface condition kept in its homogeneous form (7) and 
rn is the free surface correction related to the linearized 
pressure Po jk of (11) in the absence of the body. 
1701 
