Malavard 



the given boundary value problem, can be considered as the sum of elementary 

 potentials induced in the field by a convenient source distribution. We can then 

 write 



f q(Xi,yi,z.)(n^ + nj da , 



where q is the singularity density of the simple layer potential at points 

 (x.,y.,z. ) of the surface S, dy is an elemental surface around this point, and 

 fis and fir are the singular and regular parts of the potential of an immersed 

 source of unit strength. In the two-dimensional case, these two parts are given 

 by the following classical expressions 



n, = Re _I-[iog(^-^.) + Vo^a-ii)] 



27T 



Rei 



00 



dK - 1776 



K - K„ 



where i = x+ iz, ^. = xj + iz., and K^ = g/v 



The 4> function is also in two parts: s6^ and 4>j. 



where <t>^ corresponds to the singular part of fi^, i.e., 



I qn.do- . 



Its value is the same on both sides of 2, while its normal derivatives are dis- 

 continuities. The difference between its normal derivatives {'dcpl/^n - 30~/3n) 

 represents the source flow q. In the present case the superscripts + and - 

 correspond to the external and internal domains defined by 2 . 



The potential function ^^ must then satisfy the following boundary conditions 



(a) on z = , - 



3n 



(b) for |x| ^ CO , grad cp^ = 



(16) 



30; B0; 



(c) on X, ^— - - — j. q(x.,y.,z.) , 



with the external normal derivative given by 



408 



