Line dipole with axis parallel to walls: 



C r n(z-ih) 7T(2+ib-ia)\ 



■■ — I coth + coth 



2 1 2a 2a J 



Line dipole perpendicular to walls: 



iC r n(z-ib) 



«> = — coth - coth 



2 L 2a 



7T(s + ib-ia) '] 

 2a J 



[49c] 



[49d] 



The walls are assumed to be at y = i a/2 as before, where a/2 > |6|. If no walls are present, 

 there are two rows of singularities displaced a distance 2b relatively to each other, the spac- 

 ing in each row being 2a. A, T, and C are real; the volume emitted per unit length from the 

 source is 2nA, the circulation around the vortex is F, the line-dipole moment is aC/n. 



Expressions for cf), ijj, u, and v are easily constructed by substituting first y - b, then 

 y + b - a, (or y, and 2a for a, in formulas given in Sections 44, 45, 46, or 48, and combining 

 the two terms thus obtained. For the line vortex the formulas for the conjugate flow of Sec- 

 tion 44, not those of Section 45, are to be used, with A - r/2ff; the corresponding complex 

 potential is u; = iAln sinh (nz/a). Similarly, for the fourth case, the conjugate flow of Sec- 

 tion 46 reversed in sign is to be used, with a complex potential iB coth {nz/a); B is to be 

 replaced by C/2. (Reference: Jaffe'^'', Caldonazzo^^). 



50. TWO LINE DIPOLES IN OPPOSITION; 

 DIPOLE AND A WALL 



w = fj. I— j , ft and c real and c > 0, 



\ z -c z + C' 



/ cos (d^-a) cos {0^+a)\ /sin(0j-a) sin(a2+a)\ 



[50a] 



[50b,c] 



where the significance of f,, r,, ft, 6L is adequately shown in Figure 77. 



Figure 77 — A line dipole at (c,0) and 

 its image in a wall along the y-axis. 



116 



