where 



T^ = x^ + y'^, T^ = {x-h)^ + y^, r^ = {x+hf- + y^ 



Thus at infinity q -» 2A/t, as in the flow due to a single source of double strength at 0. On the 

 wall a; =0, r = |y| and q = 2A \y\/{h? + y^)\ is a stagnation point. 

 The components of velocity are, from Equation [43b], 



^ /cos 0, cos Q\ /sin 6. sin 0, \ r . , 



u = A[ 1 + A v^ a[ ^ + —^ [43f,g] 



The net force per unit length on the wall due to the Bernoulli term in the pressure is 



4i4^ y^ npi 



-f 1 '^^-I'l m 



dy = ^— - [43h] 



The source can be said to attract the wall. 



[For notation and method; see Section 34; Reference 2, Sections 8.31, 8.41; also 

 Reference 5] 



44. ROW OF EQUAL SOURCES OR VORTICES; SOURCE MIDWAY BETWEEN WALLS 

 OR ON ONE WALL; CONTRACTED CHANNEL 



w = - Aln. sinh Lf, a and A real, [44a] 



a 



1 / 27ra; 2n-y\ ^ ,, 

 qS = --i^ In cosh cos [44b] 



2 \ a a I 



i}, = - A tan"i tan — / tanh — [44c] 



\ a a I 



from hyperbolic formulas in Section 32 and In (re'*) = \n r + i 6. In <;& a constant term is dropped. 



In 0, tan~^ is to be interpreted so as to vary continuously with x and y. Then tan~^ 

 and y vary in the same sense if x is held fixed at a positive value, whereas they vary in op- 

 posite directions if x is kept negative. While x remains positive, tan~^ may be assumed to 

 vanish with y; then, continuity being assumed, either tan~^ and n y/a are both positive angles 

 in the first quadrant, or they are both negative angles in the fourth. The effect of letting x 

 become negative is easily seen if tan( ny/l) is liept finite and not zero. For example, let both 

 tan~^ and n-y/2 be positive and in the first quadrant, with a; = a;^ > 0. Then, if x is decreased 

 to -ajj without change in y, tan~^ increases to n^ - tan~^ [tan (^jy/a) /tanh (TT-ajj/a)]; whereas. 



100 



