= G(y/2 +(x + Z)i/2). By substituting Equation 4-19 into the periodic boundary condition
(Equation 4-17), A can be simply determined as A = 1/(2)r) .
The complex function G(;) can be expressed as an integral along the channel boundary
based on Cauchy integral formula,103
G(;) = I -' x (x)dx + I -G P 8'(x)dx
2i (x)- 2i 8(x)- (4-20)
where G9(x) = G(o(x)), GP(x) = G(fl(x)), and the complex functions ((x) = (h + ix)/2 and
/f(x) = (-a cosx + ix)/2 are the profiles of top and bottom walls in the complex plane.
Substituting the complex function q!(x, y) given by Equation 4-19 into the boundary
condition in Equation 4-18, and employing Cauchy-Riemann equations,
G(() =i G(;)
9x Sy (4-21)
We have
d
-d-[G'(x)]= 0
dx (4-22)
at the top flat wall and
1 d
-a sin(x) + 2 9[G (x)] = 0
2i dx (4-23)
at the bottom sinusoidal wall. In order to determine uniquely the complex function G(4),
additional equations are needed to complete Equations 4-22 and 4-23. We define a series of
analytic complex functions
T(;) = exp(2n<;), (4-24)
where n e Z. Note that ],(4) G(;) is an analytic function, since (4O) and G(;) are both
analytic. Based on Cauchy integral theorem, the integral of an analytic complex function on a
closed contour (path ABCDA in Figure 4-2b) is zero,