where R'(x) = R(9(x)), R (x) = R(/(x)), Q'(x) = Q(Q(x)), and Q (x) = Q(/(x)).
Substituting Equation 4-33 into Equations 4-15 and 4-16, the boundary conditions at the top flat
wall yield
d d
9[Rl[(x)]+ h d9[Q(x)] = 0
dx dx (4-39)
and
K d d 1
Bh h2 + 3[RR(x)] + [Q(x)]+h + 3[Qx)]= E (x)
4; dx dx 2 (4-40)
And at the bottom periodical wall, we have
B (-a cos(x))2 K (-a cos(x))3 + 291[R (x)] + 2(a cos(x)) [QP(x)] = 0
12;r (4-41)
and
B (-a cos(x)) -(-a cos(x))2 + d- [R (x)] + 9[Q (x)] + (-a cos(x)) d3[Q (x)]
4; dx dx
= 1E (x) + a2 sin2
2 (4-42)
Taking Cauchy integrals for F](4) R(;) and Fn(). Q(;') along a contour ABCDA
shown in Figure 4-2b, the integral in this closed contour is zero according to the properties of an
analytic function. Since the integral along BC cancels the integral along DA, the sum of
integrals along path AB and CD is zero:
exp(2np(x))RI (x)p'(x)dx + exp(2n/3(x))R8 (x)/'(x)dx = 0
(4-43)
and
Sexp(2n((x))Q (x)p'(x)dx + exp(2nf/(x))Q (x)/f'(x)dx= 0
(4-44)
Equations 4-39 to 4-44 form a complete set of equations to solve Ro(x), RP(x), Qo(x),
and Q((x). R(;) and Q(;) are then obtained from equations Equation 4-37 and 4-38, and
subsequently substituted into equation 4-29 to yield the desired Vy(x,y).