The lift and moment on a flat plate 191 
and with a = b = 3d, we have 
M/M, = 1c) (1+3sin?@) 
24 
m* (2p\4 ie Wy 
“rm | (GEAR TEPO SSRN G) sh c000 (68) 
For numerical comparison with previous sections, we take the same 
numerical case: 
6 = 10°; 2p/d = 0-2 
a/d 0-3 0-4 0-5 0-6 0-7 
M/M, 0:938 0-965 0-982 0-998 1-011 
PLANE BOUNDARY AND FREE SURFACE 
10. Although the problem is not, perhaps, of practical interest, we may 
note that the same method can be extended to the case when one boundary, 
say the lower, is a rigid plane while the other, upper, boundary is a free 
surface; we note, again, that for a free surface the boundary condition is 
taken here in an approximate form. 
Considering, as in § 2, the image systems formed by successive reflexions 
in the two surfaces, we see that these infinite series of images now consist 
of terms of alternate signs; summing these series we obtain 
dw ie gntin! A,, -} F*¥(k) e—tkz—2Ka a8 F*( Be, k) Cat oy 
GB e etl 4 1 + e-2«4 us 
o {F(- Klemens pare F(k) ES e—2Kd 
+f peated dx. (69) 
It may be verified directly that (69) satisfies the boundary conditions 
ig =c; 2=2+14, 
dz 
(70) 
1@ =0; z=x—1b. 
The expressions for the lift and the moment are 
Kk) FA(c)e-2t + F(—n) F(— nye 
Y=pcl'-2 = [pea dk, (71) 
kK) F*(x) e-*¢ + F'(k) F*(—k) e240 
{F'(k — F’(—k) F(k)}e-*4¢ 
: oe eee ate dij] yen (22) 
452 
