Ogilvie 



So now we have z as a function of t, , as well as F and w as 

 functions of t,. 



There are three parameters in this solution, a, b, and c, 

 none of which has been determined yet. By letting \t,\ -* oo , Green 

 came to the conclusion that the flow far away is a uniform stream 

 as required only if: 



c = - cos O! and V(i - c ) = sin a. (5-2) 



(Both statements are necessary to avoid an ambiguity in sign.) Also, 

 one can use the z(t,) formula to evaluate z at the leading edge of 

 the plate: 



-ia 

 z(-l) = - ie . 



(Compare Figs. (5-1) and (5-4).) This provides a relationship 

 among a, b, and c. Buc there are no more conditions to be found 

 unless we introduce more information about the physical problem. 

 For example, we could use the solution with unspecified values of 

 a and b, and work out the formula for lift on the plate. (Milne- 

 Thomson gives the formula.) If then we fix the value of lift, we 

 have another condition on a and b. However, this is rather a back- 

 wards way of going at the problem. We are most likely to want to 

 solve the entire problem just to find the lift and other interesting 

 physical quantities , and so we have not gained much if we must 

 assume the value of the lift as a given datum. 



There is another anomaly in this result: The value of h 

 (See Fig. (5-1)) has not been used in any way. In the formula for 

 z(C,), let C, = I , with |C | very large. Then every value of z com- 

 puted in this way gives a point on the free surface far away from the 

 planing surface. With a considerable amount of tedious algebra, 

 one can eliminate ^ and express y as a function of x (at least 

 asymptotically, as |x| -* oo). The first term is the most inter- 

 esting: 



aV(l - C ) r , I I J. 4. 4.1 



v ~ TT — ; — r- log X T constant! . 



y Tr(b + c) L 5 I I J 



Thus, far away from the planing surface, the free surface apparently 

 drops off logarithmically to - oo. The slope of the free surface 

 approaches zero («= l/|x|) and so there is no violation of our assump- 

 tion that the flow at infinity is simply a streaming motion parallel to 

 the X axis. But obviously the assumption that the trailing edge was 

 located at a height h below the free-surface level at infinity was 

 quite meaningless, and it cannot be enforced in the solution. 



780 



