OgiZvie 



where L is the range of x covered by the lifting surface (the 

 length of L being generally the chord length), and s(x) is the half- 

 span at cross section x. On H (the projection of the wing on the 

 plane y = 0) , the normal velocity component, <^y, is known, either 

 by direct application of the body boundary condition or by matching to 

 a near-field solution, and we obtain the usual integral equation for a 

 lifting surface. 



We shall not be concerned here with the various methods of 

 attempting directly to solve this integral equation, either by analyti- 

 cal or numerical methods. In fact, analytical methods do not exist, 

 so far as I know, except for a few special geometries , such as 

 elliptical planforms. The pair of equations (2-28) and (2-29) forms 

 a remarkable analogy to a standard boundary-value problem in two 

 dimensions which is analyzed thoroughly by Muskhelishvili [ 1953] , 

 One three-dimensional case has been solved analytically by a method 

 that has some similarity to the standard methods for the 2-D prob- 

 lenn; this was done by Kochin [ 1940] . Even his circular-planform 

 wing led to so much difficulty, it seems unlikely that it will be 

 generalized to other planforms. 



Analytical solutions have also been obtained for circular and 

 then elliptic planforms by formulating the problem in terms of an 

 acceleration potential in coordinate systems appropriate to such 

 shapes of figures. This was all done long ago. See Kinner [ 1937] 

 and Krienes [ 1940] . 



There are many nunaerical techniques for obtaining approxi- 

 mate solutions of this problem. However, I ignore these and proceed 

 to analyze a special configuration which can be treated approximately 

 as a limiting case of the general lifting- surface problem. 



2.2. High- Aspect- Ratio Wing 



It is an interesting historical fact that Prandtl's boundary- 

 layer solution really contains the essence of the method of matched 

 asymptotic expansions, but Prandtl failed to observe that the same 

 technique would work in his lifting-line problem. In the boundary- 

 layer problem, he really required the matching of two complementary, 

 asymptotically valid, partial solutions. It was probably Friedrichs 

 [ 1955] who first recognized that the high- aspect- ratio lifting-surface 

 problena could be treated the same way. Van Dyke [ 1964] discusses 

 the derivation of lifting-line theory in some detail from the point of 

 view of matched asymptotic expansions. My presentation is not 

 different from Van Dyke's in any startling ways. There are some 

 differences, partly because I have in mind applications to planing 

 problems eventually, partly because I am not an aeronautical (or 

 aerospace) engineer at heart. 



The conventional approach to solving the problem of a wing of 



696 



