The performance of a wing in subsonic flow can be calculated by transforming the geometry to a corresponding "incompressible flow wing". The transformed wing has lower aspect ratio. The method of Slender Wing theory allows us to calculate the aerodynamics of low aspect ratio wings. This in turn leads to Slender Body theory.

Going to supersonic flow, Von Karman's method of locating a source distribution along the axis leads to calculations of body shape for low wave drag. Combining this with concepts from slender wing/body theory gives results for wing-body combinations, and two simple results for the shapes that promise the lowest wave drag. The Karman Ogive gives the front shape (i.e., the distribution of cross sectional area along the axis) for the minimum-drag body with a truncated base of given area. The Sears-Haack body gives the cross-sectional area distribution along the axis, for the minimum-drag body, closed at both ends, for a given volume. Looking at the concepts of slender wing-body theory, one realizes that these area distributions are what matter - that the shapes need not be axi-symmetric. So these concepts provide ways to optimize the shape of an entire aircraft, for minimum wave drag.

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