A Geometric Problem Related to the Optimum Distribution of Lift on a Planar Wing in Supersonic Flow

Abstract

The problem studied may be regarded as a problem of geometry. Its simplest form (loosely stated) is then as follows: A mountain rises up from the x-y plane. Determine the exact shape of the mountain knowing only the cross-sectional area of every possible cut which can be made through the mountain with a vertical plane. In a more complicated version of the problem, the given information might be restricted to the cross-sectional area of every cut which can be made by a vertical plane inclined less than 45° to the y-axis. This latter case has direct applications to certain minimum drag problems in supersonic flow. The shape of the mountain corresponds to the (unknown) shape of the optimum lift distribution on a planar wing. The cross-sectional area of a cut is the integrated value of the lift along a straight line crossing the wing plan form. For a restricted range of line inclinations, these optimum integrated lift values can sometimes be determined directly. Here it is assumed that they are given. The problem in its simplest form was originally solved by Radon, who found solutions for a large class of such problems. The derivation presented here may perhaps be more readily understood.