angle of the Mach cone, also known as Mach angle, and M is the Mach number). In this case the "perfect shape" is biased rearward; therefore, aircraft designed for lower wave drag at supersonic speed usually have wings towards the rear. Sears–Haack body A superficially related concept is the Sears–Haack body, the shape of which allows minimum wave drag for a given length and a given volume. However, the Sears–Haack body shape is derived starting with the Prandtl–Glauert equation which approximately governs small-disturbance subsonic flows, as well as Ackeret Theory, which closely describes supersonic flow. Both methods lose validity for transonic flows where the area rule applies, due to assumptions made in their derivations. So although the Sears–Haack body shape, being smooth, will have favorable wave drag properties according to the area rule, it is not theoretically optimum. History Germany The area rule was discovered by when comparing a swept wing with a w-wing with extreme high wave drag while working on a transonic wind tunnel at Junkers works in Germany between 1943 and 1945. He wrote a description on 17 December 1943, with the title Anordnung von Verdrängungskörpern beim Hochgeschwindigkeitsflug ("Arrangement of Displacement Bodies in High-Speed Flight"); this was used in a patent filed in 1944. The results of this research were presented to a wide circle in March 1944 by Theodor Zobel at the Deutsche Akademie der Luftfahrtforschung (German Academy of Aeronautics Research) in the lecture "Fundamentally new ways to increase performance of high speed aircraft." Subsequent German wartime aircraft design took account of the discovery, evident in slim mid-fuselage of aircraft including the Messerschmitt P.1112, P.1106 and Focke-Wulf 1000x1000x1000 type A long-range bomber, but also apparent in delta wing designs including the Henschel Hs 135. Several other researchers came close to developing a similar theory, notably Dietrich Küchemann who designed a tapered fighter that was dubbed the "Küchemann Coke Bottle" when it was discovered by US forces in 1946. In this case Küchemann arrived at the theory by studying airflow, notably the interference, or local flow streamlines, at the junction between a fuselage and swept wing. The fuselage was contoured, or waisted, to match the flow. The shaping requirement of this "near field" approach would also result from Whitcomb's later "far field" approach to drag reduction using his Sonic area rule. United States Wallace D. Hayes, a pioneer of supersonic flight, developed the transonic area rule in