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Since a radially symmetric body is needed to generate the conical shock, these "'''conical waveriders'''" generally start out with a pointed nose cone on a cylindrical body, and smoothly turn into delta shape at the rear. Unlike the caret wing, the conical designs smoothly curve their wings, from horizontal near the center, to highly drooped where they meet the shock. Like the caret wing, conical waveriders have to be designed to operate at a specific speed to properly attach the shock wave to the wing's leading edge, but unlike them the entire body shape can vary dramatically at different speeds, and sometimes have wingtips that curve upward to attach to the shockwave.{{cn|date=November 2011}}
Conical waveriders have higher performance than carets, but generally require extremely long body shapes that are impractical for real-world aircraft. Further development of the conical sections, adding canopies and fuselage areas, led to the "'''osculating cones waverider'''", which develops several conical shock waves at different points on the body, blending them to produce a single shaped shock. The expansion to a wider range of compression surface flows allowed the design of waveriders with control of volume,<ref name="Jones_Lifting_Configs"/> upper surface shape, engine integration and centre of pressure position. Performance improvements and off-design analysis continued until 1970.<ref>Pike, J. [http://www.jackpike.co.uk/ConeWing340.html "Experimental Results from Three Cone-Flow Waveriders"]. Agard Conference Proceedings 30, Hypersonic Boundary Layers and Flow Fields, [[Royal Aeronautical Society]], London, Ref. 12, p. 20, 1-3 May 1968.</ref><ref>Pike, J. [http://www.jackpike.co.uk/OffCaret.html "The Pressure on Flat and Anhedral Delta Wings with Attached Shock Waves"]. The Aeronautical Quarterly, Vol XXIII, Part 4, Nov. 1972.</ref>
During this period at least one waverider was tested at the [[Woomera Test Facility|Woomera Rocket Range]], mounted on the nose of an air-launched [[Blue Steel missile]], and a number of airframes were tested in the wind tunnel at NASA's [[Ames Research Center]]. However, during the 1970s most work in hypersonics disappeared, and the waverider along with it.{{cn|date=November 2011}}
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