Bodies of Minimum Wave Drag
Summary
Drag at supersonic speeds include the drag components at subsonic flow conditions (skin friction drag and induced drag) and a wave drag, characteristic of supersonic flows only. This drag component is in fact due to the energy radiated by the aerodynamic system through the shock waves.
The wave drag is on turns divided in (wave) drag due to volume and (wave) drag due to lift. It can be proved that the wave drag is proportional to the volume and inversely proportional to the fourth power of the body length.
Two bodies are of particular theoretical (but also practical) interest: these are the Sears-Haack body (Sears, Haack; 1947), and the von Kármán ogive. An elegant derivation of the drag of these bodies is available in Ashley-Landahl (1965).
Sears-Haack Body
The Sears-Haack body is a slender body of revolution pointed at both ends, that corresponds to the minimum wave drag (for given volume and length) of a linear distribution of sources at supersonic speeds.
von Kármán's Ogive
The von Kármán ogive is a slender body of revolution pointed at one end
only (ogive). This is also a body of minimum wave drag. Fig. 1 shows the comparative
thickness distribution of the Sears-Haack body and the von Kármán ogive
(the x axis has two different scales).
The wave drag of the Sears-Haack body scales with 
(V=volume, L=length). Therefore, for given
volume it is convenient to have very long bodies. Doubling the length L at
constant V will reduce the wave drag by a factor 8.
Related Material
Selected References
- Miele A (editor). Theory of Optimum Aerodynamic Shapes, Academic Press, 1965.
- Lomax H, Heaslet MA. Recent Developments in the Theory of
Wing-Body Wave Drag, J. Aero. Sci., Vol. 23, No. 12, pages 1061-1074,
1956.
- Ashley H, Landahl M. Aerodynamics of Wings and Bodies, Addison-Wesley Publ. Company, Reading, Mass. 1965.