Sears–Haack body
The Sears–Haack body is the shape with the lowest theoretical wave drag in supersonic flow, for a given body length and given volume. The mathematical derivation assumes small-disturbance (linearized) supersonic flow, which is governed by the Prandtl-Glauert equation. The derivation and shape were published independently by two separate researchers: Wolfgang Haack in 1941 and later by William Sears in 1947.
The theory indicates that the wave drag scales as the square of the second derivative of the area distribution, (see full expression below), so for low wave drag it's necessary that be smooth. Thus, the Sears–Haack body is pointed at each end and grows smoothly to a maximum and then decreases smoothly toward the second point.
![D_{{\text{wave}}}\sim [S''(x)]^{2}](https://wikimedia.org/api/rest_v1/media/math/render/svg/71ea7f9638bdd52c3701c306402fada83c07b8db)
The cross-sectional area of a Sears–Haack Body is
the volume of a Sears–Haack Body is
the radius of a Sears–Haack Body is
the derivative (slope) is
the second derivative is
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