I am aware what the area rule states - my question is what is defined as "as smooth as possible area distribution" - Is there a formula for the curve?
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Since Drag is proportional to S'', the key is to derive a shape that has a minimum S''.
The result is the 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 area is given by:
$$ S(x) = \frac {16V}{3L\pi}[4x(1-x)]^{3/2} = \pi R_\text{max}^2[4x(1-x)]^{3/2}$$
And the radius is given by:
$$ r(x) = R_\text{max}[4x(1-x)]^{3/4} $$
