 Sears-Haack Body
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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, $D_{\text{wave}}\sim [S''(x)]^{2}$ (see full expression below), so for low wave drag it is necessary that $S(x)$ 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.

## Useful formulas

The cross-sectional area of a Sears-Haack body is

$S(x)={\frac {16V}{3L\pi }}[4x(1-x)]^{3/2}=\pi R_{\text{max}}^{2}[4x(1-x)]^{3/2},$ its volume is

$V={\frac {3\pi ^{2}}{16}}R_{\text{max}}^{2}L,$ $r(x)=R_{\text{max}}[4x(1-x)]^{3/4},$ the derivative (slope) is

$r'(x)=3R_{\text{max}}[4x(1-x)]^{-1/4}(1-2x),$ the second derivative is

$r''(x)=-3R_{\text{max}}\{[4x(1-x)]^{-5/4}(1-2x)^{2}+2[4x(1-x)]^{-1/4}\},$ where:

• x is the ratio of the distance from the nose to the whole body length (this is always between 0 and 1),
• r is the local radius,
• $R_{\text{max}}$ is the radius at its maximum (occurs at center of the shape),
• V is the volume,
• L is the length.

From Slender-body theory[further explanation needed], it follows that:

$D_{\text{wave}}=-{\frac {1}{4\pi }}\rho U^{2}\int _{0}^{\ell }\int _{0}^{\ell }S''(x_{1})S''(x_{2})\ln |x_{1}-x_{2}|\mathrm {d} x_{1}\mathrm {d} x_{2},$ alternatively:

$D_{\text{wave}}=-{\frac {1}{2\pi }}\rho U^{2}\int _{0}^{\ell }S''(x)\mathrm {d} x\int _{0}^{x}S''(x_{1})\ln(x-x_{1})\mathrm {d} x_{1}.$ These formulae may be combined to get the following:

$D_{\text{wave}}={\frac {64V^{2}}{\pi L^{4}}}\rho U^{2}={\frac {9\pi ^{3}R_{max}^{4}}{4L^{2}}}\rho U^{2},$ $C_{D_{\text{wave}}}={\frac {24V}{L^{3}}}={\frac {9\pi ^{2}R_{max}^{2}}{2L^{2}}},$ where:

• $D_{\text{wave}}$ is the wave drag,
• $\rho$ is the density of the fluid,
• U is the velocity.

## Generalization by R. T. Jones

The Sears-Haack body shape derivation is correct only in the limit of a slender body. The theory has been generalized to slender but non-axisymmetric shapes by Robert T. Jones in NACA Report 1284. In this extension, the area $S(x)$ is defined on the Mach cone whose apex is at location $x$ , rather than on the $x={\text{constant}}$ plane as assumed by Sears and Haack. Hence, Jones's theory makes it applicable to more complex shapes like entire supersonic aircraft.

## Area rule

A superficially related concept is the Whitcomb area rule, which states that wave drag due to volume in transonic flow depends primarily on the distribution of total cross-sectional area, and for low wave drag this distribution must be smooth. A common misconception is that the Sears-Haack body has the ideal area distribution according to the area rule, but this is not correct. The Prandtl-Glauert equation, which is the starting point in the Sears-Haack body shape derivation, is not valid in transonic flow, which is where the area rule applies.