# 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.[1]

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's 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:

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

The volume of a Sears–Haack Body is:

$V = \frac {3\pi^2}{16}R_{max}^2 L$

The radius of a Sears–Haack Body is:

$r(x) = R_{max}[4x(1-x)]^{3/4}$

The derivative (slope) is:

$r'(x) = 3R_{max}[4x(1-x)]^{-1/4} (1-2x)$

The second derivative is:

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

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

$R_{max}$ is the radius at its maximum (occurs at center of the shape)

V is the volume

L is the length

$\rho$ is the density of the fluid

U is the velocity

From Slender-body theory:

$D_\text{wave} = - \frac {1}{4 \pi} \rho U^2 \int_0^\ell \int_0^\ell S''(x_1) S''(x_2) \ln |x_1-x_2| \mathrm{d}x_1 \mathrm{d}x_2$

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 formulas may be combined to get the following:

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

## 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 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.

## References

1. ^ Palaniappan, Karthik (2004). Bodies having Minimum Pressure Drag in Supersonic Flow – Investigating Nonlinear Effects (PDF). 22nd Applied Aerodynamics Conference and Exhibit. Antony Jameson. Retrieved 2010-09-16.