The Sears–Haack body is the aerodynamic body shape with the lowest theoretical wave drag

Wave drag

In aeronautics, wave drag is a component of the drag on aircraft, blade tips and projectiles moving at transonic and supersonic speeds, due to the presence of shock waves. Wave drag is independent of viscous effects.- Overview :...

. Aircraft designed to operate at high subsonic or supersonic speeds have their cross-sectional areas designed to match as closely as possible the proportions of Sears-Haack body.

By Whitcomb's area rule

Area rule

The Whitcomb area rule, also called the transonic area rule, is a design technique used to reduce an aircraft's drag at transonic and supersonic speeds, particularly between Mach 0.75 and 1.2....

, the derivative of cross-sectional area gives wave drag. Thus, the Sears–Haack body is pointed at two ends and grows to a maximum and then decreases toward the second point.

Under the area rule, shapes with the same cross-sectional area at each point along their length as this shape have the minimal amount of wave drag, and the overall shape of many aircraft designed with transonic

Transonic

Transonic speed is an aeronautics term referring to the condition of flight in which a range of velocities of airflow exist surrounding and flowing past an air vehicle or an airfoil that are concurrently below, at, and above the speed of sound in the range of Mach 0.8 to 1.2, i.e. 600–900 mph...

 flight considerations have cross-sectional areas that approach this form (despite appearances).

The derivation and shape were published independently by two separate researchers: Wolfgang Haack

Wolfgang Haack

Wolfgang Siegfried Haack was a German mathematician and aerodynamicist. He and William Sears independently discovered the Sears–Haack body in 1947.-Life:...

 in 1941 and later by William Sears

William R. Sears

William Rees Sears was a notable aeronautical engineer and educator.-Career:William R. Sears was born in Minneapolis, Minnesota, the son of William and Gertrude Sears. He earned his BS degree from the University of Minnesota in 1934...

 in 1947.

Useful Formulas

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 2nd derivative is:

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 radius

is the radius at its maximum (occurs at center of the shape)

V is the volume

L is the length

is the density

U is the velocity

From Slender-body theory

Slender-body theory

In fluid dynamics and electrostatics, slender-body theory is a methodology that can be used to take advantage of the slenderness of a body to obtain an approximation to a field surrounding it and/or the net effect of the field on the body...

:

alternatively:


These formulas may be combined to get the following:

External links

• Haack Minimum Drag Rifle Bullet Site down - http://web.archive.org/web/*/http://www.lima-wiederladetechnik.de/Englisch/Haack_minimum_drag_bullet.htm
• Geschoßformen kleinsten Wellenwiderstandes by W. Haack, Bericht 139 der Lilienthal-Gesellschaft (1941)