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National Advisory Committee for Aeronautics, Technical Notes - On the Boattail Bodies of Revolution Having Minimum Wave Drag

The problem of determining the shape of slender boattail bodies of
revolution for minimum wave drag has been reexamined. It was found that
minimum solutions for Ward's slender»body drag equation can exist only
for the restricted class of bodies for which the rate of change of cross—
sectional area at the base is zero. In order to eliminate this restric-
tion, certain higher order terms must be retained in the drag equation
and isoperimetric relations. The minimum problem for the isoperimetric
conditions of given length, volume, and base area is treated as an example.
According to ward's drag equation, the resulting body shapes have slightly
less drag than those determined by previous investigators.

An approximate expression for the wave drag of slender bodies of
revolution having zero rate of change of cross-sectional area at the
base was first given by Vbn Karman (ref. 1). By using this expression,
together with the calculus of variations, several investigators (refs. 1
to 5) have determined minimumswave—drag bodies for various isoperimetric
conditions. Later, Ward (ref. h) derived the slender-body approximation
for the drag of bodies with a nonzero rate of change of cross—sectional
area at the base.

In reference 5, Adams considered several minimumawave—drag problems
on the basis of Ward's equation. In each case he concluded that the
minimum-drag body had zero slope at the base. This conclusion implied
that the minimum shapes for ward's equation are the same as those for
Vbn Karman's. Recently, Parker (ref. 6) presented a different expression
for the wave drag of slender bodies and showed that the optimum body
having given length and base area has a finite slope at the base. Clearly,
this result is not in agreement with that obtained by Adams.

In the present paper, the problem of determining minimum-drag boat-
tail bodies of revolution on the basis of linear theory is reexamined
with particular emphasis on the choice of drag equation, isoperimetric
relations, and method of calculating the body shape. The minimum problem
for the isoperimetric conditions of given length, volume, and base area
is treated as an example.