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National Advisory Committee for Aeronautics, Report - Non-Lifting Wing Body Combinations with Certain Geometric Restraints Having Minimum Wave Drag at Low Supersonic Speeds

Several variational problems involving optimum wing and
body combinations having minimum wave drags for different
kinds of geometrical restraints are. analyzed. Particular
attention is paid to the eject on the wave drag of shortening the
fuselage and, for slender aa-ially symmetric bodies, the eject
of firing the fuselage diameter at several points or even of firing
whole portions of its shape.

Recently several authors have used linearized theory to
study the wave drag of wing-body combinations traveling
at supersonic speeds (see, e. g., refs. 1 to 5). These studies
have clearly demonstrated the importance of finding the
wave drag of a whole airplane rather than the separate
wave drags of its various parts (Wings, fuselages, etc.),
since the magnitude of the interference terms can pre-
dominate. In efiect, this means that various optimization
problems for bodies—such as the problem of finding the body
shape having a minimum wave drag for a given volume—
should be re-examined when interfering Wings or other bodies
are in the same flow field. In many cases the solution to
the new problem difi’ers from the body-alone problem only
in interpretation.

The purpose of this report is to study minimum wave—drag
combinations which satisfy a few of the many possible
geometric restraints pertinent to the interests of airplane
designers. An attempt has been made to analyze the
various problems in a unified manner so that extensions to
other kinds of restraints can be deduced.

Many of the discussions and derivations contained in the
following are carried out on the assumption that the reader
is familiar with the concepts presented in reference 4 which
should be considered as a first part to this report. In
particular, an acquaintance with the solutions to the wave
equation referred to as “multipoles” is assumed, together
with Hayes’ invariance principle and the consequent multi—
pole distributions equivalent to a wing in the sense that
both induce the same momentum spectrum at infinity.