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National Advisory Committee for Aeronautics, Report - The Use of Source Sink and Doublet Distributions Extended to the Solution of Boundary Value Problems in Supersonic Flow

A direct analogy is established between the use of source-sink
and doublet distributions in the solution of specific boundary-
ralue problems in subsonic wing theory and the corresponding
problems in supersonic theory. The concept of the “finite part”
of an integral is introduced and used in the calculation of the
improper integrals associated with supersonic doublet distri-
butions. The general equations dereloped are shown to include
sereral previously published results and particular examples
are gice-n. for the loading on rolling and pitching triangular
wings with. supersonic leading edges.

The problem of finding pressure distributions over airfoils
of arbitrary shape and plan form or of finding airfoils which
have arbitrary pressure distributions is one of the most
fundamental problems in aerodynamic theory. At the pres-
ent time the most important and satisfactory approach to
problems of this type is provided by the methods of so-called
thin-airfoil theory. The essential assumptions in this theory
are that the perturbation velocities induced by the airfoil
are small relative to the free-stream velocity and that the
boundary conditions can be specified in a fixed reference
plane.

Under the assumptions of thin—airfoil theory the theoreti-
cal analysis of a problem in wing theory resolves itself into
the task of determining the solution of a second—order linear
partial differential equation with prescribed boundary
conditions. In the case of purely subsonic flow, Laplace‘s
equation in three dimensions must be considered, while in
purely supersonic flow the differential equation which arises
is algebraically equivalent to the two—dimensional wave
equation of mathematical physics. The classical solutions
of these two equations have been developed along two dis-
tinct lines: First, by use of orthogonal functions which can
be derived in terms of the boundary conditions, and alter-
natively by means of Green’s theorem which in turn utilizes
a known particular solution of the partial differential equa-
tion together with the given boundary conditions.