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National Advisory Committee for Aeronautics, Report - A Vector Study of Linearized Supersonic Flow Applications to Nonplanar Problems

A vector study of the partial-differential equation of steady
linearized supersonic flow is presented. General expressions,
which relate the velocity potential in the stream to the conditions
on the disturbing surfaces, are derived. In connection with
these general expressions the concept of the finite part of ,an
integral is discussed.

A discussion of problems dealing with planar bodies is given
and the conditions for the solution to be unique are investigated.
Problems concerning nonplanar systems are investigated, and .
methods are derived for the solution of some simple nonplanar
bodies. The surface pressure distribution and the damping in
roll are found for rolling tails consisting of four, sit, and eight
rectangular fins for the Mach number range where the region of
interference between adjacent fins does not affect the fin tips.

In the presentation of the theory of the flow of an idealized
incompressible fluid, vector methods can be used to reduce
greatly the mathematical manipulations involved. The
study of steady linearized supersonic flow may also be aided
by the use of vector methods. Two types of approaches,
however, can be used. Perhaps the more obvious is to make
use of common vector methods as was done in reference 1.
The other vector method, which was introduced by Robinson
in reference 2 and is used in this report, appears to be more
suited to the study of the linearized partial-differential equa-
tion of steady supersonic flow. This method allows a deriva-
tion of a hyperbolic scalar potential and a hyperbolic vector
potential along hues analogous to the derivation sometimes
used (ref. 3, ch. VIII) in dealing with common scalar and
vector potentials.

The present report presents a vector derivation of many
general results which have been found by various methods
and are given in the published literature on the linearized
partial-differential equation of supersonic flow and also
presents some results which are not found in the literature.
The general results of Hadamard (ref. 4, p. 207), Puckett
(ref. 5), and Heaslet and Lomax (ref. 6) are found as special
cases of a general expression for a scalar potential, and the
results found by Robinson (ref. 2) are obtained by the use
of a vector potential. The derivation of the scalar potential
doubtlessly helps to clarify the concept of the finite partof
an integral.