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National Advisory Committee for Aeronautics, Report - On the Kernel Function of the Integral Equation Relating the Lift and Downwash Distributions of Oscillating Finite Wings in Subsonic Flow

This report treats the kernel function of an integral equation
that relates a known or prescribed downwash distribution to an
unknown lift distribution for a harmonically oscillating finite
wing in compressible subsonic flow. The kernel function is
reduced to a form that can be accurately evaluated by separating
the kernel function into two parts: a part in which the singular-
ities are isolated and analytically expressed and a nonsingular
part which may be tabulated. The form of the kernel function
for the sonic case (Mach number of I ) is treated separately. In
addition, results for the special cases of Mach number of 0
(incompressible case) and frequency of 0 (steady case) are given.

The derivation of the integral equation which involves this
kernel function, originally performed elsewhere (see, for example,
NAUA Technical Memorandum .979), is reprode as an
appendix. Another appendir gives the reduction of the form of
the kernel function obtained herein for the three-dimensional
case to a known result of Possio for two-dimensional flow. A
third appendix contains some remarks on the evaluation of the
kernel function, and a fourth appendix presents an alternate
form of expression for the kernel function.

The analytical determination of air forces on oscillating
wings in subsonic flow has been a continuing problem for the
past 30 years. Throughout the first and greater part of
this time, efforts were directed mainly toward the determina—
tion of forces on wings in incompressible flow. These efforts
have led to important closed-form solutions for rigid wings
in two—dimensional flow (ref. 1), to solutions in terms of
series of Legendre functions for distorting wings of circular
plan form (refs. 2 and 3), and to many approximate, yet
useful, results for wings of elliptic, rectangular, and tri-
angular plan form (see, for example, refs. 4 to 12).