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National Advisory Committee for Aeronautics, Technical Notes - On the Kernel Function of the Integral Equation Relating Lift and Downwash Distributions of Oscillating Wings in Supersonic Flow

This paper treats the kernel function of the integral equation that
relates a known or prescribed downwash distribution to an unknown lift
distribution for harmonically oscillating wings in supersonic flow. The
treatment is essentially an extension to supersonic flow of the treatment
given in NACA TN 3131 for subsonic flow. For the supersonic case the
kernel function is derived by use of a suitable form of acoustic doublet
potential which employs a cutoff or Heaviside unit function. The kernel
functions are reduced to forms that can be accurately evaluated by con-
sidering the functions in two parts: a part in which the singularities
are isolated and analytically expressed, and a nonsingular part which
can be tabulated.

The kernel is treated for the two-dimensional case, and it is shown
that the two-dimensional kernel leads to known lift distributions for
both steady and oscillating two—dimensional wings. The kernel function
for three-dimensional supersonic flow is reduced to the sonic case and
is shown to agree with results obtained for the sonic case in NACA
TN 5151, and the downwash functions associated with "horseshoe" vortices
in supersonic flow are discussed and expressions are derived.

In reference 1 the kernel function of an integral equation relating
a known or prescribed downwash distribution to an unknown lift distri—
bution for a harmonically oscillating finite wing of arbitrary plan form
was treated for compressible subsonic flow. The purpose of the present
paper is to extend this treatment of the kernel function to supersonic
flow.

The kernel functions under consideration arise when linearized-
boundary-value problems for obtaining aerodynamic forces on oscillating
wings are reduced to integral equations involving the distribution of
pressure or wing loading as the unknown. In such integral equations
the kernel functions play the important role of aerodynamic influence
functions in that they give the normal induced velocity or downwash at
any one point in the plane of the wing due to a unit pressure loading
at any other point in the plane of the wing.