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

This report treats the kernel function of the integral equation
that relates a known or prescribed doumwash 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 NAC'A Report 1234
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 cutofi' or Heaviside unit function. The kernel
functions are reduced to forms that can be accurately evaluated
by considering 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 NAOA Report 1234, and
the downwash functions associated with “horseshoe” cortices in
supersonic flow are discussed and expressions are derived.

In reference 1 the kernel function of an integral equation
relating it known or prescribed downwash distribution to an
unknown lift distribution for a harmonically oscillating finite
wing of arbitrary plan form was treated for compressible
subsonic flow. The purpose of the present report 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 aerody-
namic 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.