Abstract

A method is developed for solving the mixed boundary-value problem in which a harmonic function is to be determined in a semi-infinite region with a plane boundary when the value of the function is zero on the part of the plane outside a certain circle and the normal derivative has prescribed values inside this circle. By this method, which depends on a repeated use of Abel's integral equation, the values anywhere on the boundary plane of the function and its normal derivative are expressed in terms of the known values of the normal derivative within the circle. In linearized aerodynamic theory the acceleration potential of the flow of an incompressible fluid about a wing of circular planform satisfies this type of boundary condition. The preceding theory provides an infinite set of solutions that satisfy null boundary conditions, that is, whose normal derivatives vanish within the circle. These are combined to satisfy the upwash and Kutta-Joukowski conditions for the flat plate at incidence and values are obtained for the lift and moment coefficients. In an appendix we examine the connexion of the present method with some earlier methods of solving this type of problem.