Derivation of an integral equation for transonic flows
It is noted that the nonlinear partial differential equation for the perturbation velocity potential and boundary conditions describing steady inviscid compressible transonic flow past a thin two-dimensional airfoil can be transformed into a singular integrodifferential equation and that differentiation of the latter yields an integral equation. Two forms of this integral equation currently exist: one for the singularity that is enclosed in an infinitely long strip of vanishing thickness and the other for the singularity that is enclosed in a vanishing circle. In the present article, a more general integral equation is derived by enclosing the singularity in a vanishing rectangular cavity of arbitrary aspect ratio. The two existing forms of this equation are deduced as special cases distinguished by the respective values for the aspect ratio (infinity for the first form and unity for the second).