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METHODS OF SOLVING THIN-AIRFOIL PROBLEMS
The thin-airfoil problems can be treated by the usual methods of mathematical physics. The three commonly used methods are:
1. Superposition of proper singularities. Examples are the superposition of sources and sinks to obtain a symmetric body or of the source-sink doublet layer to obtain a lifting surface.
2. By a transformation which reduces a given boundary-value problem to an easier problem, or to a problem the solution of which is known. For two-dimensional incompressible flow, the conformal transformation is a powerful method, because the theory of functions of a complex variable furnishes the necessary tool to carry out the operations. The well-known Lorentz transformation for the wave equation may be regarded as an example of conformal transformation in a four-dimensional space whose metric is defined as
ds2 = r&2 + dy2 + dz2 — a2 dt2
3. Operational methods, such as Laplace transformation or the Fourier transformation.
Each method has its advantages and difficulties. Most of the airfoil problems can be (and have been) solved by all three methods. In the following presentation, however, we shall choose only the shortest ones.
