The problem of estimating the spanwise distribution of lift and moment on an unswept lifting surface of finite span executing simple-harmonic oscillations in an incompressible fluid has been studied by many authors, among them the most notable being Cicala, Lyon, W. P. Jones, Skan, Sears, R. T. Jones, Kiissner, Biot, Boehnlein, Wasserman, Reissner, Zartarian, Hsu, Ashley, Dengler, Goland, and Shen (see bibliography). Unfortunately, the problem is so complicated that, even after the standard linearization, there exists no practicable, exact solution. Moreover, the independent lines of approach initiated by various authors have led to answers that cannot be shown to be entirely equivalent. Experimental results available at present, due to their scatter, cannot discriminate definitely which one of these theories approximates best the physical reality, although generally the methods of Reissner and that of Biot and Boehnlein are favored.

For a compressible fluid, the linearized theory in a supersonic flow is quite advanced. A number of exact solutions have been obtained for some special wing planforms and modes of motion. A few solutions are known also in the high subsonic and transonic speed range, but, generally speaking, the three-dimensional oscillating-airfoil theory is still a subject for future research.