The quarter-infinite wing oscillating at supersonic speeds
Introduction. In the April, 1951 issue of the Quarterly of Applied Mathematics, two papers appeared. One, by H. J. Stewart and Ting-Yi Li1 purports to extend the method of J. C. Evvard2 for solving supersonic wing tips to the case of the oscillating wing. The other by J. W. Miles3 solves the problem of the oscillating rectangular airfoil by the Wiener-Hopf technique. In an application of their method to the rectangular airfoil,4 Stewart and Li obtain results which disagree with those of Miles. Moreover, Stewartson5 has treated the problem independently, and his results agree with Miles'. The raison d'etre of this note is to add yet another independent solution to the list of those which agree with Stewartson's and Miles'. In view of the fluxive state which this problem is in at the moment, it is felt that this would be of current interest. The Gardner method. The method presented by C. S. Gardner6 will be used to obtain the solution. Let x, y, z represent rectilinear space coordinates, x in the direction of the free stream, and y in the "spanwise" direction. Denote by U the freestream velocity, c the velocity of sound, M the Mach number and t the time. Let z = f(x, y, t) describe the surface of the wing, g = Ufx + /, , /3 = (M2 — 1)1/2, x' = x/fi, t' — (Mx — p2ct)//3. Then in terms of x', y, z, Gardner's method for the rectangular airfoil consists in