Propulsion efficiency of vibrating bodies in subsonic gas stream

An axisymmetric body in a subsonic gas stream, namely an airfoil periodically deforming without loss of symmetry and change of length, is examined from the standpoint of thrust-power relation. The corresponding surface-time integrals for the drag force and the propulsion power are formulated in a cylindrical system of coordinates and evaluated, assuming a potential flow of the gas with small perturbations by the body. A time-independent source distribution density term is added to this time-dependent solution to insure a positive airfoil cross-sectional area. The maximum theoretical efficiency, or minimum power for given thrust, is calculated on this basis and the shape of the body is sought for which the real efficiency will approach the maximum theoretical one as closely as possible. Making the length of the body invariable in time leads to a discretely varying drag coefficient.