An unsteady linearized formulation based on Oswatitsch-Keune's parabolic method is developed to analyze transonic flow past oscillating slender bodies. In contrast to the widely used integral transform method, it is shown that all solutions can be derived by a simpler method directly in the physical plane. By various expansion procedures, low-frequency solutions then are derived according to two clearly defined frequency ranges. AdamsSears' iteration is employed to account for the second-order effects. Stability derivatives are compared with available theories and data. It is found that the derivatives depend more sensitively on thickness than on the reduced frequency. Finally, a critical assessment of the present method is given.Nomenclature (x,r,6) = nondimensional cylindrical coordinates, normalized by the true body length L = freestream velocity =iL/U, nondimensional time = true time = coL/(/, reduced frequency = true angular frequency of pitch = total and perturbed potentials, both are normalized = maximum body radius/L, the body thickness ratio