Free longitudinal vibrations of non-uniform rods

Free longitudinal vibrations of non-uniform rods are investigated by a proposed method, which results in a series solution. In a special case, with the proposed method an exact solution with a concise form can be obtained, which imply four types of profiles with variation in geometry or material properties. However, the WKB (Wentzel-Kramers-Brillouin) method leads to a series solution, which is a Taylor expansion of the results of the proposed method. For the arbitrary non-uniform rods, the comparison indicates that the WKB method is simpler, but the convergent speed of the series solution resulting from the pro-posed method is faster than that of the WKB method, which is also validated numerically using an exact solution of a kind of non-uniform rods with Kummer functions.