Complex Branches of Dispersion Equations and Mindlin's Angle for Tilted Edges

Among numerous achievements of Professor Mindlin, the author was fortunate enough to witness at first hand the birth and further developments of two important themes, complex branches of dispersion equations of guided elastic waves and tilting edges to obtain an exact solution of thickness twist mode of vibration in a strip resonator. Both are significant contributions to theoretical analyses as well as to practical applications. A dispersion equation governs a relationship between frequency and wave number (propagation constant) of guided waves propagating along an elongate medium with constant cross-section. An analysis of a dispersion equation is essential to study propagation phenomena. It is also useful to study vibration (resonance) of a finite body, because a number of guided waves, which satisfy boundary conditions of major surfaces, can be combined to approximately satisfy boundary conditions of both end surfaces. Rayleigh-Lamb's dispersion equation for plates and bars and Pochhammer-Chree's equation for cylindrical rods are well known. Previous studies took only branches yielding real wave number (real branches) of a dispersion equation into consideration. At a low frequency, there is one and only one real branch. Hence an approximation is poor for vibration analysis. In 1958, Mindlin and Medick found complex branches besides a real branch in dispersion equation in the famous Mindlin's plate theory. Their dispersion equation, which corresponds to Rayleigh-Lamb's dispersion equation, is a third order algebraic equation and hence there should be three roots, one real and two conjugate complex. Subsequently an infinite number of similar complex branches were found in Rayleigh-Lamb's and Pochhammer-Chree's equations and much better approximation is obtained in vibration analysis. Peculiar modes called edge mode of vibration in a plate and a peculiar reverberant mode of propagation in a rod were identified as to closely related with complex branches