Phase and group velocity matching for cumulative harmonic generation in Lamb waves

Owing to the enhanced sensitivity of nonlinear acoustic methods to material damage, the nonlinear Lamb wave propagation is pertinent to the nondestructive evaluation of platelike structures, and it is typically manifested as generation of higher harmonics. For dispersive waves such as Lamb waves, however, the cumulative growth of harmonics requires that the primary mode and the generated higher harmonic modes possess identical phase and group velocities. In this paper, this issue of the phase and group velocity matching in Lamb waves is explored based on a systematic analysis of the Rayleigh-Lamb frequency equations. The analysis shows that for certain values of the phase velocity, the Rayleigh-Lamb frequency equations are satisfied at equi-spaced frequencies which are multiples of the smallest. Such frequencies, together with the corresponding phase velocities and the Lamb modes, are determined analytically. Four such types of Lamb modes are identified: (i) Lamé modes, (ii) symmetric modes with dominant longitudinal displacements, (iii) intersections of symmetric and antisymmetric modes and (iv) extra Rayleigh modes. For the first three types, it is also established that the primary and the harmonic modes have the same group velocity, and that the surface motion of the plate is featured with vanishing vertical or horizontal displacements. In contrast to these three types, the fourth type only exists for a special range of the transverse to longitudinal wave speeds of the solid. This type is not featured with a common group velocity, and neither of the vertical or horizontal displacement vanishes on the plate surfaces. The obtained results are summarized as tables, and demonstrated graphically on the dispersion curves for aluminum as well as iron plates.