Harmonic generation by Rayleigh–Lamb waves in isotropic elastic plates

Abstract

Harmonic generation by Rayleigh–Lamb waves of finite amplitude in homogeneous, isotropic, stress-free elastic plates is investigated theoretically. A bifrequency primary wave field is considered, in which the two waves propagate in single yet arbitrary Rayleigh–Lamb modes. Solutions for the second-harmonic and difference-frequency components are obtained via modal decomposition and use of reciprocity relations. Two conditions are required for generation and resonant amplification of these spectral components, power flow, and phase matching. Second-harmonic generation is considered first. Although phase matching is possible between a primary wave in the first symmetric mode and its second harmonic in the second asymmetric mode, the power flow is found to be zero. However, with one primary wave in each of the first symmetric and asymmetric modes, both phase matching and power flow may be achieved with the difference-frequency wave generated in the first asymmetric mode. In particular, resonant parametric amplification of the difference-frequency wave can be achieved under conditions where dispersion prevents efficient coupling from occurring between the primary waves and other frequency components. Explicit analytic solutions are presented and discussed. [Work supported by the Brazilian Ministry of Science and Technology, and the National Science Foundation.]