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.]

Time-domain analysis and experimental examination of cumulative second-harmonic generation by primary Lamb wave propagation

“…Within the second-order perturbation approximation, there is a bulk driving force of double the fundamental frequency in the solid plate [denoted by F ½2f b ðU ½f l Þ] because of the convective nonlinearity (due to the description difference between the Lagrangian and Eulerian coordinates) and the inherent nonlinear elastic properties of the solid. [13][14][15][16][17]19 The form of F ½2f b ðU ½f l Þ associated with the lth Lamb wave was given by Deng. 13 Besides the existence of F ½2f b ðU ½f l Þ in the interior of solid plate, there are traction stress tensors of double the fundamental frequency at the two surfaces of the solid plate, denoted by P ½2f S ðU ½f l Þ.…”

“…Thus, the second-order perturbation approximation is applicable to analyzing second-harmonic generation by the primary Lamb wave propagation. [13][14][15][16][17] Specifically, the second-harmonic generation can be treated as a second-order perturbation of the primary Lamb wave propagation.…”

“…[14][15][16][17] The modal expansion analysis approach means that the excited fields can be decomposed into a series of normal modes obtained in a stress-free waveguide. To ease the physical interpretation we consider here only propagating DFLW modes and neglect the possible evanescent ones.…”

“…To date, some theoretical and experimental studies of this issue have been reported. [13][14][15][16][17][18] It is found that under certain conditions the second harmonic generated by the primary Lamb wave propagation has a strong nonlinearity, i.e., its amplitude can grow with propagation distance. Recently, some studies focus on using the effect of second-harmonic generation by the primary Lamb wave propagation to assess damage in elastic plates.…”

“…[14][15][16][17] However, it should be noted that Lamb wave tone bursts with finite duration are commonly used in practical applications. For observing clear second-harmonic signals, the physical process of generation of the time-domain second harmonic should be carefully considered in advance.…”