Three-mode coupling of symmetric and antisymmetric lamb waves in plates with finite corrugations

Coupled-mode equations governing the amplitudes of the higher-order symmetric Lamb modes S₁ and S₂ with the antisymmetric mode A₂ in an infinite elastic plate with sinusoidal surface corrugation over a finite length are obtained via multiple-scales analysis. This phenomenon of three-mode coupling is observed when the wavenumbers ks₁ and ks₂ of the symmetric modes and kA₂ of the antisymmetric mode satisfy the simultaneous resonance conditions ks₁ - kA₂ = k_w and kA₂ - ks₂ = k_w, where k_w is the wavenumber of the sinusoidal corrugation. Near resonance, the coupled amplitude equations are solved exactly as an initial-value problem and it is seen that the modes are transmitted through the grating without reflection. Complete conversion from the symmetric modes into the antisymmetric mode is observed at periodic intervals along the grating when the resonance conditions are exactly satisfied. The effect of detuning away from resonance also shows propagation without reflection with periodic energy exchange. In the latter case, the modes couple without complete conversion. This phenomenon of mode conversion is confirmed by the results of an experiment on an aluminum plate with a triangular grating excited with the S₂ symmetric mode at 2.7 MHz.