Non-collinear interaction of Rayleigh–Lamb and shear horizontal waves in a finite region in a plate

Abstract

Non-collinear interaction of guided elastic waves in a homogeneous and isotropic plate with quadratic material nonlinearity is analyzed theoretically to investigate the sum and difference frequency generation from a finite interaction region of primary waves. Using a perturbation approach and the time-harmonic Green function for isotropic plates, an explicit expression is derived for the displacement field of nonlinearly generated secondary waves when two primary monochromatic straight-crested Rayleigh–Lamb/shear horizontal waves intersect at an arbitrary angle in a right cylindrical region of arbitrary cross-section and height equal to the plate thickness. The resulting displacement observed far away from the interaction region in the direction of the wavevector of driving forces (i.e., the sum or difference of wavevectors of primary modes) is shown to grow in proportion to the interaction volume when the wavenumber of secondary mode coincides with that of driving forces with nonzero energy transfer from the primary to the secondary modes. The influence of the interaction volume and the intersection angle on the secondary wave field is investigated for a special case where the interaction region is a right circular cylinder. Furthermore, the non-collinear interaction generating the secondary mode with negative group velocity is also examined.