An explanation of the origin of backward Lamb waves using the Mindlin's grid

The origin of backward Lamb waves in isotropic plates is explained and quantitatively described in terms of coupled-mode theory. The waves under consideration are interpreted as a result of enhanced interaction of longitudinal and transverse partial waves included in the plate solution. The enhancement occurs due to the phase matching achieved at the intersections of dispersion curves for the uncoupled partial waves. The uncoupled modes exist if the artificial Mindlin's boundary conditions are satisfied at the plate surfaces. The real conditions of the free surface lead to the mode coupling and the repulsion of their intersecting characteristics. The backward waves are found to occur in the vicinity of intersections of the dispersion curves of uncoupled Mindlin modes, if they are located near the axis of zero values of the wave vectors of Lamb modes. According to the results of the study, the intersection points may lie in the regions of not only real solutions of the secular equations but imaginary ones also, despite the fact that the backward waves themselves are described by merely real solutions. For small values of the wave vectors, the coupled-mode theory is developed that allows calculations to be performed analytically and with good accuracy for the dispersion curves of backward waves. The theory explains why the proximity to each other of thickness resonances of longitudinal and transverse waves in the plates is favorable for the appearance of backward waves under study. It also becomes clear why the repulsion of coupled curves occurs along the direction of the frequency axis in the real part of spectrum, rather than the axis of wave vectors like in the imaginary spectrum part.