Ray, mode, and hybrid options for source excited propagation in an elastic plate

Impact excited vibrations in a multiwave layer can be represented in terms of ray fields or of normal mode fields. At early observation times, multiply reflected ray fields can be distinguished by their different arrivals whereas at later times, their collective effect is best expressed by the normal modes. Instead of utilizing rays or modes separately, as has been customary, a recently developed hybrid theory for multiwave layered media permits both to be combined self-consistently, and in convenient proportions [Lu et al. y Wave Motion 6, 435–467 (1984)]. The hybrid formulation is based on a ray-mode equivalent whereby a given slowness spectral interval can be filled either with rays or with modes subject to a spectral remainder. Moreover, the ray proliferation due to wave coupling at each encounter with a boundary is avoided by introduction of “eigenrays,” which are composed of self-similar combinations of the various wave fields. The general theory is applied to the dilatational and shear waves in an elastic plate. High-frequency motion at given range is calculated in terms of ray fields, normal mode fields, eigenrays, and hybrid combinations, using exact generating spectral integrals and also simplified asymptotic approximations. The numerical results confirm the validity of the ray-mode equivalent and establish conditions where the hybrid scheme offers an attractive option.