Global long wave approximations for elastic wave guides

It is well known that the long-wave approximations for thin elastic structures can be separated into local low-frequency and high-frequency ones. The former are given by more conventional plate and shell theories justified near zero frequency, whereas the latter correspond to 2D equations valid over the vicinities of thickness resonances. In this contribution, we develop a more general long-wave procedure for a broad band surface loading resulting in 2D equations involving transcendental expressions of the spectral frequency parameter. These equations are reduced to the aforementioned local approximations in the vicinities of zero and thickness resonance frequencies. As an example, SH waves in a layer loaded along its faces are considered. Comparison with the exact solution of the problem is presented. Similar global approximations could also be constructed for periodic media, due to the analogy for asymptotic schemes for thin and periodic structures.