A two-dimensional theory for the analysis of surface acoustic waves in finite elastic solids

Abstract

The analysis of surface acoustic waves in elastic solids started from a semi-infinite isotropic elastic body with solutions and techniques dated back to Lord Rayleigh. These solutions have explained the surface acoustic wave phenomena and guided its engineering applications in many fields. Research work followed have been using the same semi-infinite model with solution techniques for approximate and exact results in both analytical and numerical manners for many application problems involving finite elastic solids. On the other hand, we have noticed that various two-dimensional theories, notably plate theory by Mindlin, have been derived to study the bulk wave propagation in finite elastic solids like plates and bars with satisfactory results. The lack of such a two-dimensional theory has made the analysis of surface acoustic wave propagation in various waveguides primitive and difficult because the precise solutions cannot be obtained through any analytical effort and numerical solutions are also difficult to obtain because of the complicated nature of problems. To meet the need of a simplified analytical method for surface acoustic waves in finite elastic solids, we start the derivation from the well-known three-dimensional solutions of semi-infinite elastic solids. Using exact solutions as the basis for the two-dimensional expansion, we found that the usual procedure of dimension reduction works perfectly in this case because the unknown wavenumber is eventually removed in the expansion process, and the depth of the solid is considered through integration constants that are exponentially decaying functions of the wavenumber, which is to be specified. In addition, we also found the two-dimensional equations give the exact phase velocity without corrections of any kind in the limiting case in which the solid is semi-infinite. With these equations, two-dimensional solutions of mechanical displacements and phase velocity can be obtained analytically from four coupled differential equations that are similar to Mindlin plate equations. We demonstrated the applications of the two-dimensional theory with the analysis of surface waves in isotropic elastic rectangular strips. The velocity and displacement results offer a chance to understand surface acoustic waves in finite solids and the presence of overtone modes and transition zones.