A variational-approximation technique, which may be interpreted as a coupled-mode theory, is derived for elastic-vibration problems. The theory is valid for the strong coupling that occurs in isotropic and anisotropic elastic material as a result of Poisson's ratio, shear-shear interaction, and/or shear-extensional interaction effects. As an example, the coupled-mode theory is briefly applied to the two-dimensional extensional vibrations of a thin rectangular plate. Calculations for several of the lower branches of the frequency spectrum compare very closely to experimental results obtained with hot-pressed ferroelectric ceramics and to published experimental results. The theory predicts the existence of so-called edge modes, which have been related by past analyses to the complex branches of the dispersion relation. The technique described in this work calculates the resonant frequencies and displacement distributions for these edge modes; real propagation constants and a few zero-order approximations for extensional vibrations are used. The technique produces accurate results, yet physical insight is not obscured by mathematical complexities.