New Mathematical Formulation for Piezoelectric Wave Propagation

Elastic waves in a general anisotropic piezoelectric medium are discussed in terms of an eight-dimensional vector formalism. The eight-dimensional state vectors have the physical significance that their first three components represent the local particle displacement, the second three components represent the stresses on an arbitrarily selected plane, and the seventh and eight components represent the electric displacement normal to this plane and to the scalar electric field potential, respectively. As an application of the formalism, the dispersion relations for bulk waves and surface waves in ${\mathrm{Bi}}_{12}$Ge${\mathrm{O}}_{20}$ are derived, and the dispersion relations are given for a system consisting of a vacuum followed by an arbitrary piezoelectric film on an arbitrary semi-infinite piezoelectric substrate. Expressions are given for the propagator in arbitrary heteroepitaxial structures, and the boundary conditions associated with electrical or mechanical excitation of such structures is discussed.