Computation of plate wave dispersion diagrams and surface wave velocities without explicit boundary conditions

For the evaluation of the dispersion relation of surface acoustic waves (SAW) or plate (Lamb) waves, it is generally necessary to form a determinant of the boundary conditions and to seek its zeros as a function of the wave vector for a fixed frequency. Finding all zeros can be a numerically difficult problem. The plane wave expansion (PWE) method is used in the field of phononic crystals to formulate eigenvalue problems to compute dispersion diagrams for solid-solid compositions. We discuss in this paper how the boundary conditions can be included implicitly in the form of the PWE solution, thus leading to an efficient eigenvalue problem. The solutions of the eigenvalue problem represent waves propagating in the plate with a given wave vector along the surface. Furthermore, SAW velocities can be estimated from the slowest wave for large wave vectors. The PWE numerical algorithm is fast and accurate. Examples for a single plate and a multilayer plate are given, and extension to piezoelectric materials is discussed. The method can be of value for numerical codes requiring a generic method for wave dispersion that does not require an initial guess for the solution.