A robust formulation of SAW Green´s functions for arbitrarily thick multilayers at high frequencies

This paper presents a robust formulation of SAW Green´s functions for arbitrarily thick multilayers at high frequencies. The formulation is an alternative to that based on the transfer matrix method, which suffers from numerical instabilities when the frequency and/or thickness parameters become large. This numerical difficulty can be attributed to the mixture of exponentially growing and decaying terms during the transfer matrix calculations. To be more instructive, the numerical instability is delineated in terms of upward-bounded and downward-bounded waves within each layer. In accordance with such boundedness association, a recursive scheme not involving any growing terms is developed based on the scattering matrices to eliminate the instability. The resulting reflection matrix method is extremely concise and preserves the simplicity and convenience of the transfer matrix method. Using the reflection matrices, the generalized Green´s functions that relate the particle velocity and the rate of electric potential change to the surface stress and charge are formulated succinctly. These Green´s functions are useful for having incorporated the electrical properties of the vacuum above the surface. Numerical computations are exemplified to demonstrate the instabilities of the transfer matrix method and to justify the robustness of the reflection matrix formula.