A unified formalism using effective surface permittivity to study acoustic waves in various anisotropic and piezoelectric multilayers

A unified formalism is presented that uses the effective surface permittivity (ESP) to study surface acoustic waves (SAW) in layered substrates and guided waves in layered plates. Based on known mathematical tools, such as ordinary differential equation and transfer matrix, a generalized surface impedance (GSI) concept is developed and exploited to investigate the acoustic propagation in various anisotropic and piezoelectric layered structures. The ESP function, originally defined for the surface of a homogeneous and semi-infinite piezoelectric substrate, is extended to both the top surface of and an interface in a layered half space, as well as to either surface of a finite-thickness plate. General ESP expressions for all mentioned configurations are derived in terms of an equivalent GSI matrix. It is shown that, when using the appropriate GSI matrices, the same form of the ESP expressions applies no matter whether the structure is a homogeneous half space alone or coated with a layered plate or a layered plate alone. GSI matrices are explicitly given in terms of the bulk partial mode solutions for a substrate and via the transfer matrix for a plate. Modified GSI matrices for structures consisting of both a plate and a substrate are also specified. Analytical development is fully detailed to suit program implementation. To illustrate its versatility, the formalism is also applied to two-substrate configurations, allowing one to analyze guided waves in a plate sandwiched between and interfacial waves existing along the boundary of two different media. Numerical examples are given to illustrate the spectrum features that the ESP shows for various structures. Deduced ESP expressions allow one to locate directly all piezoelectrically active waves in any structure including at least one piezoelectric layer. Acoustic modes that are not piezoelectrically active and those in non-piezoelectric materials can be also obtained by using the intermediate results, such as derived GSI matrices.