A continuous representation of the acoustic surface-wave mode shape in arrays of reflecting grooves

A variational analysis of the reflection of surface waves by arrays of reflecting grooves is presented. The variational condition results in a system of linear algebraic equations relating the amplitudes of the waves on both sides of the edge of a groove. The influence of transverse moding, which is a result of the finite lateral dimension of the grooves, is included in the treatment, but scattering into bulk waves is not considered. Application of the linear algebraic equations across both edges of a typical groove results in a system of transmission equations, which constitutes a linear difference system for the periodic array. A closed-form continuous representation of the mode shape along the transmission path is obtained from the solution of the system of linear difference equations for the periodic array. The continuous representation is particularly convenient for use in the evaluation of perturbation integrals that arise in the calculation of the change in frequency of surface-wave resonators due to biasing states and for many other purposes. When the resonant surface-wave mode has been found, the reduced resonant frequency and phase velocity are obtained by means of a perturbation procedure. Calculated results are presented for grooved arrays on ST-cut quartz.