The power extracted from a piezoelectric surface wave in quartz by an arrangement of electrodes on the surface is determined by a perturbation technique, which is extremely accurate for low-coupling materials. The mechanical portion of the surface wave is found by neglecting the piezoelectric coupling, which is very small in quartz. The elastic solution then yields both volumetric and surface source terms, which are generated by the piezoelectric coupling and drive the steady-state electrostatic solution. A technique is introduced for reducing the remaining electrostatic problem to Laplace's equation, subject to such boundary conditions that the solution is amenable to conformal mapping techniques for several interesting, and possibly useful, electrode arrangements. Surface waves propagating along the digonal axis for all rotated Y cuts are considered, as well as those propagating in all directions along the surface of an X-cut crystal. A reciprocal relation between the excitation and detection problems is derived, and is used to determine the amplitude of a surface wave generated by the application of an alternating voltage to the surface electrodes when the solution to the detection problem for the same arrangement of electrodes is known.

Coquin, G. A., & Tiersten, H. F. (1967). Analysis of the Excitation and Detection of Piezoelectric Surface Waves in Quartz by Means of Surface Electrodes. The Journal of the Acoustical Society of America, 41(4B), 921–939.