An analysis based on the approximate equation for the thickness vibration established by H.F. Tiersten (1976, 1979, 1982, 1986) and coworkers is presented. A semi-algebraic method of resolution are investigated for partially or totally contoured resonators. In this method, the eigensolutions at V=0 are obtained by separation of the approximate equation in transformed rectangular or polar coordinates for the different regions of the resonator. A totally numerical solution of the approximate equation by the finite-element method is then proposed. The method can be used for practically any type of thickness-mode resonator. It is applied here to some typical examples of contoured resonators. The results are compared with those obtained with the previous method and with experimental results.
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