The differential equations of motion of a free, rectangular, isotropic plate are solved for the longitudinal and flexural cases upon the assumptions that the displacements are independent of one of the Cartesian coordinates and that the displacements parallel to this coordinate are negligible. The flexural and longitudinal cases are considered separately. The frequency expressions predict the effects due to mechanical coupling of two or more longitudinal modes or of two or more flexural modes. The effects of the lateral dimension of the plate upon the frequency of the longitudinal harmonics are also predicted. It is shown that in the longitudinal case only the odd harmonics of both of the modes in which the wave trains are propagated alongXandY,respectively, form a mechanically coupled system. Similarly, only the even harmonics of the flexural case form a mechanically coupled system. The law by which the displacements of two individual modes are combined to form two coupled modes is illustrated by means of photographs of interference patterns formed by aYhyphen;cut plate of crystalline quartz which is vibrating flexurally. The mathematical methods which are developed in this paper can be employed in considering the coupling of the longitudinal modes with the flexural modes of vibration. The twohyphen;dimensional case here considered, i.e., that in which the displacements are limited to theXYplane, is of special importance since it can be shown that in quartz and tourmaline the differential equations of motion and the boundary conditions become identical in form to those of the rectangular isotropic plate when it is assumed that the displacements are restricted to theXYplane and are independent ofz.The applications of this theory will be considered elsewhere.
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