Abstract
It was observed about 30 years ago, by Bechmann, that many of the undesirable overtones of a quartz resonator plate disappear as the area of the electrode plating is reduced. The cause of the phenomenon was discovered by Mortley and rediscovered recently by Shockley. Their explanation makes it possible to bring the matter within the framework of equations of motion of anisotropic plates. One-dimensional and two-dimensional solutions, with appropriate boundary conditions on a pair of parallel boundaries, lead to explicit formulas for Bechmann's Number—the ratio of length of electrode to thickness of plate below which certain overtones are absent. Other overtones, traceable to reflections from all four edges of a free, rectangular plate, are more difficult to treat mathematically owing to the usual obstacle which is present even in the elementary theories of plates. However, in an AT-cut, quartz plate, the modes that prevent a general solution are relatively unimportant and their omission makes it possible to obtain closed solutions that include some of the additional undesirable modes.
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