It is generally known that surface acoustic waves, or Rayleigh waves, have different mode shapes in infinite plates. To be precise, there are both exponentially decaying and growing components in plates appearing in pairs, representing symmetric and antisymmentric modes in a plate. As the plate thickness increases, the combined modes will approach the Rayleigh mode in a semi-infinite solid, exhibiting surface acoustic wave deformation and velocity. In this study, the two-dimensional theory for surface acoustic waves in finite plates is extended to include the exponentially growing modes in the expansion function. With these extra equations, we study the surface acoustic waves in a plate with different thickness to examine the coupling of the exponentially decaying and growing modes. It is found that for small thickness, the two groups of waves are strongly coupled, showing the significance of including the effect of thickness in analysis. As the thickness increases to certain values, such as more than five wavelengths, the exponentially decaying modes alone will be able to predict vibrations of surface acoustic wave modes accurately, thus simplifying the equations and solutions significantly.

The resonant frequencies of thickness-shear vibrations of quartz crystal plates in rectangular and circular shapes are always required in the design and manufacturing of quartz crystal resonators. As the size of quartz crystal resonators shrinks, for rectangular plates we must consider effects of both length and width for the precise calculation of resonant frequency. Starting from the three-dimensional equations of wave propagation in finite crystal plates and the general expression of vibration modes, we obtained the relations between frequency and wavenumbers. By satisfying the major boundary conditions of the dominant thickness-shear mode, three wavenumber solutions are obtained and the frequency equation is constructed. It is shown the resonant frequency of thickness-shear mode is a second-order polynomial of aspect ratios. This conforms to known results in the simplest form and is applicable to further analytical and experimental studies of the frequency equation of quartz crystal resonators.

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Based on Mindlin plate theory, the frequency spectrum of face shear mode in AT-cut quartz plates and its frequency temperature behavior are studied. The results of numerical simulation are compared with experiment data. Both exact three-dimensional plate equations and Mindlin first-order plate equations are used to calculate the spectrum respectively, which can help engineers to decide suitable dimensions of ATcut quartz plate to avoid the couple of thickness-shear and face-shear. The temperature behavior characteristic is quite an important characteristic of quartz resonator. In the present paper, the frequency temperature behavior of face-shear mode is also studied. Linking the frequency spectrum and temperature behavior results can enable engineers not only to get suitable dimensions of AT-cut quartz plate at room temperature but also to get suitable dimensions at certain temperature range. The measurement data for rectangular, AT-cut quartz plates is also demonstrated. 基于 Mindlin 板的理论，得到 AT 切割石英晶体板面剪切振动的频谱图，以及面剪切振动的频率温度特性结果， 并且与实际样品的量测值进行比对。本文分别以三维精确理论和一阶板的理论对 AT 切割石英晶体板 Z 轴方向 面剪切振动进行理论推导，计算出的频谱图可帮助晶片设计人员选择合适的晶片尺寸来避开厚度剪切振荡和面 剪切振荡的偶合。温度特性对石英晶体谐振器来说至关重要，本文通过理论推导得出了 Z 轴方向的面剪切振荡 的频率温度特性结果，这部份的结果结合频谱图可以帮助晶片设计人员找到在一定的温度范围内都无偶合情况 出现的晶片尺寸。实际的样品量测的结果也被附加于论文中，用于和理论计算结果的比较。 关键词：面剪切，一阶板的理论，频率温度特性 1. 引言 AT 切割石英晶体板因为其良好的频率温度特性被广 泛地应用于压电谐振器的制作中。它所利用的振动 形态是位移与 1 x (diagonal axis)方向平行的厚度剪切 振动模态。然而石英晶体是各向异性的材料，晶体 板的振动除了厚度剪切振动模态以外还存在很多不 被需要的寄生模态。在实际的石英晶体谐振器的设 计中，如何选择好合适的石英晶体板的尺寸来避开 寄生模态和厚度剪切振动模态的偶合，是设计人员 最关心的问题。面剪切振荡是最早期被发现的寄生 模态之一，关于它的理论推导早在 70 年代就已经由 Mindlin 提出。[1]Mindlin 对面剪切振动模态的推导 是基于三维理论的基础上进行的，为了能用较为简 单的表达式来表达出解的形式，板的模型进行了简 化，并且为符合自由边界条件，还假设板的横截面 978-1-4244-9821-5/10/$26.00 ©2010 IEEE

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