## 4 Discussion  
It can be hypothesized that for the resonators with the wrap-around electrode in the shape and dimensions according to the scheme in Figure 2 there will be an increase in the resonance frequency compared to the fully electroded resonators of the same dimensions. The differences between a fully electroded resonator and a resonator with the wrap-around electrode, which are also very plausible causes of the increase in resonance frequency, are:  
* Inhomogeneous polarization, especially in the area of electrode wrap-around, resulting in inhomogeneous material properties;
* Inhomogeneous distribution of the excitation electric field in the resonator.  
One may also ask why in the case of the mesh, or--more precisely--of the wrap-around shape according to variant (a), the growth of the resonance frequency is about \(0.2\%\) higher than in the case of the wrap-around according to variant (b). A discretization error cannot be ruled out even though the difference in the calculated resonance frequency between variants (a) and (b) is less than an order of magnitude lower than a tenth of a percent in the case of the calculated resonance frequency.  
However, we can hypothesize that the difference in the calculated frequency is caused by the different amount of polarized volume of the transducer in the direction of its thickness. This, in accordance with the results given in [3], is  
* A total of \(88\%\) of resonator volume in configuration (a);
* A total of \(90\%\) of resonator volume in configuration (b).  
This reasoning is consistent with the conclusions made, for example, in [12]. The effect of electrode size on the resonance frequency of symmetrically electroded discs is studied there. The increase in resonance frequency with decreasing electrode area is shown. Compared with the problem of the resonance frequency of a thickness oscillation resonator with a wrap-around electrode studied here, there are two major differences:* In [12], axially symmetrically electroded discs are studied, in contrast to resonators with a wrap-around electrode, which are symmetric only in one plane;
* In [12] the radial oscillations of the disc are studied, in contrast to the thickness oscillations studied in resonators with a wrap-around electrode.  
For longitudinal oscillations of piezoelectric rods and thickness oscillations of plates, a relation for the electromechanical coupling coefficient valid for this type of oscillations [1; 2], can be derived. The condition is full electroding of the opposite circular bases. For the thickness oscillations of the plate with full electrodes [2], the following applies  
\[k_{t}^{2}=\frac{\pi}{2}\frac{f_{r}}{f_{a}}\text{cotg}\left(\frac{\pi}{2}\frac{ f_{r}}{f_{a}}\right). \tag{29}\]  
As mentioned in the introduction to this chapter, the relationship between \(k_{eff}\) and the electromechanical coupling coefficient of a given type of oscillation can be expressed. In the case of the thickness oscillation and its electromechanical coupling coefficient \(k_{t}\), the following is true [2],  
\[k_{t}^{2}=\frac{\pi}{2}\sqrt{1-k_{eff}}\text{cotg}\left(\frac{\pi}{2}\sqrt{1- k_{eff}}\right). \tag{30}\]  
For comparison, let us calculate the coefficient \(k_{t}\) for the fully electroded resonator as well as for resonators with a wrap-around electrode. The calculation will be performed, although we know that for resonators with a wrap-around electrode it will only be approximately valid. The results of the calculation are summarized in Table 3.  
The calculated values of \(k_{t}\) are higher than the corresponding values of \(k_{eff}\) in all of the considered cases, but the trend is the same for both quantities. In the case of resonators with wrap-around electrodes, \(k_{t}\) is lower than in the case of a fully electroded resonator.