1. Quartz transducers. Because quartz transducers are not perfectly acoustically rigid, the distance l between the transducers does not correspond to their geometric separation. For a substance such as water, whose density ρ and sound speed cₛ are known, the reflection coefficient r = rₛ = rₑ is found to be approximately 0.8. This value depends not only on the properties of the liquid but also on the incident frequency. Consequently, the sound wave penetrates into the transducers and the effective distance l becomes variable.
2. Transducer geometry. As discussed later, transducer geometry has an enormous influence on resonator characteristics. The thickness L of a transducer makes it a resonant system in its own right. Resonances in the vicinity of the transducer’s natural frequencies cannot be used (Fig. 4.3).
3. Losses due to diffraction, radiation and other effects. In cylindrical resonators, effective attenuation losses arise from diffraction at the finite cell boundary and radiation from the outer transducer surfaces near the cell resonances. Numerous other attenuation mechanisms exist, but the complexity of the resonator means that they can only be estimated approximately.
4. Electrical crosstalk. Insufficient shielding of the transducers leads to electromagnetic interference at high frequencies.
Items 1, 2 and 3 are purely mechanical and depend individually on the measurement range, the sample and the resonator materials. The geometry of a cylindrical resonator alone implies the presence of subsidiary modes in addition to the principal mode. These modes are particularly troublesome during measurement. A relationship between the principal and subsidiary modes is therefore required. Labhardt [35] gives the following expression for the position of the m-th radial mode relative to the n-th principal mode in a plane–plane resonator:
The sensitivity of Eq. (4.14) to transducer geometry becomes apparent when one of the transducers is slightly concave.
Original mathematical notation is preserved in the extracted image assets.