For a slightly concave transducer with radius of curvature k, according to [28]:

Another consideration by Labhardt concerns the contact between the transducers and the liquid under investigation. As noted above, the reflection coefficient is less than unity, extending the propagation path of the wave between the transducers; this requires a correction. According to [35], the frequency dependence of the transducer impedance leads to:

with

Here fₙ is the cell resonance frequency, l is the cell length, c is the sound speed in the liquid, ρ_fl. is the density of the liquid, f_Q is the transducer resonance frequency, c_Q is the sound speed in the transducer and ρ_Q is the transducer density.

This relationship is essential for the measurement procedure and accompanies every measurement.

All other mechanical losses—including diffraction, boundary-material effects and radiation—are highly specific to each resonator. According to [35] and [36], diffraction effects in a plane–plane resonator at a given cell resonance frequency fₙ can be expressed by the following relationship:

Original mathematical notation is preserved in the extracted image assets.