Errors using the spatially averaged free-field pressure approximation for description of the receiving properties of piezoelectric transducers

In modelling of ultrasonic measurement systems and calculation of diffraction corrections for such systems, it is often assumed that the spatially averaged free-field pressure over the front surface of a piezoelectric transducer, ⟨p⟩, can be used to calculate its voltage output. It is of interest to investigate whether errors due to this ⟨p⟩-approximation are acceptable for piezoelectric transducers applied in high-precision ultrasonic measurements in fluids. 3D-axisymmetric finite element (FE) modelling is used to investigate such errors in a transmit-receive measurement setup employing two identical transducers. The approximate output voltage is found by Vout = ⟨p⟩MV and compared to the output voltage from FE modelling of the transmitter-medium-receiver system, Vout,ref, using the appropriate boundary conditions at the two transducers. MV is the open-circuit free-field receiving voltage sensitivity of the receiver for normally incident plane wavefronts, calculated assuming spherical reciprocity. Two examples of piezoelectric transducer pairs are studied, operated in the frequency range of their lower radial modes. It is shown that the ⟨p⟩-approximation may introduce notable deviations between Vout and Vout,ref, of up to approximately 40° for the phase angle and 2 dB for the magnitude. The deviations depend on the fluid medium, separation distance, frequency, and the transducer’s construction and vibrational characteristics.