Measurement sensitivity distribution analysis of continuous wave ultrasonic Doppler transducers based on reciprocity theorem

Continuous Wave Ultrasonic Doppler (CWUD) technique has diverse applications in a range of fields. In such applications, measurement sensitivity distribution (MSD), which describes the effective sensing region of the transducer pair, has a significant impact on the measurement performance. However, existing methods fail to accurately characterise MSD, let alone enable its further visualisation. In this paper, an MSD visualisation method for CWUD based on the reciprocity theorem is proposed. Firstly, the energy density difference (EDD) distribution and the sensitivity distributions for compressibility and density, which characterise MSD, were derived. Then, the acoustic pressure and particle vibration velocity in the acoustic field were obtained through finite element analysis. The EDD distribution and the sensitivity distributions for compressibility and density were calculated and used to visualise the MSD. Finally, the method was applied to MSD visualisation in gas kick detection, and experimental results validated its effectiveness in accurately characterising and visualising MSD. The method is not constrained by the structure of the measured object and does not require any simplification. Through straightforward calculations, it accurately characterises and visualises MSD under various operating conditions, providing guidance on optimising measurement conditions.