An exact analytical model of frequency-variable piezoelectric stack transducers is proposed, and their dynamic characteristics are studied in this article. Based on the linear piezoelasticity theory, the dynamic analytical solution is first derived, and then its correctness is validated by comparing it with the results of a special example in the previous literature and the ones of the experimental study. The effects of the tuning resistance and the layer number of the active element on the dynamic characteristics are discussed. Numerical results show that tuning the resistance and the layer number of the active element can enable the multi-frequency characteristics of the piezoelectric stack transducers. A proper layer number of the active element can minimize the short-circuited resonance frequency and the open-circuited anti-resonance frequency. These findings provide guidelines to design and optimize the piezoelectric stack transducers, which have promising potential in developing the multi-frequency Langevin transducers for some underwater sound and ultrasonic applications, such as ultrasonic cleaning, ultrasonic chemistry, and sonar radiators.

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