Ultrasonic piezoelectric transducers used in classical nondestructive testing are producing in general longitudinal vibrations in the MHz range. A simple mechanical model of these transducers would be very useful for wave propagation numerical simulations, avoiding the existing complicated models in which the real components of the transducer are modeled by finite elements. The classical model for longitudinal vibrations is not adequate because the generated longitudinal wave is not dispersive, the velocity being the same at any frequency. We have adopted the Rayleigh-Bishop model, which avoids these limitations, even if it is not converging to the first but to the second exact longitudinal mode in an elastic rod, as obtained from the complicated Pochhammer-Chree equations. Since real transducers have significant vibrations damping, we have introduced a damping term in the Rayleigh-Bishop model, increasing the imaginary part and keeping almost identical real part of the wavenumber. Common transducers produce amplitude modulated signals, completely attenuated after several periods. This can be modeled by two close frequencies, producing a “beat” phenomenon, superposed on the high damping. For this reason, we introduce a two-rod Rayleigh-Bishop model with damping. Agreement with measured normal velocity on the transducer free surface is encouraging for continuation of the research.
Strongly nonlinear potential theory on metric spaces
