Equivalence of Gaussian and piston ultrasonic transducer voltages

A plane-wave decomposition of collimated beams and electromechanical reciprocity relations are used to demonstrate fundamental differences and unusual similarities about transducer fields and transducer voltages at ultrasonic frequencies under various conditions. It is shown that the voltage induced by a transmitting acoustic piston transducer (constant particle velocity over the transducer surface) radiating into an ideal fluid medium on a second identical piston transducer, operating as a receiver, is nearly identical to the voltage observed when the two transducers have instead a Gaussian radial surface velocity distribution. The strong similarity in induced voltage begins when the two devices are separated by only several acoustic wavelengths, still well within the nearfield of both transducers, and the similarity increases with separation. Cases of transducer velocity distributions different from piston, such as Lorentzian or triangular, yield results nearly identical to the piston case. Transducers of differing size are also treated. It is further shown that an "equivalent transducer" can be derived for any combination of radiators, whose field is identical to the voltage measured using the pair. Moreover, with this concept the calculation of voltage in two-probe experiments is as simple as well-known approximations to the Rayleigh integral for a single transducer. These results have substantial consequences for calculations, either analytical or numerical, that predict the voltage measured in two-transducer experiments.