Some basic relations for ultrasonic fields from circular transducers with a central hole

Ultrasonic transducers with a central hole are increasingly in use in high-intensity therapeutic ultrasound and similar medical applications. The hole is intended for the addition of a diagnostic device. Some fundamental properties of the fields produced by such transducers are investigated here theoretically, based on a Rayleigh integral algorithm. The approach is restricted to those cases where the Rayleigh integral can be fully solved to yield closed-formula results that can be easily used by the reader. This means a concentration on points on the field axis which, on the other hand, is most important under the aspect of the safety of the patient (maximum amplitudes). Closed-form expressions describing the influence of the central hole on the acoustic pressure, on the particle velocity and on the time-averaged intensity are presented. The relation between the true intensity and the intensity expression derived from the square of the acoustic pressure is discussed in some detail, an aspect which is important in ultrasonic measurement practice where the local intensity value is mostly assessed based on the square of a hydrophone signal.