A Taylor Series Expansion for Time Savings in Accurate Computation of Focused Ultrasound Pressure Fields

Abstract

A model for the continuous wave (cw) pressure beam distribution of a focused axially symmetric ultrasonic radiator with constant radius of curvature involves integration of e^ikr/r over the surface of the radiator. (k is the complex wave number and r is the distance from a radiating area element to the field point.) A single integral form exists, and it is this form that is expanded in a Taylor series in frequency. Thus, when representing a pulse as a superposition of cw beams, the need to do numerical integrations for each one of a large number of frequencies is eliminated. Accuracy of the truncated Taylor series depends on the coordinates of the field point as well as on the difference between the frequency of interest and the reference frequency. Accuracy criteria for a particular application are also presented. The computer time savings for our applications correspond to a factor of about 60 with accuracy being maintained.