Abstract
A simplified difference equation technique has been developed to calculate the three‐dimensional evolution of radially symmetric complex ultrasonic fields. Localized changes in the magnitude of the field with respect to distance along the axis of propagation are first assumed to be small compared to those with respect to radial displacement. The complex Helmholtz equation in cylindrical coordinates then may be simplified by separating variables and approximating these variables by truncated series expansions. The resulting difference equation determines complex field values at discrete locations in a plane from corresponding adjacent locations in the preceding plane. Stability criteria which limit both the step sizes between consecutive planes and between calculated field values in one plane are discussed. The technique has been applied to the propagation of ultrasonic fields generated by two‐dimensional focusing and non‐focusing uniform and Gaussian profile piezoelectric transducers and theoretical and experimental data are presented. [Work supported in part by NASA and NSF.]
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