Calculation of the transient acoustic wave field emitted by a focused transducer with an arbitrary rim

Abstract

Closed-form analytic expressions are derived for the transient acoustic wave field emitted by a focused transducer with an arbitrary rim. The radiating part of the transducer is a spherical surface bounded by a simply connected closed curve of arbitrary shape. Starting from the Kirchhoff–Huygens representation of the emitted acoustic wave field, the expression for the acoustic pressure is transformed into a line integral along the rim of the transducer by employing the Maggi–Rubinowicz transformation in the Kirchhoff theory of diffraction by a black screen. The resulting line integral for the transient acoustic pressure is evaluated numerically to study the shape of the beam emitted by the transducer in its dependence on the shape of the rim and to analyze the resolving power of the transducer in ultrasonic applications. For a focused transducer with a circular rim, a closed-form analytic expression is derived for the transient acoustic pressure on its axis. These results serve as a check on the numerical results obtained for the more general cases. For all cases, the acoustic pressure at the focus admits a closed-form analytic representation. [Adrianus T. de Hoop performed this research as a Visiting Scientist with Schlumberger-Doll Research, Ridgefield, CT.]