Spherical caps are widely used to produce efficiently focused ultrasonic fields. Due to the curvature of the source surface, no exact analytical expression of the radiated field is known. The approximate model of Williams and O’Neil [J. Acoust. Soc. Am. 17, 219-227 (1946); 21, 516–526 (1949)] is expected to be limited to high-frequency, slightly focusing sources. In order to determine more precisely the limits of that model, and to dispose of an efficient tool for describing precisely the focused field, a new numerical method is presented. It is based on two expansions of the pressure field into spherical harmonics in two conveniently chosen domains. The expansions are then matched. Validity of the method is assessed by making a comparison with the exact analytical solution for the planar case. Then, the sound field radiated by a sharply focusing radiator is investigated. Comparison of the results of the numerical procedure with the approximate model indicates a much wider domain of validity of the model than generally thought. More precise conditions for the applicability of the model are derived, that follows from simple considerations of geometrical acoustics.
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François Coulouvrat (1993) studied this question.