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Analytical velocity potential solutions for transducers with a focal length of 15 cm, operating at 2 MHz with radii of 5 cm (black) and 2 cm (red). A complex pressure subtraction of the small radius (2 cm) transducer from the large one (5 cm radius) provides an estimate of the profile for a third annular transducer (dashed line) with 5 cm outer radius and 2 cm inner hole radius. The focal peak is lower, (red arrow) and the amplitude of the first sidelobe is raised (green arrow).
