A set of planes parallel to the intermediate plane divides the source into a set of consecutive rings, with the center of each ring located on the dividing plane at a distance zi from the intermediate plane. The furthest point on the transducer is a distance d from the intermediate plane.
Ring–Bessel method dividing (a) solid or (b) phased-array transducer into rings of arc width ΔR. Shown between the dashed circles is the surface of the ith ring rotated onto a plane parallel to the intermediate plane. The center of the ring has radius Ri . All rings are circular and fill the entire transducer surface.
Sampling of the transducer surface into rings (left) and points (right). When the ring intersects an element of a phased array, those points on the ring are given the velocity amplitude and phase of the element; points outside an element are given values of zero. The actual number of points is much greater than shown here.
(Color online) Normalized pressure amplitudes calculated by the Ring–Bessel, Rayleigh–Sommerfeld, and Field II techniques along a center line through the focal zone in the (a) transverse plane at a distance of 13 cm from the transducer and (b) axial plane for a spherically curved solid transducer.
Face view of the 128-element phased-array transducer with randomly placed circular elements. The depth of the array in the z direction from front to rear is 2 cm.
(Color online) Upper panels: Axial slices of the magnitude of pressure through the center of focus produced by the steered phased-array transducer of Fig. 5 using (a) Rayleigh–Sommerfeld and (b) Ring–Bessel calculations from the source to the intermediate plane (at 8 cm), followed by AS propagation. Lower panels: The normalized pressure amplitude calculated by both techniques along a line through the focal zone in the (c) transverse plane and (d) axial plane for the steered phased-array transducer.
Comparison of calculation times and mean difference.
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