The geometrical definition of the double curved transducer.
Geometrical definition of the angle β.
The integration angles , , , and , which are determined by the spherical wave passing the boundaries of the transducer. The transducer is seen from the back side toward the positive z-axis.
A sphere’s crossing with a transducer (solid line) and virtual crossings (dotted line). View seen from the transducer’s back side in the direction of the z-axis. (a) The sphere has not yet crossed the side edges. The sphere is symmetrically placed at xp = 0. (b) The sphere has crossed the edges. Dotted lines indicate spherical waves outside the active aperture. The sphere center is offset to the left, which makes the left virtual arc significantly longer than the right virtual arc. See supplemental material (Ref. 27).
The integration angles plotted for a point located in zone 1 of a transducer. The angles show that the wave crosses the left edge before it crosses the right. Start and end times are and , respectively.
The curve shape of the integrand in Eq. (56) at different time steps. It is clearly seen how the curve increases asymptotic forward infinity at φ min and φ max. Notice that a full symmetric case is shown for the plot. Symmetry is always the case, however, depending on the value of φ min and φ max one or both spikes at the start and end of the integration domain may not be present.
(a) A comparison between P(θ) and T(θ) at two time instants, T1 and T2, where T1 < T2. (b) The difference between P(θ) and T(θ) at the two time instants. Notice how φ min and φ max include a wider angle difference for T2 and how the error has increased significantly.
Results of simulating a single point in front of a double curved transducer. (a) Full pulse profile. (b) Zoom onto (a) to magnify the difference.
Results of simulating a sharp spiking spatial impulse response from a double curved transducer. E field = 0.49%, E 1T = 3.580%, E 3T = 0.006%, and E 2p = 0.83%. Only a few data points are shown from each curve. (a) Full pulse profile. (b) Zoom onto (a) to magnify the difference.
Results of simulating a convex nonelevation focused array.
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