U.S. Navy CBT Papers
 P. H. Rogers, and A. L. Van Buren, "New Approach to a Constant Beamwidth Transducer," J. Acous. Soc. Am., vol. 64, no. 1, pp. 38-43 (July 1978).
The theory of a broadband constant beamwidth transducer which is to be used primarily as a projector is presented. The transducer is a spherical cap of arbitrary half angle α shaded so that the normal velocity is equal to U0Pν(cos ϑ), where Pν is the Legendre function whose root of smallest angle occurs at ϑ=α. The required value for ν, the order of the Legendre function (which is not, in general, an integer) can be obtained to within 1% for α?1 radian from the approximation ν?0.5[(4.81/α)−1]. The transducer is shown to have uniform acoustic loading, extremely low sidelobes, and an essentially constant beam pattern for all frequencies above a certain cutoff frequency. Under piezoelectric drive the transducer is shown to have a flat transmitting current response over a broad band.
 J. Jarzynski and W. J. Trott, “Array Shading for Broadband Constant Directivity Transducer,” J. Acous. Soc. Am., vol. 64, no. 5, pp. 1266-1269 (Nov. 1978).
The theory of a broadband constant directivity transducer is developed. The transducer is an array of isophase, omnidirectional elements on an acoustically transparent spherical surface. It is shown that, with appropriate amplitude shading of the array elements, the beam pattern has no side lobes and the directivity is constant at all frequencies above a cutoff frequency (determined by the beam width and the radius of the sphere.) A shading function is derived, which consists of a simple linear combination of powers of cos ϑ, and several beam patterns are calculated.
 A. L. Van Buren, L. D. Luker, M. D. Jevnager, and A. C. Tims, "Experimental Constant Beamwidth Transducer," J. Acous. Soc. Am., vol. 73, no. 6, pp. 2200-2209 (June 1983).
The theory of a broadband constant beamwidth transducer (CBT) which is based on Legendre function shading of a spherical cap was described in a previous report [P. H. Rogers and A. L. Van Buren, J. Acoust. Soc. Am. 64, 38–43 (1978)]. Theoretical calculations showed the CBT to have uniform acoustic loading, extremely low side lobes, virtually no nearfield, an essentially constant beam pattern for all frequencies above a certain cutoff frequency, and a flat transmitting current response over a broadband for piezoelectric drive. In this paper we present experimental results obtained for a prototype CBT based on fifth‐order Legendre function shading. The transducer is a spherical cap with a diameter of 30.5 cm and a total cap angle of 50°. Measurements on the CBT showed it to possess very low side lobes and a nearly constant beamwidth over the frequency range from 20 to 140 kHz for ambient pressures up to 3.4 MPa. We also present formulas useful in the design of a CBT which is to be used as a transmitter and/or receiver. The formulas involve simple algebraic and trigonometric expressions which can be readily evaluated using a pocket calculator.
 K. C. Benjamin, A. K. Walden, and A. L. Van Buren, “Design and Development of a Constant Beamwidth Transducer for Sub-Bottom Acoustic Profiling,” OCEANS '97. MTS/IEEE Conference Proceedings, vol. 2, pp. 1054 – 1059, (Oct. 1997).
The design, fabrication, and acoustic calibration for a new Constant Beamwidth Transducer (CBT) is presented. Although designed for a sub-bottom profiling application, the transducer may be used whenever a spatially constant sound beam is desired over a relatively wide frequency range. The CBT design is based on the theoretical work presented earlier by Van Buren et al. (1983) and relies on an axis-symmetric velocity distribution acting over a spherically curved surface. The velocity distribution follows a Legendre shading function that is accomplished by dividing the surface electrode pattern into several discrete concentric rings. Design theory, fabrication, and measured results for a prototype transducer are presented.