Delay-induced spherical wavefronts plus Legendre weighting yield frequency-independent beamwidth.

Evidence

These results are important because they also provide a link between array loudspeakers and constant-directivity horns. Figure A6.9, reproduced from Keele, 2002, shows how the delays for a planar CBT array effectively move the driver from its position on a flat surface to a point on a circular arc. That is, it provides a delay that makes the wavefront at the planar array that is spherical. According to the Fourier theory we have described earlier, the Fourier transform of the combination of phase shifts, due to a spherical wavefront and Legendre weighting, results in a frequency independent constant beamwidth above a certain cutoff frequency. Furthermore, the cutoff frequency is a function of both the required directivity angle and the length of the array. — David M. Howard, Jamie Angus, Acoustics and Psychoacoustics, 2919

These results are important because they also provide a link between array loudspeakers and constant directivity horns. Figure A6.9, reproduced from Keele (2002), shows how the delays for a planar CBT array effectively move the driver from its position on a flat surface to a point on a circular arc; that is, it provides a delay that makes a wavefront at the planar array that is spherical. According to the Fourier theory described earlier, the Fourier transform of the combination of phase shifts—due to a spherical wavefront and Legendre weighting—results in a frequency-independent constant beam­ width above a certain cut-off frequency. Furthermore, the cut-off frequency is a function of both the required directivity angle and the length of the array. — David Howard, Jamie Angus, coustics and Psychoacoustics, 2903

Categorization

• XREF: This is the payoff of the earlier Fourier-transform treatment promised in prior chapters — directivity becomes a Fourier relationship, the book’s core through-line.
• DEEPTHINKING: The claim that directivity is a frequency-independent property once you engineer the right wavefront shape is a genuine paradigm shift for how one thinks about beamforming.
• DEFINE: Spells out the causal chain — phase shifts from a spherical wavefront combined with Legendre weighting, Fourier-transformed, produce constant beamwidth.
• corroborated: 2 independent passages support this claim — David M. Howard, Jamie Angus, Acoustics and Psychoacoustics, 2919; David Howard, Jamie Angus, coustics and Psychoacoustics, 2903