Smaller drivers radiate more omnidirectionally at a wider frequency range

Evidence

The acoustical measurements shown are components of the spinorama: the conventional on-axis frequency response, measured at 2 m, the sound power output, and the directivity index, the difference between the two. Directivity index (DI) is 0 for an omnidirectional source, and here we see that this little 6.5-inch (165 mm) woofer approximates that quite well up to about 150 Hz. One can see the directionality progressively increase to about 7 dB at 2 kHz, above which the crossover network progressively attenuates the woofer and the tweeter takes over. Being small, the tweeter exhibits much better dispersion, about 4.5 dB around 4 kHz, and then it too becomes progressively more directional, reaching a DI around 9 dB at the highest frequencies. — Toole, Floyd E, Sound reproduction- the acoustics and…, 1478

Categorization

• DEFINE: Defines directivity index as the difference between on-axis and sound-power responses, with 0 dB meaning omnidirectional.
• XREF: Connects to typical loudspeaker design tradeoffs - baffle-step and driver-size-directivity relationships familiar from any loudspeaker design text.
• QUESTION: What is the exact driver-size-to-wavefront relationship that sets where a driver transitions from omnidirectional to directional?