Horn size physically constrains the frequency range of directional coverage

Evidence

As an example, if one wanted a constant directivity horn that had a cover­ age of 90°  40° from 500 Hz then the mouth would have to be at least 0.62 m for the 90° direction and 1.28 m for the 40° direction. This is not small! You can have a smaller horn but you must recognize that it will not have this coverage angle at 500 Hz. The sine of the 6 db  point is inversely proportional to the frequency below Fθmin; thus at an octave below Fθmin  will be approximately double that desired, and so on, until  becomes greater than 90°, which implies the speaker is omnidirectional and has no directivity. — David Howard, Jamie Angus, coustics and Psychoacoustics, 2495

Categorization

• QUESTION: The excerpt ends mid-explanation of how directivity degrades below Fθmin — the octave-down doubling is described in the following paragraph. What is the exact mathematical relationship with the mouth dimension?