I'm a biologist who has been having trouble with the above concept; I searched this forum and found some info from the speaker/audio perspective but I'm interested in animal emitters so I'm not sure how similar the concepts are.
First, based on Jakobsen et al. (2013) who quote older work I haven't gotten copies of yet (which the authors say is based on piston/baffle type models, so potentially the same assumptions which go into speaker design), the directionality of a sound beam will increase with sound frequency and with emitter size (larger bats assumed to have larger emitters; wavelength info at bottom). This means that to maintain a beam of given directionality (in black), a smaller bat will have to use higher frequencies.
I should clarify that by 'increased directionality' I mean a narrower sound beam. I have seen on speaker acoustics sites that the same term often means 'a wider beam' (reaching more of the room/audience).
Anyway, no explanation was provided as to why this is true. At first it seemed counterintuitive that a larger emitter would be more directional, but after looking at diffraction animations I got from similar questions at this site, I think it's because the size of the emitter needs to be considered relative to the wavelenth of the emitted frequency:
So, here's what I'm thinking: if a sound's wavelength is equal to or smaller than the emitter/opening, it will pass through with minimal diffraction and therefore be directional, whereas if the sound's wavelength is larger than the emitter/opening, it will diffract and spread out more (becomes less directional).
Is this a correct interpretation, or have I missed something? Any clarifications very much appreciated.
EDIT: I know about Huygen's principle, but admittedly have difficulty with it. If I imagine many points in a slit all emitting circularly, the emitters at the ends of the slit still aren't summing with waves from other point emitters at the end-side, and this is the case regardless of the size of the slit. But perhaps, since there are more point emitters in a larger slit, that means more energy is moving forward in the summed straight wave, and therefore the increased directionality is also due to greater power?