# Directionality of sound depending on emitter size and wavelength

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?

• Yes, your interpretation is exactly right. Of course, its a continuum of directionality improvement as the emitter size grows compared to the wavelength (although certainly not linear). – Digiproc Jan 25 '17 at 17:21

Your argument is correct, though the bat presumably emits sound through an orifice, i.e. a 2D surface, rather than a slit. However the sound diffracts in basically the same was as for a slit.

If we make the usual physicists assumption that everything is spherical, or in this case a circular disk, the diffraction pattern created by a circular orifice is an Airy disk. If we take the first minimum to give the radius of the beam of sound then the half-angle of the beam is given by:

$$\sin\theta \approx 1.22 \frac{\lambda}{d}$$

where $\lambda$ is the wavelength and $d$ is the diameter of the orifice.

• thanks. In the click-through to "Airy disk" I don't see mention of Huygen's principle, but I assume the rings are caused by destructive interference? I know assumptions always oversimplify, but the diffraction pattern seems very different from what are typically described as two 'lobes' on the sides of a bat's sound beam, something I also don't understand. Are there infinite lobes, but the others further out from the main beam are just weak due to more destructive interference? airbattle.co.uk/Assets/ai_radar_1.jpg – Bzrs Jan 25 '17 at 17:33
• @Bzrs: the sound emitted through a circular orifice would propagate out as a cone with the half-angle I gave in my answer. The sound intensity profile is the Fourier transform of the orifice - the Huygen's construction is a mathematical technique for calculating this. I know nothing about bat echo location, but maybe the orifice they use isn't circular, or do they project though two orifices e.g. both nostrils? – John Rennie Jan 25 '17 at 17:37
• there are both oral emitters and nasal emitters among bats--the nasal emitters have large nose leaves or complicated facial folds that are hypothesized to further shape the sound beam. I have never considered Fourier transforms for anything other than sounds, so FT of an orifice is a new concept. Thanks for your answer! – Bzrs Jan 25 '17 at 17:51
• @Bzrs: see this Wikipedia article though it may go into more detail than you want. – John Rennie Jan 25 '17 at 18:01
• Claim on Skeptics.SE: "Motorcycle exhausts openings are facing towards the rear of the bike, and it's obviously to the back where the gases and all the noise are directed. Assuming that the noise a motorcycle makes travels in an omnidirectional manner is just wrong, because noise is air (or other gases, for what's worth) in movement. With the air/ gas jest directed towards the rear of the bike, it's there where all the noise goes." – Keith McClary Apr 5 at 4:29