This takes us back to the interesting "r^2" comment in the recent twitter spat (#913). Perry was correct in my view, to point out 1/r^2, and Matthew Ocko made quite the dick of himself trying to explain it away as Twitter-speak, a bit of a "covfefe" moment. Maybe it is Twitter-speak, but I don't see any other reference to it. Either way, clearly this spat wasn't his finest hour, not least by trying to erase it ever happened.

Engineers generally refer to r-squared loses, not 1/r^2. What Perry said makes no sense. She said the r-squared issue doesn't matter with a tightly focussed beam, but r-squared loses apply regardless of the beam width.

Ocko didn't say "R^2 losses", he said "R^2 math" and then said it was Twitter-speak. Had he said "R^2 losses" I'd agree.

On the point of the tightly focussed beam, in the far field, I agree 1/r^2 applies, but considering the aperture size, this is still in the Fresnel near field.

I don't think we know the frequency yet, but it has to be at least 40 or 50kHz. Let's say 50kHz. That means the wavelength is <7mm. How many wavelengths of near field do you think you will get? This is not a large emitter trying to produce a plane wavefront, where only the ends of the wavefront lead to divergence. Its an array trying to produce the tightest beam it can. It will diverge in an r^2 manner from a couple of wavelengths out.

You're assuming a single point source, this is a phased array, so there are interference effects dependent on both wavelength, as you say, but also the aperture size.

Consider the complex interference patterns of the ~ 1,000 element radiator, when phased appropriately there are going to be some very hot spots where the wavefronts interfere constructively by design, and where they converge it will also depend on the distance from the radiator.

Although there is a transition region where both near field and far field effects are considered relevant, the near field is typically defined as the Fraunhofer distance, 2*D^2/L where D is the largest dimension of the radiator and L is the wavelength.

Given your frequency, and a maximum linear aperture dimension of, say, 0.5m:

c=340 (speed of sound m/s)

D=0.5 (maximum aperture dimension)

f=50000 (frequency)

L=0.0068 (wavelength)

gives the extent of the near field as 73.5m, so I'd suggest for this application's use cases, and in the demonstrations given, we're well inside the near field where we should be considering interference effects well ahead of far field concerns.

Edit: My experience is in RF, it looks like acoustics refer the the near field differently, i.e. D^2/(4L). This gives a near field distance of 9.2m, so the use cases are still well within the near field with the amended definition (from

https://en.wikipedia.org/wiki/Phased_array_ultrasonics)