I think I am on to something.
I've opened a book
"introduction to acoustics" by Rienstra and Hirschberg. There is a section for pulsating-sphere emitter. And man it's difficult to follow. Lots of math, with heavy reliance on symbols defined in earlier sections. And no pictures or practical examples to quickly relate to.
introduction to acoustics wrote:For a compact sphere the first term is dominating. We find exactly the result which we could anticipate from (2.59), the second derivative to time of the volume of the sphere is the source of sound. ... The second term of (6.9) is dominating for large sphere radii. In such a case the action of the wall movement is that of a piston which generates plane waves.
Nontheless, I think I managed to pick and decipher the two key pieces of math, for to derive frequency response of a sphere densely covered by point loudspeaker-like sources. (and after deriving it myself, and rereading the chapter, I realized the result was actually provided in the book, I just didn't recognize it).
For how I derived it, see attached .docx file. I just state here, that I got that for a sphere densely covered by small drivers with heavy cones, the frequency response of the sphere is not going to be flat, even if the drivers are small enough to be considered point-sources, and have ideal lack of directionality.
Specifically, the response is flat up to a corner frequency fc = c/(2*pi*R), where c is speed of sound and R is the radius of the sphere. (It is exactly when the circumference of the sphere equals one wavelength.) Below that frequency, the amplitude should fall like 1/f.
For my sphere, the corner frequency is about 1100 Hz. Go figure. So I multiplied the frequency response of the driver by this sphere-factor, and here's what I got:
- theory testing.png (93.63 KiB) Viewed 5559 times
Apart from a small hiccup between 1.2-1.5k, the calculated response fits well with the measured one. From 2.8k, they deviate strongly, with real thing having much more output than expected. I would guess, this is because my drivers are too far apart, and above 2.8k they just stop collaborating to make that piston-like radiator, and behave more like a scattered set of point sources. Then probably going into beamy mode at circa 9k.
So I feel like I have a satisfactory explanation to that "BUT WHY" question.
