Because I've seen the OEM data sheets that Eminence provides to manufacturers. Axial response only. And I know that your not going to get better than 45 degrees when the diaphragm diameter is three wavelengths. This is the age of the internet, there are no secrets, and you can find anything you want to with a few mouse clicks, including this:
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Check out page 29.
Refering to page #29 do you like discuss Fig. 2.7. the characteristics for Circular Ring Source or the math for Plane Circular Surface Source formula 2.28 ?
Note: characteristics for circular piston source are illustrated on page 30, see Fig. 2.8.
By the way a very nice book. Very nice to see state of the art in the year of 1940.
The circular piston surface source is the model to do lots of math in acoustic. But I would not claim a real loudspeaker shows exactly same characteristics like a plane piston surface source.
Sometimes I'm wondering that TB does not discuss further stuff all around acoustic science.
For the record, acoustic science starts where the speaker ends.
Sourced to the off axis considerations, as an engineer just this moment I'm considering if the sheet on top of a steinway grand piano fits the needs when opened to adjust the sound to the audience. Does the piano player really bother about off/on axis just the moment when playing? Probably not. Does this misshappen formed plank mounted on top of a grand piano really fit the needs to provide a well ballanced off axis characteristic of all the keys on a piano keyboard? I'd doubt that it's really perfect but, It's good since around 150 years, seems like there is no need to make any improvement to this old lady.
Does this all have anything to do with the subject?
IMO yes because it is related to acoustic science.
Sourced to the HT crossover,
I find it best to have a crossover for the HT set to around 4kHz. It's 4kHz because of the response of the woofers.
You may ask why? I'm playing an ampeg svt and sometimes like this midrange/treble growl when the power amp is overloaded. Crossover setting around 3kHz (or below) gives an unpleasant sound coming out from the HT. The woofers do this job in a far more pleasant way. That's why I prefere full range for the woofer (electro-mechanical LPF) and let the HT come into play where the woofer ends. Of course I know I have dispersion and comp filter effects.
I could do it the different way and make the cab perfectly working under acoustic science considerations but very likely then would not have much fun with the svt when the power amp is running into overload range. To me it's a trade off. A trade off of beeing perfectly applied to acoustic science consideration or having fun with the svt.