Is there an ideal back loaded horn shape?

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Now that oblate sphéroïde waveguides have brought up the topic of polar response in a two way, I was wondering if there is an idea shape and response to a backloaded horn.

I am also curious if smooth curves in a backloaded horn have better performance, as opposed to speakers where forms are shaped by multiple straight cuts of wood.
 
The ideal flare & response is the one which meets the desired design criteria.

Depends what you call 'better.' A continuous flare promotes higher efficiency across the full operating BW of the horn. Whether you actually want that in practice with a back-load is another matter entirely. LF wavelengths are too long to be significantly affected; the effects are more noticable as frequency increases and wavelength decreases. Since, generally speaking, you want to attenuate output from a back-horn above a certain point (250 - 300Hz, give or take), this is not especially beneficial from my POV.
 
Well tractrix is the one with most efficiency for a given length, but has the most honky horn sound of the normally used expansions (the one that comes closest to the old mechanical gramophone horns).
Linear expansion has long been established to be the best from a sound quality PoW. Virtually no colouration within a given frequency window. It just takes longer to reach as deep as the other possible expansions.
Exponential expansion is somewhere in between these two with regards to length vs. effectiveness ratio.

But as Scottmoose said, with backloaded horns the expansion type has little influence on the sound. Even parabolic expansion, which would be bad for a front loaded horn, can work well with the longer frequencies. Just look at all the folded tube horns, and even TQWTs that are sometimes closer to parabolic than linear.
 
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