Question regarding the tuning frequency formula for flared ports

Hi everyone

I have been reading the paper "Maximzing Performance from Loudspeaker Ports" that covers the interesting benefits of flared ports. I really want to apply this knowledge to my upcoming project, but I am having trouble with the formula for the tuning frequency, and I am hoping that someone can help.

I have no experience calculating speaker enclosures or ports manually from equations. I have been using WinISD to do the calculations for me. With that said; It would be great if someone could walk me through the process of deriving the tuning frequency from the formula.

The paper can be found here and the formula can be found on page 32. If you are not familiar with the paper I suggest reading from "5. Study # 3: Modeling flared ports and prediction of tuning" on page 30 (or the whole paper, it's great) to get an understanding of the equation.

Other than generally not being able to get any sensible numbers from the formula (I have attached a screenshot of my calculations), I specifically have a question about the comment below the formula "The only difficulty in using this formula is finding the best fit flare radius to a given profile, but even this is relatively easy using the built in optimizers in most spreadsheet software". I don't get why any sort of optimisation is needed? If I define L_act and know that I want an NFR of =0.5 I can just solve for r_fit, or am I understanding it wrong?

Screenshot 2024-04-04 at 21.31.31.png
 
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