Why not just use the Hornresp Horn Segment Wizard to find the required volumes?

In the attachment, S1 is the area at the throat of a segment, L12 is the length of the segment, F12 is the horn flare cutoff frequency and T is the horn flare parameter (0.5 in your case).

Why is it different, if the original T is also 0.5?

The Horn Segment Wizard results would suggest otherwise

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Hornresp does the same thing that TL.app does when splitting the hyp/ex horn into segments. The individual segments do not add up to equal the original hyp/ex flare.

This is really difficult to explain so I'll attempt to do it in a way that can be followed along with pics.

First I'll start with a really big hypex horn with T=0.5. I'll use 100 for S1 and 10000 for S2 and 500 for L12 so we can use even numbers and get some dramatic results.

In image 1 you can see that TL.app calculates the volume of that flare to be exactly 976.1445 liters, circled in red. Now let's skip down to image 3 and Hornresp calculates the exact same volume for this flare, circled in red.

NOW we split the horn up and start getting into some real trouble. In image 2 I told TL.app to split the horn in half. Note that the flare total volume (both segment 1 and segment 2 added together) has changed, circled in red. It went from 976.1445 liters to 904.08011 liters. The only thing that changed was that I split it in half. Now I have two flares, each with T=0.5 and they do not add up to the same volume as the original unsplit T=0.5 flare. Everything else is the same, throat area, mouth area and length.

Also note that in image 2 the cross sectional area at the split is 786.56 circled in green and the first segment volume is 76.803238, circled in orange.

Now let's begin the split procedure in Hornresp. Image 4 (in text) shows the Horn Data for the

**original unsplit** flare. (One segment, S1=100, S2=10000, L12=500). I only need to know the volume at the exact midpoint, so I used 250 cm segment lengths. The cross sectional area at the midpoint is reported by Hornresp Horn Data as 786.560736 sq cm, circled in green. Now look back at image 2, the area at the end of the first segment (the midpoint) is 786.56, circled in green. So both TL.app and Hornresp agree that the cross sectional area at the midpoint of the flare should be 786.56.

Unfortunately that's where the similarities to the original flare end. Let's continue the split procedure in Hornresp. Referring now to image 4 (right after the text Horn Data) I've specified the inputs for the first flare, S1=100, S2=786.56, L12=250 and T=0.5. Hornresp reports the volume for that segment as 76.803 liters, circled in orange. Look at image 2 again, circled in orange the first segment is 76.803238. So both Hornresp and TL.app agree on the volume of the first segment, but this is not correct compared to the original unsplit flare.

(This is not shown anywhere in the images but TL.app says the volume of the second segment in image 2 is 827.27687. Hornresp specifies the volume of a S1=786.56, S2=10000, L12=250 and T-0.5 segment as 827.277 liters. So again, TL.app and Hornresp agree that the second segment is the same volume, but again this is not correct compared to the original unsplit flare.)

If we take the two split segment volumes (Hornresp and TL.app both agree that they are 76.803 and 827.277 liters respectively), we end up with 904.08. Now look back at image 2, circled in red the total SPLIT flare volume is 904.08.

The volume of the SPLIT flare (904.08) is NOT the same as the volume of the original UNSPLIT flare (976.145). Not in TL.app and not in Hornresp.

As I said before the reason is because two back to back T=0.5 flares do not equal a single T=0.5 flare, and I'm pretty sure that is correct. EACH T=0.5 flare is increasing hyperbolic towards the end of the segment, so two back to back flares start out somewhat hyperbolic at the beginning of the first flare, then get more and more hyperbolic approaching the end of the first segment, then at the start of the second segment we are again at just somewhat hyperbolic and getting increasingly more hyperbolic towards the end of the second flare. Imagine two LeCleach flares back to back, they would not equal a single LeCleach flare in the same way that two back to back 0.5 T flares do not equal a single 0.5 T flare. This LeCleach example is easier to visual because at the end of each segment the LeCleach flare begins to flare so much that it folds backwards so if you put two LeCleach flares back to back it would not equal a single LeCleach flare.

Now let's try this another way, instead of letting Hornresp calculate F12 like I just did, we'll let it calculate AT like you did in the attachment picture.

In this case we need to specify F12, so let's go back to the original unsplit flare in image 3. Hornresp calculates the original flare as 28.34. So referring to images 6 and 7, I've input the start areas of the two segments, the length, the flare T=0.5 and F12=28.34. Clearly this is even more wrong than the previous method, as in both cases the calculated mouth area AND segment volume is too low.

So clearly we would have to use different F12 and/or different flare T for the two segments to equal a single 0.5 T flare. But at this point it's getting beyond me and I could fool around with different F12 and different flare T until the volumes match up but if I can't prove they are mathematically the same as a single 0.5 T flare then it probably won't be.

if you are not able to blow this picture up enough to read clearly the original link is here.

Image - TinyPic - Free Image Hosting, Photo Sharing & Video Hosting
If you still can't blow it up enough with that link to see it clearly I'll host it somewhere else, let me know.

EDIT - I attached the pic to the bottom of the post as an attachment as well, that should allow it to be blown up to full size so it should be legible.