Introductory notes:
The portion of a trombone known as its bell is a variable radius flared tube horn.
It has a straight axis of symmetry, here to be placed along the x-axis.
The bell may be modeled by revolve, of one Bessel curve, or several sequential Bessel curves.
The length of the bell can be 502 mm.
The throat of the bell can have a radius of about 10 mm.
The mouth of the bell can have a radius of about 108 mm.
The bell flare constant, gamma, could be 0.7, for a single Bessel curve, a good match to an actual bell.
The Bessel curve of link1 has a fitting parameter b = 0.0063, which makes the dimensions come out well.

https://www.researchgate.net/publication/220058446_Trombone_Synthesis_by_Model_and_Measurement

There is a lot of complicated calculus math for sound vibration, but the equation for the resulting Bessel curve is a simple equation.

Link2 is a thesis of 175 pages, and link3, based on link2, produces 4 Bessel curve segments, and a 3D printable model of a trombone.
An actual trombone bell would have several Bessel curve segments, which is said to be related to construction mandrels, historically used to produce an actual trombone bell.
Optimum sound vibration calculus may require several Bessel curves.
For a bell created from several Bessel curves, each curve would have its own gamma value, with sequential radii.
Some parameters can be calculated from each other...

http://www.acoustics.ed.ac.uk/wp-content/uploads/Theses/Braden_Alistair__PhDThesis_UniversityOfEdinburgh_2006.pdf

https://github.com/pieterbos/PrintBone

A script or node to produce Bessel curves is intended.

- Brian

From a a person with trombones and tubas in their youth, THANKS!!!

That is some serious work.

I was fortunate enough to have attended a high school that inherited experimental brass instruments from the military bands. So, while I have to admit my math could not keep up, the discussions of the outcomes of traditional methods of creating experimental instrument brought back some memories of the WIDE range of performance of the various instruments we were provided. (Obviously, we got the not-quite-rousingly-successful experiments.)

That was a herculean effort and it was cool to be introduced to it!

Tom

Here is an ALPHA version of the BesselHorn curve, as a Script. A Node could be done.
It seems to work fine in MoI Beta 5 as well as MoI4.
Clicking on a parameter label restores its default value. (Thanks to Max for technique.)
cm was used for the Units.

Modification was made to the EggOvoid script. Egg build options need to be removed :-)

Some minor cleanup needs to be done.
Limited testing done.
It may be desire-able to modify the Limits of the sliders...
Error catching for weird parameter values is not done in the script. Have not had any crash so far.
Had to "relearn" Alert coding. (Alerts are commented out.) (The Alert would not show 1.0, just 1)
Had to "relearn" method of establishing input xValue range. Method shown in comments.

- Brian

The script .js and .htm files should be placed in the MoI commands folder in AppData.
(I use Max's customUI to run such scripts.)

Revolve of the profile curve is left up to the user of the script :-)

Hi Brian,
Still working on Pyriform and avianform curve?? Those options don't do anything yet...

Hi Burr,

So far I left the two lines of the build options, in the event I decided to have the script perform some other curves.
Exponential horn is a possibility, but there is only ONE flare value for a given set of r0, r1, and length, so a flare slider makes no sense.
Maybe some semi-fancy coding could make flare parameter and flare slider not show up for exponential...

So that is one reason why the name Alpha was used.
Some minor code modification and deleting the Build options would eliminate the leftover build options.

-Brian

Trombone Bell:

Version 3 of BesselHorn curve script has some minor changes.

1. Added Conical horn curve line. It is just a line, which revolves to a cone, or frustum.
Flare slider has no effect.

2. Added exponential horn curve. It is calculated with positive x values, and so lies to the right of the y-axis, automatically.
There is only one exponential curve between the two endpoints, so flare slider has no effect.
eFlare is the calculated flare for the exponential curve, for given parameters.

3. Shifted Bessel and Conical curves to right of y-axis. So all 3 curves have the same endpoints, for given set of parameters,
and there is less jumping around with changing parameters.

In the event a multi curve horn is created, the multiple curves would need to be placed sequentially, end to end.
A nodeeditor version used multiple times, each with its own parameters, might permit multiple joined segments, in a single node program. (Or multiplex mode...)

- Brian

Link to 18 equations:

http://www.audioheritage.org/vbulletin/showthread.php?18435-Geometrical-Exponential-Horn-Calculations

- Brian