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[Phys-L] FW: Bessel Horn

From: Don <dgpolvani@gmail.com>
Sent: Thursday, June 3, 2021 2:29 PM
To: 'Brian Whatcott' <betwys1@sbcglobal.net>
Subject: RE: [Phys-L] Bessel Horn



Brian,



I submit that for a Bessel horn (as in my post of this morning):



V = (pi/(2*gamma - 1))*(2*x_0*(y_0^2 - y_1^2) - L*y_1^2)



R = sqrt((2*x_0*(y_0^2 - y_1^2) - L*y_1^2)/(L*(2*gamma -1)))



is considerably simpler both to check and evaluate than (my posted results of yesterday):



V = (pi*b^2/(2*gamma - 1))*(1/(2*x_0)^(2*gamma - 1) - 1/(2*x_0 + L)^(2*gamma - 1))



R = sqrt(( b^2/(L*(2*gamma - 1)))*(1/(2*x_0)^(2*gamma - 1) - 1/(2*x_0 + L)^(2*gamma - 1)))



It continues to be unfortunate that on this list we can’t use standard mathematical symbols and notation. If we could, the difference would seem more dramatic - just as it appears to me on my yellow lined pad. I belong to that passing generation of physicists who tried very hard to get expressions in simplest terms due principally to the labor involved in numerical evaluations (before easy access to personal PCs) and the increased ease of checking and interpreting the results. In graduate school, the personal computer was just being invented. When I started at Westinghouse in 1969, the engineering department had one HP “personal computer” for 300 engineers. It had an 80-character display and a separate x-y plotter to plot results, We used a sign-up sheet to schedule 30 min sessions on the computer. It paid big dividends in saved time to have your expressions thoroughly checked for correctness and easy to enter into the computer. This was facilitated by having your expressions in simplest possible terms.



Don Polvani



From: Brian Whatcott <betwys1@sbcglobal.net <mailto:betwys1@sbcglobal.net> >
Sent: Thursday, June 3, 2021 12:42 PM
To: phys-l@phys-l.org <mailto:phys-l@phys-l.org>
Cc: Don <dgpolvani@gmail.com <mailto:dgpolvani@gmail.com> >
Subject: Re: [Phys-L] Bessel Horn



Don proposed a simpler formulation for the Bessel horn.



Not sure that I accept that ~



V = (pi*b^2/(2*gamma - 1))*(1/(2*x_0)^(2*gamma - 1) - 1/(2*x_0 + L

)^(2*gamma - 1))

This yields an equivalent cylindrical radius (R) for a cylinder of the same

length (L)

R = sqrt(( b^2/(L*(2*gamma - 1)))*(1/(2*x_0)^(2*gamma - 1) - 1/(2*x_0 + L

)^(2*gamma - 1)))



~ is simpler than the original formulation! <g>



I recall that when Oliphant's group at Birmingham U were developing the high-power magnetron

they used acoustic analogies for microwave transmission lines, quarter wave stubs, circulators etc.

In this spirit, I suggest that the essence of coupling the mouthpiece of the trumpet to the bell is

a matter of providing a smooth slope for the characteristic impedance of the sound path until it is

high enough to couple efficiently to the impedance of free air. I note that musicians say that

minor differences in bore size affect the "mellowness" versus "brightness" so there is a filtration

of overtones in effect too.

In waveguides the cross section dimensions seem to be more important than volume.

Against that speculation, I note that trumpets use constant diameter tubing for part of their sound path,

while cornets are conical, end to end, apparently.

On Thursday, June 3, 2021, 10:37:44 AM CDT, Don via Phys-l <phys-l@mail.phys-l.org <mailto:phys-l@mail.phys-l.org> > wrote:





I believe the results I presented for the Bessel horn in my previous post

are correct (see below for previous post). However, they can be

considerably simplified. In thinking about checking the dimensions of the

horn volume and equivalent cylindrical radius, I realized the following.

Start with the Bessel horn radius (y) from Dan Beeker:







y = b*(x + x_0)^-gamma (where I assume, as Dan

now says, that the bell end occurs at x = x_0)



y_0 = b*(2*x_0)^-gamma (radius of the bell = 2.25

in for average trumpet)



y_1 = b*(2*x_0 + L)^-gamma (radius of the bore = 0.23 in

for average trumpet, L = positive length along x-axis between bell and bore

radii = about 10 in as my estimate)







Then the volume of the Bessel horn rotated about the x-axis between the bell

and bore radii and the equivalent cylindrical radius (R) for this same

volume are given more simply by:







V = (pi/(2*gamma - 1))*(2*x_0*(y_0^2 - y_1^2) - L*y_1^2)



R = sqrt(V/(pi*L)) = sqrt((2*x_0*(y_0^2 - y_1^2) - L*y_1^2)/(L*(2*gamma -

1)))







Using the example values of my post below:







x_0 = 1 in, y_0 = 2.25 in, y_1 = 0.23 in, L = 10 in, gamma = 1.272831







I get the same volume and equivalent radius reported below:







V = 6.139897*pi in^3



R = 0.783575 in







But the dimensions of V and R are a lot easier to check, and the numbers are

a lot easier to compute!







Don Polvani







From: Don <dgpolvani@gmail.com <mailto:dgpolvani@gmail.com> >

Sent: Wednesday, June 2, 2021 7:45 PM

To: Phys-L@Phys-L.org <mailto:Phys-L@Phys-L.org>

Subject: RE: [Phys-L] Phys-l Digest, Vol 198, Issue 1







For the Bessel horn, I think the resolution to the +/- sign of epsilon (aka

gamma) has simply to do with the relative positions of the trumpet bell and

trumpet bore along the x-axis. For the bell closer to the origin than the

bore, the minus sign is correct. For the bell farther from the origin than

the bore, the plus sign is correct. I am defining the "bore" as that radius

(smaller than the bell radius) which stays "constant" throughout the rest of

the trumpet.







I have used (as Dan Beeker originally stated):







y = b*(x + x_0)^-gamma







This produced a nice "trumpet-like" curve (with the bell closer than the

bore to the origin). I searched the internet for typical trumpet parameters

and found:







Average bell radius = 2.25 in



Average bore radius = 0.23 in







I started the bell at x_0 = 1 in and assumed it took 10 in ( length L) to

reach the smaller bore radius.







This yielded the fitted parameters (fit at the bell and bore radii):







b = 5.436792



gamma = 1.272831







I then rotated the trumpet about the x-axis to find the volume (V) contained

between the bell and bore separated by L along the x-axis







V = (pi*b^2/(2*gamma - 1))*(1/(2*x_0)^(2*gamma - 1) - 1/(2*x_0 + L

)^(2*gamma - 1))







This yields an equivalent cylindrical radius (R) for a cylinder of the same

length (L)







R = sqrt(( b^2/(L*(2*gamma - 1)))*(1/(2*x_0)^(2*gamma - 1) - 1/(2*x_0 + L

)^(2*gamma - 1)))







Using the above numerical parameters, this yields:







V = 6.139897*pi in^3



R = 0.783575 in (so, bore radius <= R <= bell radius)







Thanks again to Dan Beeker for providing the Bessel horn information.







Don Polvani







-----Original Message-----



From: Phys-l <phys-l-bounces@mail.phys-l.org <mailto:phys-l-bounces@mail.phys-l.org>

<mailto:phys-l-bounces@mail.phys-l.org <mailto:phys-l-bounces@mail.phys-l.org> > > On Behalf Of Dan Beeker via



Phys-l



Sent: Tuesday, June 1, 2021 9:20 PM



To: phys-l@mail.phys-l.org <mailto:phys-l@mail.phys-l.org> <mailto:phys-l@mail.phys-l.org <mailto:phys-l@mail.phys-l.org> >



Cc: Dan Beeker <debeeker@comcast.net <mailto:debeeker@comcast.net> <mailto:debeeker@comcast.net <mailto:debeeker@comcast.net> > >



Subject: Re: [Phys-L] Phys-l Digest, Vol 198, Issue 1







David,







I should have been more explicit.







1. You are correct. S is the cross sectional area. In terms of radius



(a) the formula would be







a = b(x^-epsilon) = b/(x^epsilon)







The factor of 2 in th formula for S comes because the area is (pi)r^2.



Sub a into the area formula.







area S = (pi)r^2 = (pi)a^2 = pi[b/x^epsilon]^2 = pi[b^2/x^2epsilon]







pi*B^2 morphs into the constant b.







2. As far as whether the exponent epsilon is positive or negative, I'll

have to



ponder this. I think epsilon should be positive so the function gets

smaller as one



moves away from the large end of the bell.







3. And I see a typo in







If epsilon = 0 you have a cylindrical horn. If x = 1 you have a conical

horn.







Should read:



If epsilon = 0 you have a cylindrical horn. If epsilon = 1 you have a

conical horn.







4. And I believe I made another typo which may explain the confusion.







where a is the bore radius of the horn and x0 is the small end of the

horn.







Should read



where a is the bore radius of the horn and x0 is the large end of the

horn.







So embarrassing.







Dan



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