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# 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> 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.

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

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

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> > 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>

Cc: Dan Beeker <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.

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.

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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