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

["Don" <dgpolvani@gmail.com>]

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> On Behalf Of Dan Beeker via

> Phys-l

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

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

> Cc: Dan Beeker <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