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# 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

Date: Mon, 31 May 2021 21:41:59 +0000
From: David Bowman <David_Bowman@georgetowncollege.edu>
To: "Phys-L@Phys-L.org" <Phys-L@Phys-L.org>
Subject: Re: [Phys-L] rms / conic / arithmetic / geometric averages
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Regarding where Dan B wrote:

... From page 214 of "The Physics of Musical Instruments"
2nd ed., the Bessel horn is defined by
S=Bx^(-2*epsilon) where

x is the geometric distance measured from the reference point
x = 0,

If epsilon = 0 you have a cylindrical horn. If x = 1 you have a
conical horn.
Dan, why does the above formula have a negative exponent rather than positive? And what is the factor of 2 in the exponent for? Is S supposed to be the surface area of the cross section at a distance x from the reference point? If S is the radius or diameter at a distance x along the tube then I don't see the point of the factor of 2 in the exponent (or the negative sign).

From page 432, applied to a brass instrument the relationship
for a bessel horn can be written

a = b(x + x0)^(-gamma)

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

b and x0 are chosen to give the correct radii at the small and
large ends of the horn and gamma defines the rate of flare.
...
Again, why isn't the above formula isn't supposed to be

a = b(x + x0)^(+gamma)

with a positive exponent?

David Bowman