Chronology |
Current Month |
Current Thread |
Current Date |

[Year List] [Month List (current year)] | [Date Index] [Thread Index] | [Thread Prev] [Thread Next] | [Date Prev] [Date Next] |

*From*: Dan Beeker <debeeker@comcast.net>*Date*: Tue, 1 Jun 2021 21:19:57 -0400

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

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

Message-ID:

<BL0PR04MB49136282A3225376A93E2518ED3F9@BL0PR04MB4913.namprd04.prod.outlook.com>

Content-Type: text/plain; charset="utf-8"

Regarding where Dan B wrote:

... From page 214 of "The Physics of Musical Instruments"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).

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.

From page 432, applied to a brass instrument the relationshipAgain, why isn't the above formula isn't supposed to be

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.

...

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

with a positive exponent?

David Bowman

**Follow-Ups**:**Re: [Phys-L] Phys-l Digest, Vol 198, Issue 1***From:*"Don" <dgpolvani@gmail.com>

- Prev by Date:
**Re: [Phys-L] rms / conic / arithmetic / geometric averages** - Next by Date:
**[Phys-L] Impossible vehicles.** - Previous by thread:
**Re: [Phys-L] rms / conic / arithmetic / geometric averages** - Next by thread:
**Re: [Phys-L] Phys-l Digest, Vol 198, Issue 1** - Index(es):