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Re: [Phys-L] rms / conic / arithmetic / geometric averages

Here is how this curve looks as a profile in inches of the last 27 inches of the tube + bell. y = 0.23 +0.002*x + b * (x+0.001 )^(-gamma)  b = 1.34606 x 10^ -29 and gamma = -20.38273
This is an instance of a positive power as David suggested.
https://imgur.com/k5Z2dR6
On Tuesday, June 1, 2021, 09:02:36 AM CDT, Brian Whatcott <betwys1@sbcglobal.net> wrote:

I played around with this expression:  y = 0.23 +0.002*x + b * (x+0.001 )^(-gamma) to reach a plausible form  at b = 1.34606 x 10^ -29 and gamma = -20.38273    On Monday, May 31, 2021, 04:42:21 PM CDT, David Bowman <david_bowman@georgetowncollege.edu> wrote:

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
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