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*From*: "Don" <dgpolvani@gmail.com>*Date*: Mon, 31 May 2021 16:55:22 -0400

Thanks, Dan! Much appreciated! Little did I know in high school that my trumpet might be approximated by a Bessel horn. Didn't even learn about Bessel functions until sophomore year in college. I have a two-part medical appointment tomorrow with an awkward one hour pause between the two parts. I can see now what I will be doing during the awkward pause.

Don Polvani

-----Original Message-----

From: Phys-l <phys-l-bounces@mail.phys-l.org> On Behalf Of Dan Beeker via

Phys-l

Sent: Monday, May 31, 2021 3:31 PM

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

Cc: Dan Beeker <debeeker@comcast.net>

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

Fletcher and Rossing wrote the book "The Physics of Musical Instruments". It is

now somewhat dated but still holds a gold mine of information on the brass (as

well as other) musical instruments. Arthur Benade is another resource. He

developed instrumentation to measure the acoustic impedance of brass

instruments. He worked for Conn Musical Instruments and was on the faculty at

(iirc) Case Western. His book "Fundamentals of Music" is not as technical as

Fletcher and Rossing's book but both give great introductions to the acoustics of

musical instruments. Both provide a myriad of references for those truly

interested in the topic.

The goal of most any musical instrument is for the harmonics produced to mimic

the harmonics of a string fixed at both ends. Or a pipe of the same for that

matter. Only a few geometries will do this, the cone in particular. Interestingly a

bessel horn meets the acoustic requirements.

It is a handy place to start modelling a brass instrument as the mathematical

formulation is extremely flexible. 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.

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. The easiest way to come up with the

parameters is perhaps to make some measurements and try to fit the formula to

them. I've always done this by trial and error but the more mathematically

capable members of the list may come up with a more efficient method.

Acoustics of musical instruments is a fascinating subject that can quickly draw

the curious into ever deepening study. Proceed at your own peril.

Dan

On 5/31/2021 12:00 PM, phys-l-request@mail.phys-l.org wrote:

Date: Sun, 30 May 2021 15:23:18 -0400comments on my 5/29/21 post regarding the flared horn paraboloid and

From: "Don" <dgpolvani@gmail.com>

To: <Phys-L@Phys-L.org>

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

Message-ID: <000801d75589$3f2c0910$bd841b30$@gmail.com>

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

Thanks again to David Bowman for his (5/29/21) clear, detailed, and helpful

spherical segment equivalent cylindrical radii.

correct despite the ambiguities of the English language.

1) Flared Horn

David is quite right. I neglected the minus sign option when taking

the square root and ended up mistakenly revolving the outer side of

the parabola about the y-axis (which produces a bulbous horn) instead

of the inner side (which produces a flared horn). Again, a plot of my

expression for x as a function of y would have shown this error (I did

plot y vs x but not my chosen x vs y) I agree with his results for

the equivalent cylindrical radius (R) for the flared horn formed by

the inner side of my chosen parabola. (See David's response below for

my notation and more expressions)

R = sqrt((R_1^2 + 2*R_1*R_2 + 3*R_2^2)/6)

2) Spherical Segment with Two Bases

I'm glad that we have reached consensus that my results of 5/24/21 are

of the flared bell of a trumpet. I have found internet articles for bell sizes, bore

Finally, I have not been able to find a mathematical expression for the shape

sizes, and how trumpets are made but no mathematical expression for the

shape of a trumpet bell. If any of you know of such an expression, I would be

most appreciative if you would send it to me.

_______________________________________________

Don Polvani

-----Original Message-----

From: Phys-l <phys-l-bounces@mail.phys-l.org> On Behalf Of David

Bowman

Sent: Saturday, May 29, 2021 10:23 PM

To: Phys-L@Phys-L.org

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

Regarding Don P's 'average' radius calculation for his new updated

paraboloid of revolution:

1) My results for a "proper" flared (parabolic) horn (similar to a

trumpet bell) are as follows:

Parabola : y = a*(x - c)^2

a = h/(R_1 - R_2)^2

c = R_1

V = pi*h*((17*R_1^2 - 14*R_1*R_2 + 3*R_2^2)/6)

R = sqr((17*R_1^2 - 14*R_1*R_2 + 3*R_2^2)/6)

Where: h = vertical height between bottom and top of the paraboloid

of revolution

R_1 = radius (larger) of bottom base

R_2 = radius (smaller) of top base

V= volume of paraboloid of revolution

R = radius of equivalent cylinder with same

height and same volume

As a high school trumpet player (long, long ago!), I am curious if

anyone on the list knows the exact mathematical shape of a trumpet

bell?

Forum for Physics Educators

Phys-l@mail.phys-l.org

https://www.phys-l.org/mailman/listinfo/phys-l

**References**:**Re: [Phys-L] rms / conic / arithmetic / geometric averages***From:*Dan Beeker <debeeker@comcast.net>

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