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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 -0400
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 comments on my 5/29/21 post regarding the flared horn paraboloid and spherical segment equivalent cylindrical radii.

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 correct despite the ambiguities of the English language.

Finally, I have not been able to find a mathematical expression for the shape of the flared bell of a trumpet. I have found internet articles for bell sizes, bore 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?