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

[David Bowman <David_Bowman@georgetowncollege.edu>]

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?

I suspect that Don's (volume & 'average' radius) calculation above is not for the paraboloid he may have intended.  But is rather for a paraboloid formed by revolving the 'other half' of the parabola he describes above around the y axis.  In order for his new parabola to actually describe a flared bell horn when revolved he needs to revolve the 'inside' half of the above parabola whose inverse form is: x = R_1 - (R_1 - R2)*√[(y/h)].  *That* flared paraboloid has an average radius:
R_avg = √[(R_1^2 + 2*R_1*R_2 + 3*R_2^2)/6].

But the paraboloid Don describes above having the formula:
R_avg = √[(17*R_1^2 - 14*R_1*R_2 + 3*R_2^2)/6]
is for revolving the other 'outer' half of the parabola whose inverse form is: x = R_1 + (R_1 - R2)*√[(y/h)].

Both halves, i.e. x = R_1 ± (R_1 - R2)*√[(y/h)], when solved for y = y(x), give Don's above full parabola function, i.e. y = h*((R_1 - x)/(R_1 - R2))^2.  But the revolved half having Don's above average radius formula does not make for a flared bell paraboloid.  Rather it is, again, for a blunted/bulbous one.  For both half versions the parabola's vertex is at the outer edge of bottom base (i.e. the one with radius R_1) at the coordinates (x_v, y_v) = (R_1, 0).  But the half of the parabola for which Don's above volume & average radius formulas apply actually has a top base with a radius of 2*R1 - R2 rather than his desired radius of R_2.  This makes the top even wider than the bottom (because R_1 > R_2), and again that wrong/outer half paraboloid has a blunted/bulbous shape rather than a flared one.  The top base formed from the 'inner' half of the parabola does, indeed, have a upper end radius of R_2 and has an overall flared shape, but its average radius is
R_avg = √[(R_1^2 + 2*R_1*R_2 + 3*R_2^2)/6].

I would point out that even both of Don's updated paraboloids (those formed from each of the 2 halves of the parabola) again have the parabola vertex at one of the bases where, this time, the slope of the figure is now zero (rather than infinite as in his earlier version), and again both ends of the horn are thus treated asymmetrically, and this asymmetry shows up in the average formulas.  In his latest paraboloids the parabola has essentially been turned sideways compared to his earlier paraboloid and compared to my paraboloidal horn.

When, in an earlier post, I mentioned my power law solid of revolution being flared for p > 1 what I had in mind is for the independent variable to be along the symmetry axis direction, and for the power law to open up away from that axis as a power law as one moves along that axis from the neck toward the flared bell.

(distance from axis) = (constant)*(distance along axis from sharp tip)^P

where p > 1 for a flared end (& p = 2 for my paraboloidal horn).  The vertex/singular point of the power law is *on* the symmetry axis corresponding to the sharp point zero radius tip end.  The actual sharp tip singular point is on a portion of the symmetry axis that is the cut off so the narrow end has a nonzero finite radius at the 'neck' end of the horn.  This makes the whole horn be made of regular nonsingular points of the power law/parabola where where the slope is finite and nonzero everywhere on the horn, and the corresponding formulas are symmetric in both ends (and thus don't care which end is the wide/flared end and which one is the narrow/neck end).

Don continues:

> 2) My results for the zone and segment of two bases of a sphere,
> as far as I can see, do not assume that one base is in one
> hemisphere and the second base is in the other hemisphere.
> David, if I interpret his comment correctly, seems to feel that
> the two bases must be in different hemispheres to result in the
> R shown.  My derivation does not assume this ("opposite
> hemisphere") restriction.  Numerical evaluations of my result
> also show that the two bases may be in the same or different
> hemispheres and still yield the correct equivalent R.  Also,
> when the two bases are in the same hemisphere then:
>
>                min(R_1,R_2) <= R <= max(R_1,R_2)
>
> But, I agree that this inequality need not be satisfied when the
> two bases are in different hemispheres.  However, in that case,
> we still have (at least, in the numerical case that I tried):
>
>                R < R_sphere
> Perhaps, I misunderstood David's comment about the opposite
> hemisphere requirement?

I agree that Don's formula is correct for the situation for when both ends are in the same hemisphere and for the situation for when the ends are in opposite hemispheres.  I didn't mean to require that the later situation must always be the case.  I just thought that was the situation Don had in mind because he wrote

> > This case explicitly involves the height h in the expression
> > for R.  It does check in the hemispherical case
> > (R_1 = R_sph, R_2 = 0, h = R_sph ) and full sphere case
> > (R_2 = R_1 = 0, h = 2*R_sph).

Because both of Don's given examples of a hemisphere and the whole sphere can be understood as limiting cases of having both ends in opposite hemispheres (even the hemisphere case can be understood this way because the equator can be thought of as belonging to both hemispheres) I just thought Don had that ends-in-opposite- hemispheres situation in mind.  Apparently I misread Don's intent.  Since Don's formula is also correct when both ends are in the same hemisphere, and since in that situation min(R_1, R_2) ≤ R_avg ≤ max(R_1, R_2) does indeed hold, the formula *does* have this central tendency property for *that* situation.  But in the other ends-in-opposite-hemispheres situation that property can break down, and does so maximally in the whole sphere case (R_1 = R_2 = 0, h = 2*R_sphere).

> All my results simply find the radius of a cylinder of the same
> height and same volume as the object in question.  This
> "equivalent" cylindrical radius may, or may not, agree with
> someone's definition of the "average" radius, but it will always
> produce the same volume for the same height.

True.

> Don Polvani

David Bowman