A Quesstion for the Theoreticians

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The Ghost - 15 Oct 2005 01:07 GMT

I am looking for an analysis of the frequencies of radial modes in a
circular cavity (room). The specific case of interest to me is one in
which the ratio of cavity height to cavity diameter is approximately 0.1,
and the highest frequency of interest is such that the maximum ratio of
cavity diameter to wavelength is on the order of 2.0. The height of the
cavity is much smaller than a quarter wavelength at all frequencies of
interest.

More specifically, I am interested in an exact formula for the frequency of
the first minimum in acoustic impedance looking radially from a point at
the center of the cavity. The acoustic impedance at all of the cavity
walls can be considered to be infinite.

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salmonegg@sbcglobal.net - 15 Oct 2005 01:21 GMT

On 10/14/05 5:07 PM, in article
aLX3f.205270$nF6.171703@fe04.news.easynews.com, "The Ghost"
<theghost@hotmail.com> wrote:

> I am looking for an analysis of the frequencies of radial modes in a
> circular cavity (room). The specific case of interest to me is one in
[quoted text clipped - 8 lines]
> the center of the cavity. The acoustic impedance at all of the cavity
> walls can be considered to be infinite.

I have some difficulty understanding exactly what you want to know.

Nevertheless, a cylindrical cavity of the kind you describe is easily
covered by partial differential equation that are separable by standard
techniques. There probably are formulas in various handbooks.

That said, I cannot imagine any room actually being a true resonator unless
it has hard surfaces made from concrete or the like.

Bill

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The Ghost - 15 Oct 2005 02:10 GMT

> On 10/14/05 5:07 PM, in article
> aLX3f.205270$nF6.171703@fe04.news.easynews.com, "The Ghost"
[quoted text clipped - 18 lines]
> covered by partial differential equation that are separable by
> standard techniques.

In business-related R&D time is of the essence. Accordingly, if the
analysis has already been done, I am looking to find it rather than waste
my time re-inventing the wheel.

> There probably are formulas in various handbooks.

I am open to and grateful for any/all specific references that you are able
to provide.

> That said, I cannot imagine any room actually being a true resonator
> unless it has hard surfaces made from concrete or the like.

The cavity walls are hard metal (brass/aluminum/steel).

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Greg Locock - 15 Oct 2005 02:43 GMT

>> On 10/14/05 5:07 PM, in article
>> aLX3f.205270$nF6.171703@fe04.news.easynews.com, "The Ghost"
[quoted text clipped - 13 lines]
>>> impedance at all of the cavity walls can be considered to be
>>> infinite.

Check out RD Blevins "Natural Frequencies and Mode shapes". I don't know
if he has that specific case, my copy is elsewhere, but it's always a
good place to start.

Cheers

Greg

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Angelo Campanella - 15 Oct 2005 21:48 GMT

>>I am looking for an analysis of the frequencies of radial modes in a
>>circular cavity (room). The specific case of interest to me is one in
[quoted text clipped - 3 lines]
>>cavity is much smaller than a quarter wavelength at all frequencies of
>>interest.

Morse "Vibrations and Sound", pages 398-401 details cylindrical modes in
rooms. Where N is the radial mode number and M is the lateral mode
number, the following table provides some of the "harmonics"(?) that arise:

N=    0    1    2    3
M:
0    0    1.217    2.233    3.238
1    o.586    1.697    2.714    3.726
2    o.972    2.135    3.173    4.192
3    1.337    2.551    3.612    4.643
4    1.693    2.955    4.037    5.082

>>More specifically, I am interested in an exact formula for the
>>frequency of the first minimum in acoustic impedance looking radially
>>from a point at the center of the cavity. The acoustic impedance at
>>all of the cavity walls can be considered to be infinite.

I'm afraid the formula for the sound pressure is too complicated to type
here. It is best that you get a copy of Morse's pages. The same formulae
likely can be found in Morse's later texts.

Angelo Campanella

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The Ghost - 15 Oct 2005 23:56 GMT

>>>I am looking for an analysis of the frequencies of radial modes in a
>>>circular cavity (room). The specific case of interest to me is one
[quoted text clipped - 29 lines]
>
> Angelo Campanella

Thanks,Ang. I found it myself earlier today. Unfortunately I previously
checked Theoretical Acoustics (Morse & Ingard), and when I did not find it
there I assumed (incorrectly) that it would not be in Vibration & Sound.

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Robert A. Hedeen - 15 Oct 2005 03:16 GMT

>I am looking for an analysis of the frequencies of radial modes in a
>circular cavity (room). The specific case of interest to me is one in
[quoted text clipped - 8 lines]
>the center of the cavity. The acoustic impedance at all of the cavity
>walls can be considered to be infinite.

I don't have my copy of Blevins here (I generally leave that in the
office on weekends), but I DO have Beranek & Ver's Vibration and Noise
Control Engineering handy, and whaddya know, they reproduce part of
Blevins' table "Acoustical Modes & Natural Frequencies" in their
chapter on Sound in Small Enclosures.

Ghost, for a closed cylindrical volume it looks like the formula you
want is f = c G / 2 pi r, where G is a frequency-like parameter equal
to 1.8412 for the first mode. (There is an outside possibility that G
= 3.8317 because this truncated version of the table does not make it
clear which of Blevins' indices refers to the radial modes).

Robert A. Hedeen
GE Global Research
Niskayuna, NY

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The Ghost - 16 Oct 2005 00:05 GMT

>>I am looking for an analysis of the frequencies of radial modes in a
>>circular cavity (room). The specific case of interest to me is one in
[quoted text clipped - 24 lines]
> GE Global Research
> Niskayuna, NY

Thanks Bob. Even though I have referred to Blevins on several occasions
for the resonant freqeuncies of plates, I was unaware of the section on
acoustic cavities which is in the chapter on Fluid Systems. Also, that
outside possibility is the correct one for the radial mode (one nodal
circle). The 1.8412 coefficient is for the case of one nodal diameter.

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Ken Plotkin - 15 Oct 2005 03:27 GMT

>I am looking for an analysis of the frequencies of radial modes in a
>circular cavity (room). The specific case of interest to me is one in
[quoted text clipped - 8 lines]
>the center of the cavity. The acoustic impedance at all of the cavity
>walls can be considered to be infinite.

There is a closed form solution in terms of Bessel functions radially
and harmonics circumferentially and axially.

There's a paper by Larry Pope in JASA, Vol 50, No. 3 that has the
solutions for modes in a cylindrical cavity. It's part of an analysis
of sound transmission into a cylindrical shell, so you might have to
dig a bit to extract just the cavity part.

Memory is a little rusty - I wrote a program that did a similar
calculation in the mid 70s. I recall that the cavity mode solution
was in Morse's book on theoretical acoustics.

Hope this helps. E-mail me if you have trouble finding Pope's paper.

Ken Plotkin

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Robert A. Hedeen - 15 Oct 2005 03:39 GMT

>>I am looking for an analysis of the frequencies of radial modes in a
>>circular cavity (room). The specific case of interest to me is one in
[quoted text clipped - 11 lines]
>There is a closed form solution in terms of Bessel functions radially
>and harmonics circumferentially and axially.

...

>Ken Plotkin

Good. The Bessel function mode shapes are associated with the j index
in the Blevin table, so the correct frequency parameter is 1.8412 (see
my previous posting). I assume you just want the frequency, because it
is too late a night here to try to reproduce the mode shape formula
using crappy usenet ascii text.

Robert A. Hedeen
GE Global Research
Niskayuna, Ny

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The Ghost - 16 Oct 2005 00:08 GMT

>>I am looking for an analysis of the frequencies of radial modes in a
>>circular cavity (room). The specific case of interest to me is one in
[quoted text clipped - 24 lines]
>
> Ken Plotkin

Thanks, Ken. Your comment about Morse's book prodded me to take a look at
Vibration & Sound, where I indeed found the analysis. I had previously
looked in Theoretical Acoustics (Morse & Ingard) and failed to find it, and
I incorrectly assumed that it would not be in the other book.

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Natural Science Forum / Physics / Acoustics / October 2005

A Quesstion for the Theoreticians