## SLAT TYPE HELMHOLTZ RESONATOR FORMULA

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### SLAT TYPE HELMHOLTZ RESONATOR FORMULA

Recently a very attentive person noticed an error related to published Slat type Helmholtz resonator formulas and reported this.

From: Scott Smith
Newsgroups: alt.sci.physics.acoustics
Date: 2004-02-02 15:45:49 PST

The noticed error was further investigated by the author versus diverse editions of the referred books.

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WRONG often published and in calculators used formula
fo = 2160*sqrt(r/((d*1.2*D)+(r+w)))

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CORRECT formula
fo = 2160*sqrt(r/((d*1.2*D)*(r+w)))

r = slot width
d = slat thickness
1.2 = mouth correction
D = cavity depth,
w = slat width
2160 = c/(2*PI) but rounded
c = speed of sound in inch/sec.

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DETAILS

There was some confusion about the origin of this error:
Both:
Master Handbook of Acoustics
Handbook for Sound Engineers
were referred as possible sources.

I took liberty to check different editions of the books themselves
Master Handbook of Acoustics - F. Alton Everest - editions 2, 3 and 4
Handbook for Sound Engineers - Glen M. Ballou editions 2 and 3

The formula with the included error is:
2160*sqrt(r/((d*D)+(r+w)))

This error does NOT originate from "Master Handbook of Acoustics" but from "Handbook for Sound Engineers"

2160*sqrt(r/((d*D)*(r+w))) [+ sign to be substituted by *]
Both Handbook for Sound Engineers: editions 2 and 3 still show the faulty version.

Scott Smith reported his RIGHTFUL CONCERN about the spreadsheets he found on the net and reported at least 3 of them based on the wrong formula.

I took the liberty to check any further and came to the conclusion that for "Slat type Helmholtz resonators" I could not come up with even 1 single correct calculator, neither HTML, nor Java or Excel calculators.
So this problem seems even worse then reported by Scott Smith.
The faulty "Slat type Helmholtz resonator" calculators on the net are NOT the exception but the rule!!!

Even the calculators on highly respected sites as SAE: :(
http://www.saecollege.de/reference_mate ... mholtz.xls
http://www.saecollege.de/reference_mate ... encies.htm
but also others as e.g.: :(
http://homepages.tig.com.au/~audio/elec ... lmholz.htm
http://www.mhsoft.nl/Helmholtzabsorber.asp
http://www.mindspring.com/~c_campbell_2 ... mholtz.xls
are wrong.

This error does not appear in the "Master Handbook of Acoustics - F. Alton Everest"
So why are that many calculators wrong, knowing That F. Alton Everest is also a standard work in the studio world?

Lets analyse both approaches:

1) Handbook for Sound Engineers - Glen M. Ballou - editions 2 and 3

fo = 2160*sqrt(r/((d*D)+(r+w))) (note this is still the Wrong formula)

d = the effective depth of the slot in inches, which is approximately (1.2) x (thickness of the slot in inches)
Important to note is that the factor 1.2 is an approximation.
The correct mathematical modelling of the mouth correction for Slat type Helmholtz resonators is a rather complex business.

2) Master Handbook of Acoustics - F. Alton Everest - editions 2, 3 and 4

fo = 216*sqrt(p/(d*D))

fo = resonance freq.. in Hz
p = perforation percentage (noted as values 1 to 100%)
D= airspace depth in inches
d = thickness of slat

p = 100 * r/(w+r)
r = slot width
w = slat width

One notices that F. Alton Everest ignores this (in reality complex) mouth correction.
It is not that uncommon that lots of books, even respected works only present stylized versions of formulas.
Any work is written in function of a specific target group.

As such works as the "The Master Handbook of Acoustics" will go less deep in mathematics as more specialized works.

While the aforementioned noted factor 1.2 for the mouth correction in the "Handbook for Sound Engineers" is only an approximation, it's still better to use this approximation than ignoring the mouth correction.

What is this mouth correction?
A Helmholtz resonator is a mass-spring system, which is comparable with a panel or membrane resonator.
The system is based on a mass which vibrates in resonance on a spring.
The ratio of the mass versus the dynamic stiffness of this spring defines the resonance frequency.
The air layer in the cavity acts as a spring with a certain dynamic stiffness mainly defined by its volume.
The larger the Volume, the weaker the spring becomes (lowering resonance frequency) and vice versa.

For a panel resonator it's easy to imagine what the mass is: the panel.
The heavier this mass becomes the lower the resonance frequency and vice versa.
As such a panel resonator is mainly defined by the combination of both properties.
This isn't complete, since angle of incidence, weakness of spring, damping etc. will influence the resonance frequency and the Q-factor.

For a Helmholtz resonator this mass is represented by the mass of the air enclosed by the neck or slot of the resonator.
However this apparent mass extends outside the exact geometrical boundaries of this neck or slot.
This is covered by the mouth correction, which is in fact a correction factor increasing those geometrical boundaries.
In reality this phenomenon is much more complicated than the simple factor, used by the traditional formulas.
As such the distance between those necks or slots (interaction) and others will influence this correction.
For practical use however the standard formulas are a good approach.

I can only assume that it is the lack of this mouth correction in Everest's formulas why Ballou's formula (correctly printed in Michael Rettinger's book on Studio Acoustics) is used.

In fact one can call it incredible that so many calculators on the net are wrong, even from well respected organizations as SAE, which is and presents itself as:
"The largest Institute for Multimedia, Audio Education and Digital Film Education worldwide"

NONE of the calculators I found mention the source of the used calculation method.

And here Some EXTREMELY important comments, in relation to the error reporting by Scott Smith, are due

From: Tony Woolf
Newsgroups: alt.sci.physics.acoustics
Date: 2004-02-04 10:46:33 PST

...I found your post a useful reminder that published equations can have errors.
For this reason and others, it is dangerous to pick up and use equations that you are not familiar with and do not understand.
Your post also made me realize that is not nearly so easy to see an error in an equation in spreadsheet format as it is in conventional notation, and I think this is also worth remembering ...

From: Noral Stewart
Newsgroups: alt.sci.physics.acoustics
Date: 2004-02-04 17:56:50 PST
<SNIP>
.........that many of us in this newsgroup know that errors exist in even the most respected works and we watch out for them.
However, those in some other newsgroups might be more inclined to take anything in print as gospel and spread it.
I can point out errors in almost every book I have. For some, I even have lists for various editions.
Errors are there for a variety of reasons.
<SNIP>
Unfortunately, we are plagued by many people who trust equations and and data they find without verifying them or without asking the conditions that make the data valid.
These same people buy computer programs with limitations and possibly errors they do not understand, and suddenly think they are an expert.

Note from the author:
It is indeed incredible, and this is certainly valid for the studio world, how many data is copied from site to site, without any respect for the author or source of the information or used calculation methods.
Is this done just to look clever? To attract people to there site? To present themselves as experts?

The above example is a SCHOOL EXAMPLE of this type of behavior.

It most likely started with nothing more than a simple typesetting error remaining unnoticed.

As it seems now, the resulting WRONG calculation formula used by so many sites became the standard rather than the exception.

Why then did other trained acousticians didn't noticed this before?
Mostly professional acousticians do NOT rely on net calculators, but on there study and books.
And here the comments by Noral Stewart are important.

For a trained acoustician a formula is a mathematical representation of a physical process.
As such it is a language in the same manner the written word is the tool for a writer to express his thoughts.

Anyhow the author wishes to express his respect to Scott Smitt for:
a) Reporting this.
b) Being critical enough to notice this error.

Further the author expresses his respect:
1) Tony Woolf (Europe: UK acoustician - http://www.tonywoolf.co.uk )
2) Noral Stewart (US: NC acoustician - http://www.stewartacousticalconsultants.com )

3) J.F. Oros (member http://forum.studiotips.com )
4) Jack Hildwine (member http://forum.studiotips.com )
5) Jeff D. Szymanski (Chief Acoustician: http://www.auralex.com )
For checking the exact content of the by the author lacking editions of Everest's and Ballou's books.

Eric Desart
http://www.acoustics-noise.com
Eric.Desart
Moderator

Posts: 2461
Joined: Tue Feb 10, 2004 4:29 am
Location: Antwerp, Belgium

Hi Eric:

You probably don't cruise AVS, so I thought I'd mention there's a question at the end of http://www.avsforum.com/avs-vb/showthre ... did=367793

---------------------------------

formerly HTbuph
Devout buckeye fan!

Registered: Apr 2003
Location: Vero Beach, FL
Posts: 209

Are these correct (from Everest)?

diaphram absorber
f = 170/((m*d)^0.5)

f - frequency
m - panel density (lbs/ft2)
d - airspace depth (in)

perforated panel absorber
f = 200* (p/(d*(t+0.8*D)))^0.5

f - frequency
p - perforation %
d - airspace depth (in)
t - panel thickness (in)
D - hole diameter (in)

---------------------------------------
Regards
Bob Golds
"The only thing we regret in life is the love we failed to give."
"Be a rapturist -- the backward of a terrorist. Commit random acts of senseless kindness, whenever possible" - Jake Stonebender
Bob

Posts: 4360
Joined: Sun Feb 08, 2004 4:37 am

Bob,

I've little time now, so come back to it later or leave it to other cleverder guys.

Every time I see a formula I must recalculate from Metric to find those constants back:
I'm a metric SI guy.

The panel formula is correct in such a manner that it assumes the backside of the panel to be an infinite mass and the sound is a plain wave with straight incidence (which is true for axial modes and about at long distance from sources for traveling waves).
If not: as a heavy panel on a light drywall this changes.
Also the resonance frequency is very angle of incidence dependent.

In fact for any resonator one must be careful for the exact tuning when it matters.
All formulas you'll see always assume stylized boundary conditions.

I'll check those things later.

For the panel resonance you can use in the time between my Mass-spring Excel file on my site.

In fact the formula for a panel resonator is exactly equal as the formula to calculate the mass-spring of a drywall.

The other one I must check myself.
Eric.Desart
Moderator

Posts: 2461
Joined: Tue Feb 10, 2004 4:29 am
Location: Antwerp, Belgium