## Rectangular plane sound source

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### Rectangular plane sound source

I found this equation to calculate the sound power level of a rectangular sound source:

PWL = SPL + [10 log (a*b/r^2)] - 10 with the condition r > b/Pi
PWL = Sound power level
SPL = Sound pressure level

I used this equation in my analysis and found that the SPL I got was greater than the PWL. Is this possible? Shouldn't the SPL decrease as the sound travels farther away from the source?

I can't seem to find the reference for this equation so I can't check its validity. Does anyone here come across this equation? Is there any other equation to calculate the sound pressure level of a rectangular sound source? Or can anyone recommend some reference books for me to check up on? Thanks in advance.
seiko

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Joined: Mon Apr 04, 2011 1:09 am

### Re: Rectangular plane sound source

I have these data somewhere from a mathcad simulation for rectangular drivers we wrote years ago.
I'll look 'm up.

Can you give an example of your calculations?
If you view life with the knowledge that there are no problems, only opportunities, you are a marketing manager.......this is my personal philosophy
bert stoltenborg

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### Re: Rectangular plane sound source

Let say a rectangular source (PWL = 116 dB) with the dimension of 3m x 8m, the SPL is calculated at a distance of 18.5m from the source.

a = 3m
b = 8m
r = 18.5m

Using the equation I meantioned in the 1st post,

PWL = SPL + [10 log (a*b/r^2)] - 10
SPL = PWL - [10 log (a*b/r^2)] + 10
= 116 - [10 log (3*8/18.5^2)] + 10
= 116 - (-11.54) + 10
= 137.54 dB

I'm doubting the validity of this equation. I hope someone can enlighten me
seiko

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Joined: Mon Apr 04, 2011 1:09 am

### Re: Rectangular plane sound source

I was thinking you pointed to something else.

If you have to correct a sound power value for surface S it's 10log(S).
So in your example it's 10log(24) = 13.8 dB.

116 dB + 13.8 dB = 129.8 dB.

On a distance of 18.5 meters you subtract 10log(4*pi*r^2) (or 20log(r)+11) = 36.3 dB

At 18.5 meters you have 93.5 dB.

You also have to correct for Q (frequency dependent, of course) and at larger distances for air absorption, ground effects, refraction etc.

These calculations are normally used for sound isolation calcs.
Of course if a rectangle has a sound power of 116 dB (normalized to 1 meter), the SPL on distance r is just 20log(r)+11 dB lower at very large distances. In the near field the rectangle behaves as a line source.
Last edited by bert stoltenborg on Mon Apr 04, 2011 4:49 pm, edited 1 time in total.
If you view life with the knowledge that there are no problems, only opportunities, you are a marketing manager.......this is my personal philosophy
bert stoltenborg

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### Re: Rectangular plane sound source

Sound power is a bit silly in practical life.
Over here in the pot smoking country they permit for let's say a factory a sound power of X.
Then you have to calculate if at a given point (often a zone around an industrial area) the sound power doesn't exceed 50 dB(A).
IMHO this is totally stupid as there is no mentioning of direction; if I have a high Q line array I can fulfill the permission for power and at the same time exceed the 50 dB(A) because I radiate not spherical but in a narrow direction.
If you view life with the knowledge that there are no problems, only opportunities, you are a marketing manager.......this is my personal philosophy
bert stoltenborg

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### Re: Rectangular plane sound source

So from your post, I can deduce that

SPL = PWL + 10 log (a*b) + 10 log [Q/(4*Pi*r^2)]

This seems to be the one used by my senior. The only problem I have with it is that I do not have any reference to back it up. Is there any reliable websites or reference books that I can look into for this equation? I think the reference to the surface sound power level correction would suffice.

Does this equation take into the account of directivity? A reference book stated that if the measuring point is at an angle away from the normal axis at the center of the rectangle sound source, a directivity correction has to be added to the SPL.

You said that the rectangle behaves as a line source in near field. Does this mean I should use another equation instead of the one mentioned above to get the PWL if the SPL (for example, at 1m away from the source) is given?
seiko

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Joined: Mon Apr 04, 2011 1:09 am

### Re: Rectangular plane sound source

Check that formula of yours again, I don't trust it
This is how it's described in our law, the Handleiding Meten en Rekenen Industrielawaai. (directive for measuring and calculating industrial noise)
It is in any book on acoustics as it is basic.

Near field acoustics is more difficult as you have to calculate complex numbers.
In essence for a far field source you get a 6 dB decrease in sound per doubling of distance, for a line source 3 dB.
But to do it right you have to get an understanding of the math which means read something like Beranek, Olson, Kinsler, Pierce...
If you view life with the knowledge that there are no problems, only opportunities, you are a marketing manager.......this is my personal philosophy
bert stoltenborg

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### Re: Rectangular plane sound source

Huh? The formula basically comes from what you had stated in your previous post... Hahahaha...

Correction for area source = 10 log (S) = 10 log (a*b)
Loss of power due to distance = 10 log (4*Pi*r^2)
Directivity correction = 10 log Q

So,

SPL = PWL + Correction for area source + Directivity correction - Loss of power due to distance
= PWL + 10 log (a*b) + 10 log Q - 10 log (4*Pi*r^2)

It's not correct? Now I'm confused

As for the line source, the loss caused by distance = 10 log (4*Pi*r) ?

Anyway, I'd gone through one of Beranek's book - "Noise and Vibration Control Engineering". I can't seem to find the equations for calculating the SPL of an area sound source (rectangular plane). Does he have other titles that address this topic? I'll try to look for other books by the authors you mentioned. Thanks for the help.
seiko

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Joined: Mon Apr 04, 2011 1:09 am

### Re: Rectangular plane sound source

That seems better.

The pi in the equation stands for the angle of radiation.
4 pi means a sherical radiation, 2 pi is half sperical, etc.
If you view life with the knowledge that there are no problems, only opportunities, you are a marketing manager.......this is my personal philosophy
bert stoltenborg

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### Re: Rectangular plane sound source

Beranek's Acoustics covers the theoretical stuff.
For a practical approach there is Davis sound system engineering or the JBL sound system design reference manual (was always somewhere online)
If you view life with the knowledge that there are no problems, only opportunities, you are a marketing manager.......this is my personal philosophy
bert stoltenborg

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Location: Achterhood, Netherlands

### Re: Rectangular plane sound source

If you're not sure about something theoretical, then look up Morse & Ingard or Kuttruff.

The equation you wrote seems like some messed up SPL to SWL conversion.
Why this has to do anything with a 'rectangular' source I don't understand.

Power and Pressure are inter-related through Intensity, so if you want to estimate SPL at a distance from a source having some arbitrary SWL then:

SPL = SWL + 20*log10(Q/(4*pi*r^2))

So if you measured the source at a free field in a radial distance of r, then your SWL is

SWL = 20*log10(Q/(4*pi*r^2)) - SPL (at distance r).

Hope this helps.
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jonessy

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### Re: Rectangular plane sound source

Thanks for the suggestions. I'll try to look those references up.

@jonessy, what I'm looking for is an equation to predict the sound pressure level from a rectangular plane sound source. I think the equation you suggested is the fundamental equation for SPL. I believe there are some directivity corrections or different equations for SPL from different types of sound source.

Anyway, thanks for the input. I'll try to read up more on this topic.
seiko

Posts: 5
Joined: Mon Apr 04, 2011 1:09 am

### Re: Rectangular plane sound source

Jon,

I think it's because when you measure or calculate a SPL in a big room like a factory, and you have to estimate sound levels at f.e. a house somewhere, you have to correct for the surface of the dividing construction.
It confused me to as initially I thought he wanted to know something about the radiation impedance of a rectangular surface.
There is no analytical formula for this like there is for a piston.
If you view life with the knowledge that there are no problems, only opportunities, you are a marketing manager.......this is my personal philosophy
bert stoltenborg

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Location: Achterhood, Netherlands

### Re: Rectangular plane sound source

Jon,

Isn't that Q the same as modifying Pi in the equation?
If you view life with the knowledge that there are no problems, only opportunities, you are a marketing manager.......this is my personal philosophy
bert stoltenborg

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Joined: Sun Apr 18, 2004 11:03 am
Location: Achterhood, Netherlands

### Re: Rectangular plane sound source

Yes. So for an omni source Q=1 and the radiation is 1/4*pi*r^2, which is 'classic' inverse square law 6dB/DD.
For Q=2, you get 1/2*pi*r^2, essentially radiating into half space, 3dB/DD.
Etc.. Etc..
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jonessy

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