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Author Topic: Inverse Square Law - the rest of the story?  (Read 5051 times)

Mike Karseboom

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Inverse Square Law - the rest of the story?
« on: July 20, 2015, 12:14:47 pm »

I have a rudimentary understanding of some aspects of acoustical theory and thought I had a reasonable handle on the inverse square law.


The fundamental concept, as I understand it, is that without reverberation or reflections if you have X sound pressure level at point A, then the sound pressure level Y at some distant point B will be lower by one fourth for every doubling of the distance from point A. So, the sound you hear should quieter the further you move away from point A.


What I don't understand is why the "loudness" of  sound at point B relative to point A depends also on how far the origin of the sound is from point A.  For example, using a hand held meter  at point A I might observe 65dBA average peaks (an oxymoron?) from a music stage 1000 feet away.  If someone stands 10 feet away between me and the stage and speaks, I might see 70dBA peaks from their voice.  Now if I stay in the same line with stage and person and go to point B another 200 feet away,  I would think the "loudness" of each sound would fall off at the same rate.  Yet at point B I can still hear the music with the perception of not much change in loudness while the person speaking is totally inaudible.


I guess I thought any sound that is some sound pressure level X at point A, could be considered a point source at A, and the loudness would fall off the same as you move away from point A.  But that does not seem to be the case.  What other concepts am I missing?
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Mark McFarlane

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Re: Inverse Square Law - the rest of the story?
« Reply #1 on: July 20, 2015, 12:34:04 pm »

The inverse square law is based on sound emanating from a point source equally in all directions.  'In all directions' in 3d in a constant velocity medium like air, means a sphere, so this is also called spherical spreading.  The physics (and loss by distance squared) are based on the origin (original location) of the sound, from which the sound energy emanates in all directions.  If you look at the energy density of the sound (and ignore all other energy loss factors like absorption) per unit area, as the sphere (wavefront) expands, the energy must follow the inverse square law (r**2).

If you don't start at the origin (like your example), the energy isn't traveling out in all directions, it is moving only in a smal arc, so it is not spreading spherically from this new reference point, hence no inverse square.  At great distances from the source (very large r) spherical spreading basically quits being significant and other energy loss modes take over, like absorption.

It may be easiest to think of this as energy density per unit area.  Close to source all the energy is concentrated at a single point, and it rapidly looses energy 'per square inch' of the wavefront as you move away spherically from the source.

Kind of like a fart.  Up close the smell is really obvious, but the difference between 20 and 30 feet away isn't much.
« Last Edit: July 21, 2015, 03:15:57 am by Mark McFarlane »
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Steve M Smith

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Re: Inverse Square Law - the rest of the story?
« Reply #2 on: July 20, 2015, 01:12:52 pm »

Considering the area covered is the best way to understand this.  If sound exits a speaker over a fixed angle, at any position, you can work out the area covered.  If you double the distance, the area is quadrupled.  The same amount of energy is now spread over this area so its intensity is now a quarter of what it was (works for light too).




Steve.
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Art Welter

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Re: Inverse Square Law - the rest of the story?
« Reply #3 on: July 20, 2015, 01:58:27 pm »

For example, using a hand held meter  at point A I might observe 65dBA average peaks (an oxymoron?) from a music stage 1000 feet away.  If someone stands 10 feet away between me and the stage and speaks, I might see 70dBA peaks from their voice.  Now if I stay in the same line with stage and person and go to point B another 200 feet away,  I would think the "loudness" of each sound would fall off at the same rate.  Yet at point B I can still hear the music with the perception of not much change in loudness while the person speaking is totally inaudible.
 What other concepts am I missing?
Mike,

Mark answered your question,  but to look at the simple math might be helpful, sound drops 6 dB per doubling of distance in the free field.
High frequencies are attenuated further depending on temperature and humidity, but that won't show up on the simple dB meter.
Since your stage measurement is at 1000', distance has doubled 8 times from the source. The next drop of 6 dB requires you to walk another 1000', while the drop from the voice at 10' to 20' would be 6 dB, so the voice has already dropped below the stage sound by about 1 dB just 20 feet from the person.

A 10 dB drop in level (at 1000 Hz) sounds half as loud to your ear.

Your examples:
65 dB at 1000' (59 dB at 2000')
71 dB at 500'
77 dB at 250'
83 dB at 125'
89 dB at 62.5'
95 dB at 31.25'
101 at 15.625'
107 at 7.8125'
113 dB at 3.9'

70 dB at 10' (only 81 dB at 2.5')
64 dB at 20'
58 dB at 40' (1 dB less than the stage sound at 2000')
52 dB at 80'
46 dB at 160'
40 dB at 320'

Art
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George Friedman-Jimenez

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Re: Inverse Square Law - the rest of the story?
« Reply #4 on: July 20, 2015, 04:13:45 pm »

Mike, the sound pressure level in both situations you propose drops off like 1 over r-squared, where r is the radius of a sphere with its center at the point source of the sound. The formula is the same, but the r is measured from a different point in each situation. What you are missing is the concept of a spherical wavefront of sound coming from each source. That wavefront is the surface of a 1000 foot radius sphere for the stage sound and the surface of a 10 foot radius sphere for the spoken sound. The air pressure changes everywhere else on the wave front affect the sound at your ear. If you built an infinitely large wall just behind the speaking person and put a small hole in it, you would block all the other parts of the wavefront from the stage and could then consider the sound emanating from the hole to be a new point source. Then the spoken sound and the stage sound would attenuate similarly with distance from the wall and the speaking person, like 1 over r-squared where r is now the distance from the hole in the wall not from the stage.
« Last Edit: July 20, 2015, 04:16:45 pm by George Friedman-Jimenez »
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Mike Karseboom

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Re: Inverse Square Law - the rest of the story?
« Reply #5 on: July 20, 2015, 06:38:21 pm »

Much thanks for the several different ways to describe what is actually happening. Bottom line is the "point source" must be considered from its point of origin.  Anyplace else and it is not a point source (except for the hole in the infinite wall example).


Now I know why my 65dB at the venue property line can still be heard a quarter of mile further out.  No wonder noise complaints are so common.  It seems the initial volume (SPL) of the noise has much more to do with the neighborhood impact than  the distance the neighbor is to the property line.  That is, some one across the street from the venue will experience  about 65dB and someone  another 2-3 blocks away will still hear about 59dB.  And the pulsating, "unnatural" nature of music might still be cause for complaint by the person hearing 59dB.



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--Mike
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Steve M Smith

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Re: Inverse Square Law - the rest of the story?
« Reply #6 on: July 21, 2015, 02:55:49 am »

If the question is related to noise complaints then actual dB levels are probably not relevant.  Noise as a nuisance is a lot more subjective than that.  Something which annoys one person might not even be noticed by another.


Steve.
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Keith Broughton

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Re: Inverse Square Law - the rest of the story?
« Reply #7 on: July 21, 2015, 07:24:42 am »

If the question is related to noise complaints then actual dB levels are probably not relevant.  Noise as a nuisance is a lot more subjective than that.  Something which annoys one person might not even be noticed by another.


Steve.
You make a good pont.
I have consulted with venues about noise complaints and it's far more complicated that just db levels.
In one case, when we drilled down to the root problem, the noise complaint was a tool for keeping the "undesireable type of people" out of the neighbourhood.
When subjective tests were made, the music from the venue was actually considerably quieter than the local lawn mowers.
On the lakeshore in Toronto we have a venue that was clearly in place long before the condo buildings were even constructed but people moved in THEN started to complain.
Hey, if you move into an airport area, guess what...there will be planes landing!
There is actually a sign to that effect out in Mississauga near the airport.
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Ivan Beaver

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Re: Inverse Square Law - the rest of the story?
« Reply #8 on: July 21, 2015, 11:47:52 am »

You make a good pont.
I have consulted with venues about noise complaints and it's far more complicated that just db levels.
In one case, when we drilled down to the root problem, the noise complaint was a tool for keeping the "undesireable type of people" out of the neighbourhood.
When subjective tests were made, the music from the venue was actually considerably quieter than the local lawn mowers.
On the lakeshore in Toronto we have a venue that was clearly in place long before the condo buildings were even constructed but people moved in THEN started to complain.
Hey, if you move into an airport area, guess what...there will be planes landing!
There is actually a sign to that effect out in Mississauga near the airport.
True.

Very often the actual noise from the "problem" is below the noise floor of the area.  So when the music is turned off, the SPL does not change.

That does not mean you cannot hear the music/problem.  But in terms of SPL, it is not a contributing part.

So people have to find "other" reasons.  Often it is because they  don't like the style of music or the people at the venue

As usual -simple SPL is no where near a complete tool.  It only gives 1 of many different possible answers.

And of course you have different scales-response time etc to deal with when trying state something that "appears" to be simple-such as "how loud is it"?

It depends.
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Merlijn van Veen

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Re: Inverse Square Law - the rest of the story?
« Reply #9 on: July 23, 2015, 12:38:06 pm »

If the question is related to noise complaints then actual dB levels are probably not relevant.  Noise as a nuisance is a lot more subjective than that.  Something which annoys one person might not even be noticed by another.


Steve.

Two words: "dripping faucet"

Steve M Smith

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Re: Inverse Square Law - the rest of the story?
« Reply #10 on: July 23, 2015, 12:41:06 pm »

Two words: "dripping faucet"
Or a dripping tap over here!


Steve.
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Re: Inverse Square Law - the rest of the story?
« Reply #10 on: July 23, 2015, 12:41:06 pm »


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