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# Advanced Math/Inverse Square Law with Unknown Distance?

Question
I've been trying to understand the basic mechanism behind how the inverse square law works and specifically would like to get a handle on formulas to compute this.  I've read thoroughly about 30 different pages on the web and while I get the basic idea conceptually I'm having a hard time with using the formula(s).  There seems to be many variations on the formula,  ( even when just using it for say "light" and not something more complex like radiation ), and this also complicates things for me. The main thing that perplexes me is, many of these pages say there are two distances as inputs but it seems there are actually three?  And the value of "20" keeps popping up, I can't tell what this number is referring to, is 20 supposed to be a distance or a value in decibels?

I'm using Excel to experiment with these formulas as well as some of the online calculators, I'd really like to know what the number "20" is, represented in the formulas I'm using is it distance or decibels??

I've included two similiar formulas from two separate links,
both show this number 20 being multiplied by the log value
of what appears to be the ratio of two distances... but what exactly
IS this value of 20 representing?  It can't represent the intensity
value as this seems to be already established as "Lp1" in formula one
or "Sref" in formula two ;

Lp2 =  20 log (R2 / R1) + Lp1
http://www.engineeringtoolbox.com/inverse-square-law-d_890.html
............................................
Snew=Sref + (20 * Log (Dref / Dnew)
http://www.crownaudio.com/inverse-square-law.htm

Where:
Dref = Reference Distance
Dnew = New Distance
Sref = Reference Sound Level
Snew = New Sound Level

The pesky factor of 20 comes from the definition of decibels. In this definition you have

decibels = 10log(I)

where I = intensity and log refers to the logarithm in base 10. Thus, if you have an intensity say of A1 at a distance R1 and you want to express the intensity A2 at distance R2 where you know that intensity drops off as distance squared then

A2 = A1･(R2/R1)^2

Converting this to decibels (or dB as it is often written), and using the algebra rules for logs, you get

10log(A2) = 10log[A1･(R2/R1)^2] = 10log(A1)+10log[(R2/R1)^2] = dB(A1) + 2･10log(R2/R1)

where 2･10 = 20.

So the factor of 20 comes from the factor of 10 from the definition of dB and the fact the 2 used to square the ratio of distances comes down from an exponent to a mulitiplicative factor when you take the log.

Nobody likes dBs.

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#### randy patton

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college mathematics, applied math, advanced calculus, complex analysis, linear and abstract algebra, probability theory, signal processing, undergraduate physics, physical oceanography

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26 years as a professional scientist conducting academic quality research on mostly classified projects involving math/physics modeling and simulation, data analysis and signal processing, instrument development; often ocean related

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