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Deriving a formula for calculating dB based on distance

  1. Sep 12, 2011 #1
    1. The problem statement, all variables and given/known data

    Decibel level from source at 1.0 m is 60.0 dB

    Find dB level at distance 2.0 m.

    2. Relevant equations

    dB = 10 log10 (I/I0) <- this one I know but couldn't see an application.

    Forgot to add that I0 is equal to 1.0 x 10^-12 W/m^2

    dB = dBi - 20 log10 (r2/r1), where r2 is the farther sound source and r1 is the closer one.

    3. The attempt at a solution

    I arrived at the correct answer of 54.0 dB by using the looked up formula. But I'd rather know how to derive that formula that relates the source distances myself. I tried for a while but am unable to do it. Anyone able to give me some insight on how to arrive at the second formula?
  2. jcsd
  3. Sep 12, 2011 #2


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    Homework Helper

    Welcome to PF!
    The intensity of sound diminishes with distance from the source in the same way as light or strength of gravity - because it spreads out in space. The source has a certain power in watts. You can find out the intensity (watts per meter squared) at a distance r by realizing that the power spreads out over the surface area of a sphere with radius r:
    I = P/(4πr²)
    You can use this once to find the P since you know I at distance 1 meter. After you know P, use it again to find the I at distance 2 meters.

    The decibel level is really just a comparison of two sounds as you see from dB1 = 10 log10 (I/I0). Put in the values of I and I0 to see what it is. It will, of course, be negative because the I is lower than I0. The sound at 2 m is so many decibels lower than the sound level at 1 m.

    This is a whole other definition of dB. Some intensity has been defined as being 1 dB and the formula has been used to find that the intensity of this sound at distance 1 m is 60 dB louder than the defined value. Take the 60 dB and subtract the dB1 value you calculated earlier to get this relative db value at 2 m. Say it is 6 dB lower than the 60 dB sound; then it is 54 dB relative to the original defined value.
  4. Sep 12, 2011 #3
    Appreciate the help Delphi. I can see the relationship now.
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