Deriving a formula for calculating dB based on distance

[wtfbuck]

Homework Statement

Decibel level from source at 1.0 m is 60.0 dB

Find dB level at distance 2.0 m.

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

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?

 

[Delphi51]

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.

Decibel level from source at 1.0 m is 60.0 dB

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.

 

[wtfbuck]

Appreciate the help Delphi. I can see the relationship now.