ANOther physics question i need help with!

Question

An ordinary light bulb will hum at 13 decibels one meter away in a spherical shaped wave. Explain how many decibels you would hear if you were 2 meters away from 2 light bulbs.

Answer 1

Hello Gg!

 

Decibels (dB) -- a measure of sound intensity level -- is a logarithmic unit, linking sound intensity I (given in watts per meter squared) to the human's average ability to hear "just-noticeable differences" in sound when they change by a factor of 10 in intensity up or down.  The logarithm, as defined for decibels, converts that factor of 10 into a value of 10 dB.  Note that sound intensity and sound intensity level are related, but distinct, quantities.

 

If we assume the source is isotropic -- emanating equally in all directions, which is what they mean by a "spherical shaped wave" -- then the power (P) of the source is related to intensity (I) a certain distance from the source by the expression:

 

I = P/(4πr²)   [unit = W/m²]

 

where r is the distance from the source.

 

The power is fixed by the strength of the source in disturbing the air around it.  That does not change no matter how far from the source you are.  The reason a source sounds fainter farther away is that you are intercepting less and less of that power at larger distances, as the power is dispersed through larger and larger total areas.  If you connect all the points a certain distance r from the source, you get a spherical shell of radius r, which has surface area 4πr².  So, when you hear a source get fainter, you are hearing a loss in intensity.

 

Sound intensity level (β), in decibels, is related to I by:

 

β = 10log(I/I_o)

 

where I_o represents the average threshold intensity for human hearing, 10^-12 W/m².

 

When sources (at the same location) are combined, their intensities add directly (more power through the same area).  So, we want to add the intensities of multiple sources, and see how his changes the sound intensity level in dB.

 

 

Let's say:

 

I₁ = sound intensity of one light bulb one meter away

 

Then, the total sound intensity of 2 identical light bulbs 1 meter away would thus be twice that (twice the power).

 

As can be seen in the equation above, the sound intensity goes down as the square of distance from the source.  So, as r increases by a factor of two, the sound intensity goes down by a factor of 4.

 

So the sound intensity 2 meters from two light bulbs is:

 

I₂ = 2I₁/4 = I₁/2

 

The sound intensity level for one light bulb one meter away is given by:

 

β₁ = 13 dB = 10log(I₁/I_o)

 

The sound intensity level of two bulbs two meters away is given by:

 

β₂ = 10log(I₁/2I_o) = 10log[(1/2)(I₁/I_o)]

 

I rewrote the log expression the second way because it may be a little easier to see that it is log(ab), where a = 1/2 and b = (I₁/I_o).  By the rules of logarithms, log(ab) = log(a) + log(b).  So:

 

β₂ = 10(log(1/2) + log(I₁/I_o)) = 10log(1/2) + 10log(I₁/I_o) = 10log(1/2) + β₁ = 10log(1/2) + 13 dB

 

or, for short:

 

β₂ = 13 + 10log(1/2)

 

So, you can just evaluate 10log(1/2) with a calculator, add that result to 13, and that is your new sound intensity level.

 

I hope this helps!  Let me know if you have any more questions about this, or would like to check an answer.