How Does Distance Affect Sound Intensity Levels?

[getty102]

Homework Statement

The sound intensity level of a certain sound source is measured by two listeners located at different positions along a line from the source. The listeners are located on the same side of the source and are separated by 34.8 m. The listener that is closest to the source hears the sound with a sound intensity level of 55.8 dB. The sound intensity level of the sound heard by the more distant listener is 49.8 dB.

Homework Equations

β = 10dBlog₁₀(I/I₀)
I_av=P_av/4∏r²

The Attempt at a Solution

β₂ - β₁ = 20dBlog₁₀(r₂/r₁)

10^((β₂-β₁)/20) = r₂/r₁.

r₁/r₂ = 1/10^((β₂-β₁)/20)

r₁((1/34.8)+1) = 1/10^((β₂-β₁)/20)

r₁ = 1/10^((β₂-β₁)/20)/((1/34.8)+1)

This doesn't seem to be working, I'm thinking my math is getting messed up somewhere's along the line.

 

[collinsmark]

Homework Statement

The sound intensity level of a certain sound source is measured by two listeners located at different positions along a line from the source. The listeners are located on the same side of the source and are separated by 34.8 m.

I interpret the last statement as saying that the listeners are separated from each other by 34.8 meters. It's kind of ambiguously worded, so I'm not 100% certain. But for the rest of this post I'm going to assume that they are separated from each other by 34.8 meters, such that

r₂ = r₁ + d

where d = 34.8 meters. r₁ is the distance from the source to the closer listener, and r₂ is the distance from the source to the more distant listener.

If I'm interpreting this incorrectly, let me know.

The listener that is closest to the source hears the sound with a sound intensity level of 55.8 dB. The sound intensity level of the sound heard by the more distant listener is 49.8 dB.

I'm a little confused now. The problem statement gives a lot of statements, yet no instructions on what it is we're supposed to find. The 'problem' itself hasn't been specified. In other words, what is the question?

Homework Equations

β = 10dBlog₁₀(I/I₀)
I_av=P_av/4∏r²

The Attempt at a Solution

β₂ - β₁ = 20dBlog₁₀(r₂/r₁)

10^((β₂-β₁)/20) = r₂/r₁.

So far so good.

r₁/r₂ = 1/10^((β₂-β₁)/20)

I'm not sure why would want to invert both sides of the equation at this point in the process. I don't see a purpose in that. But, okay...

r₁((1/34.8)+1) = 1/10^((β₂-β₁)/20)

Now you've lost me.

According to your above equation compared with the previous one, you substituted something in for r₂,

r_2 = \frac{1}{\frac{1}{d} + 1}

I don't think that's justified. From the way I interpret the problem statement, r₂ = r₁ + d.