Calculation for finding distance with regard to sound intensity

[Aleisha]

Homework Statement

Two amplifiers can be used to power a sound system.
Amplifier 1: Has a fixed power output of 300 W.
Amplifier 2: Has a variable power output of between 15 W and 175 W.
You may assume that the power output of the amplifier is equal to the power of the sound waves that are produced.
The speaker is disconnected from Amplifier 1 and is instead connected to Amplifier 2, which is operated at maximum power.
Previous calculated question sound level of amplifier 1= 121.15 dB
Calculate how close to the speaker must you stand for the sound level to be the same as when the speaker was connected to Amplifier 1.

Homework Equations

A=4 x Pi x r^2
I=P/A
dB=10log(I/Io)

The Attempt at a Solution

I want to calculate a distance, so the the sound decibel of amplifier 2 (121.15 dB) will equal amplifier 1.
I don't know an equation with distance that would relate to the equations above.

 

[haruspex]

I don't know an equation with distance that would relate to the equations above

Yet you managed to quote an equation which is relevant.
When sound has traveled a distance r from its source, how thinly is its power spread?

 

[Aleisha]

Yet you managed to quote an equation which is relevant.
When sound has traveled a distance r from its source, how thinly is its power spread?

Right ofcourse I didn’t realize I was looking for radius. Don’t i need to know the area to get the radius? Or could I equate the two equations together?

 

[haruspex]

Right ofcourse I didn’t realize I was looking for radius. Don’t i need to know the area to get the radius? Or could I equate the two equations together?

A wavefront from the source forms an expanding sphere. The power is spread out over the surface of the sphere. As the radius doubles, what happens to the power density?

 

[Aleisha]

The power density will increase by a factor of 4, if the radius is doubled?

 

[haruspex]

The power density will increase by a factor of 4, if the radius is doubled?

Power density is power per unit area. The power total is constant. If the radius doubles what happens to the area?

 

[Aleisha]

Ohhh the area is 1/4 of the original amplifier?

 

[haruspex]

Ohhh the area is 1/4 of the original amplifier?

No.
Consider a single pulse of sound emitted from a point source. After some time it has traveled a distance r, so the wavefront forms a spherical shell radius r centred on the source. The total energy in the pulse is spread evenly over the shell, producing a certain energy density, ρ. The same time again later it has expanded to radius 2r. What is the energy density now?

 

[Aleisha]

Twice as large

 

[haruspex]

Twice as large

The total energy is the same. It is spread over a larger area.
Do you understand the concept of density?

 

[Aleisha]

It must be half then? That’s what I thought I meant because if 1/4 because the radius its r squared not just r... I’m getting too confused haha

 

[haruspex]

It must be half then?

That is moving in the right direction, but as you say it is radius squared, not half but ...?
(I have the feeling that you have, in your mind, already given the right answer but kept writing it down wrongly.)

 

[Aleisha]

That is moving in the right direction, but as you say it is radius squared, not half but ...?
(I have the feeling that you have, in your mind, already given the right answer but kept writing it down wrongly.)

Ah ha! It’s 4 times bigger!

 

[haruspex]

Ah ha! It’s 4 times bigger!

What is four times bigger, the area or the energy density?

 

[Aleisha]

What is four times bigger, the area or the energy density?

It must be the area according to the equation A=4 x Pi x r^2

 

[haruspex]

It must be the area according to the equation A=4 x Pi x r^2

Right, so what happens to the energy density?

 

[Aleisha]

I=P/A If the area is four times bigger then the intensity (energy density) then the intensity with decrease by 4 times, correct?

 

[haruspex]

I=P/A If the area is four times bigger then the intensity (energy density) then the intensity with decrease by 4 times, correct?

Right.
So can you now answer the question in post #1?

 

[Aleisha]

Yes yes! Thank you I get the correlation now :)