Intensity of sound, power output

[jangchen]

Homework Statement

A loudspeaker, mounted on a tall pole, is engineered to emit 75% of its sound energy into the forward hemisphere, 25% toward the back. You measure an 85 dB sound intensity level when standing 3.5 m in front of and 2.5 m below the speaker. What is the speaker’s power output?

Relevant Equations

dB = 10 log(I / Io) , I = P / 4pi×r^2

I is proportionate to E.

r = 4.3xxm

So I put 0.75I

The answer is 49mW..

But I got 97.xxmW

What is wrong?

 

[TSny]

I get 49 mW. If you show the details of your calculation we can maybe spot a mistake. If I had to guess, I would say you used 4πr² for the area of a hemisphere.

 

[mjc123]

Your equation for I assumes the energy distribution is spherically symmetrical. But the question tells you that is not true. If 75% of the energy is emitted forwards (as opposed to 50% for spherical symmetry), how would that affect the I value you measure?

 

[jangchen]

If I had to guess, I would say you used 4πr² for the area of a hemisphere.

Oh, I used 4πr².. Do I have to use 2πr² ?

 

[mjc123]

If 75% rather than 50% of the energy goes forwards, then
I_forward = 1.5P/4πr²
I'm not sure what you meant by "So I put 0.75I", but if it means you used 0.75 instead of 1.5 in the above equation, then that would explain your error.

 

[TSny]

There's more than one way to think about it. Can you answer each of the following?
(1) What is the intensity I (in W/m²) at the point given?
(2) What is the total power P (in W) of sound radiated in the forward direction (hemisphere)?
(3) How is the answer to (2) related to the total power radiated by the speaker?

 

[jangchen]

Now I understand surely. Thank you!