Intensity of sound, power output

AI Thread Summary
The discussion centers on the calculation of sound intensity and power output, specifically addressing an error in using the area formula for a hemisphere. The user initially calculated the intensity using the incorrect area of 4πr² instead of the appropriate 2πr², leading to discrepancies in the power output. It was clarified that if 75% of the energy is emitted forward, this affects the intensity measurement, requiring adjustments in the calculations. The conversation emphasizes the importance of understanding energy distribution in sound intensity calculations. Ultimately, the user expresses gratitude for the clarification and understanding gained from the discussion.
jangchen
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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?
 
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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.
 
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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?
 
TSny said:
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 ?
 
If 75% rather than 50% of the energy goes forwards, then
Iforward = 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.
 
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?
 
Now I understand surely. Thank you!
 
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