How Do You Calculate Reverberation Time and Sound Levels in an Auditorium?

[roldy]

Homework Statement

The surfaces of an auditorium are 200ft x 50ft x 30ft and have an average absorptivity a=0.29.

(a) What is the reverberation time?
(b) What must be the power output of a source if it is to produce a steady-state sound pressure level of 65 dB reference 20 $$\mu$$Pa?
(c)What average absorptivity would be required if a speaker having an acoustic output of 100 $$\mu$$W is to produce a steady-state level of 65dB reference 20$$\mu$$Pa?
(d) Calculate the resulting reverberation time, and comment on its influence on the intelligibility of the speaker

Homework Equations

Reverb time=0.191V/(Sa)
S=surface area
a is the average absorptivity coefficient
V is the volume of the room
A is the absorptivity

$$W=10^{\frac{SWL}{10}}*W_o$$
W is the power output (Watts)
SWL is the sound power level
$$W_o=10^{-12}$$

$$SIL=10^{-12}*10^{\frac{SPL}{10}}$$
SIL=sound intensity level
SPL=sound pressure level

$$SWL=SIL-10log\left(\frac{4}{A}\right)$$

a=A/S

$$W=10^{\frac{SWL}{10}}*W_o$$

The Attempt at a Solution

(a)S=2(50*30)+2(200*30)+2(200*50)=35000

Reverb time=$$\frac{0.101*200*50*30}{35000*.29}=4.759$$sec


(b)First I found A
A=a*S=0.29*35000=1015

Then I found SIL from the SPL given

SIL=$$10^{-12}*10^{\frac{65}{10}}=3.16*10^{-6}W/m^2$$

Then I found the SWL.

$$SWL=SIL-10log\left(\frac{4}{A}\right)=3.16*10^{-6}-10log\left(\frac{4}{1015}\right)=24 W/m^2$$

Then I calculated the power out put from $$W=10^{\frac{SWL}{10}}*W_o$$

W=$$2.51*10^{-10}$$ Watts

Did I do this right? This seems like a really small number for this situation

(c) I just solved $$SWL=SIL-10log\left(\frac{4}{A}\right)$$ in terms of A

$$A=\frac{4}{10^{\frac{SIL-SWL}{10}}}=4*10^9 m^2$$

So then the average absorptivity is $$a=\frac{4*10^9}{35000}=1.14*10^6$$

Now this doesn't seem right if I think about it logically. If my power is really small (part b), why would I need a tremendous amount of absorptivity? I can solve part d easily but if I solved either part b or c then d will be wrong. Can anyone see if I made a mistake if any?

 

[jbsteele]

(d)Reverb time=\frac{0.101*200*50*30}{35000*1.14*10^6}=4.2secThis reverberation time is on the higher end and will significantly reduce the intelligibility of the speaker.