Uncertainty of room mode frequencies

[pitchtwit]

Hi - I'm new here - I'm just writing a paper based on an acoustical analysis of a rock/pop performance venue in a college I work at, and I'm up to my uncertainties section.

I've read up on uncertainties, and feel confident in working out uncertainties for simple additions, multiplications, divisions, squares and roots - but my equation is more complex than that.

Here's the main equation: -
[PLAIN]http://dl.dropbox.com/u/11341635/Room%20modes.png

n_x, n_y and n_z are integer values from one to infinity.

l_x = 13.640 m ± 0.005 m
l_y = 8.109 m ± 0.005 m
l_x = 6.241 m ± 0.005 m

c is the speed of sound and is calculated using the equation: -
[PLAIN]http://dl.dropbox.com/u/11341635/Speed%20of%20sound.png
(the 101.1% is due to humidity at 30 degrees C and 30 % humidity - the maximum that could have been reached during the experiment).

T was 23 degree C ± 2

I'm not sure how to deal with the integer values - I'll need to put values to each at some point - I was assuming each would be either 1, or the maximum reached during the experiment (around 15), but this would mean that the end uncertainty would be a variable number depending on what these integers were.

Assuming each integer value were 1, how do you deal with 1/x in uncertainties? I can't find any reference to this in all my guidance documents.

Then finally there's the square root to deal with. I know that to deal with a square root you usually divide the relative uncertainty by 2, but what if it's a square root of a bunch of uncertainties - like in this case.

 

[mikeph]

http://www.upscale.utoronto.ca/PVB/Harrison/ErrorAnalysis/

Nice little article. Gives you the general formula if you scroll down a little. I hope you like differentiating!

 

[Dickfore]

The key is to take the log of both sides and differentiate then.

 

[pitchtwit]

Thanks - I'm reading that document now.

What about a simpler one - I have two sets of data for reverberation time measurements - one at 500 Hz and one at 1,000 Hz. The estimated mean average for the 500 Hz set is 0.908 s with a relative uncertainty of 0.00861 s and for the 1,000 Hz set these are 0.793 s and 0.0194 s.

I need the mid-frequency reverberation time, which is a mean average based on only the 500 Hz and 1,000 Hz measurements. So in this case it is (0.908+0.793)/2 = 0.850 s. How do I work out the uncertainty for this value?

 

[pitchtwit]

I think I'm getting it. For the last example I arrived at the following equation: -

[PLAIN]http://dl.dropbox.com/u/11341635/Uncertainty%20of%20MidFrequency%20Reverberation%20Time.png

Think I've got in about as deep as I need for now.

Many thanks.