Re: [Phys-l] decibel dilemma

[John Denker <jsd@av8n.com>]

Anthony Lapinski wrote:
> While searching for some good review problems regarding sound, I came
> across this in Tipler's physics text:
>
> It says the "background" sound level in a room is 40 dB. With 100 people
> talking, it rises to 60 dB. What's the sound level when 55 people leave?

Solution:  Convert from log power units (dBa) to honest-to-goodness power
units.

40 dBa = 10,000 power units.
60 dba = 1,000,000 power units.

Assume power adds linearly, hence:
  observed power = a + b N       for some coefficients a,b        [1]
where N is the number of people.

a = 10,000 power units
b = (1,000,000 - 10,000) / 100 = 9,900 power units per person

with 45 persons present, the power is 45,550 power units

Finally we can convert that to log power units to obtain 56.585

> deltaB = 10log(100/45) = 3.46   --->   sound level is 60 - 3.46 = 56.54 dB

That's close;  it differs from my answer because it neglects
the "a" term in equation 1.  That's OK when there are "enough"
people present.


> This problem seemed trivial to me, so I took it farther.
>
> So what if 90 people leave? deltaB = 10 log(100/10) = 10   --->   now 50 dB
>
> And if only 1 person remains (99 leave): deltaB = 10log(100/1) = 20   --->
>   back to 40 dB (background)
>
> But 40 dB was the background for an EMPTY room! If ALL people leave, the
> sound level should be 40 dB.

For one person, I get (10,000 + 9900) power units, or 42.99 dBa
by direct application of equation 1.

> Something doesn't seem right.
>
> After struggling with this for some time, I found the solution manual and
> it read:
>
> "The room background noise is 20 dB less than or 1/100 of the noise of 100
> people, so it can be neglected. 

That's OK, if there are "enough" people present ... but not OK
with only one or two people present.

> Total intensity after 55 people leave
> is... 3.46 dB less..."
>
> Does this make sense? Can't we just make the 40 dB background our
> "threshold," so it would, in effect, cancel out? The 40 dB is always there
> no matter how many people are present. How does one take this into
> account? What background noise level would be "significant"?

1) See above.

2) Exponentials are tricky.  Exponentiation and addition do not commute.
When you have a sum of exponentials, it is is common -- verrry common --
to find that one term dominates the sum.  If conditions change slightly,
you find that a different term dominates the sum.

In fields such as pattern recognition, there are algorithms that
/depend/ on the approximation that a sum is equal to its largest term.

If you make too many assumptions about which term is dominant, you can
get into trouble.

I've seen professional PhD-type engineers make this mistake more times
than I can count.