Okay… that was totally incorrect… The FAA has certified methods of audio dosimetry that really do act like straight integrators. Internationally, everyone else seems to follow suit, with variations in scaling above 85 dB. But there appears to be a body of supporting research that says integration is the way to go.
Hmm… as a physicist that smells to high heaven. But I can view integration as the practical limit of an exponential envelope with a very long time constant. It looks and acts just about the same.
So instead of using 5.3 scaling on the e-foldings of my time constant, we should use something more like 1/10 e-folding scaling on the way up. That really does look just like plain integration over the first 8 hours.
(I define my e-folding value as: 1/(8 * 3600 * Fsamp))
But then, on the way back down from an exposure, resting at sound levels below 70 dB for at least 10 hours… that means we need a vastly different time constant on the way back down. Using the same time constant as the integrator would require us to wait more than 80 hours.
So fiddling around, taking astronomical liberties with my modeling, I find that using 3 e-folding scaling on the way down, with the same underlying time constant, really gives us closer to 4 e-foldings over a 10 hour period.
It looks like this: expose to 85 dBSPL for 8 hours, then relax for 10 hours.
The problem with the incorrect profile is that it climbs too quickly, the dosing at even louder levels is too rapid, and the shape of the climb is at direct odds with established protocols backed by a body of research. So without probing too much deeper, I’ll accept their claims. My new profile looks correct on the way up, and produces satisfactory recovery after another 10 hours of resting. Our half-life in recovery is around 2 hours.
But now this is really quite interesting… You see, in my original profile, I made the assumption, as I still do, that there is a restorative activity in place at all times. But in my original model it had the same strength on the way back down in recovery mode, as the climb had on the exposure side.
In effect, whenever the exposure was increasing, my old model assumed the ear would just leave hands off and let the sound have its way. Of course, when you say it like that, it seems wrong at face value.
Now, with two vastly different time constants, with a ratio of about 30:1, that says the restorative action is really strong, and it fights the exposure on the way up till it has almost minimal impact. The exposure wins out on the way up, but only very slightly, like 3% in its favor. And that’s why it looks like a linear integration profile.
We know it has to be exponential. So the restorative action on the way up is bending that purple curve into a much more shallow curve. Almost too shallow to see, so that it appears like a linear integration slope.
I find that discovery quite interesting! It appears that our ears are designed to fight damaging sounds!!