I confess I haven't read all of the postings; some I've only skimmed. You guys are funny! Well... some of you.
A couple things came to mind, perhaps going over old ground. Or things everybody already knows. (Sorry.)
The original objection to "white" noise was because the energy is distributed linearly, while our perception of sound is logarithmic. So there's the same energy from 100 to 200 Hz, and from 1000 to 1100, and 1100 to 1200, and so on.
Our ears hear the bit from 100 to 200, and compare it to the bit between 1000 and 2000, which is one white slice compared to ten. The high frequency chunk wins. By a lot.
So pink noise is equal energy in the 100 to 200 and the 1000 to 2000 bands. Nicely balanced for us humans and many of our sounds. (Including music, the prototype of all that is wonderful.)
I've heard of brown noise, but reject the premise. I don't think it applies here, anyway. Pink noise has an appropriate rumble.
If the attenuation between the source and the destination is 20dB, which it very well could be, the actual power difference is a factor if 100. So 1 watt at the destination end is equivalent to 100 watts at the source end. Something to think about.
The amount of sound leaking through a door versus through the little gap around the door can also be a little surprising. You will find most of it is through that little gap, not the broad expanse of the panel.
Were it me, then, my first guess, my current thinking, would be in line with some of the others:
- some experimentation will be required.
- Insulate the gap around the door and see what that gets me.
- Mask the sound at the destination end rather than creating a potentially uncomfortable experience at the source. Remember... the sound intensity for the listener would be the same in either case.
- Use a modified pink noise. We certainly won't need response down to sub bass; rolling off the low frequencies could be as simple as using a less-than-stellar speaker. "Low enough" is all we're after. The highs will take care of themselves.
- Consider (seriously consider) just leaving it on, rather than trying to synchronize with the main event. (Those of us with tinnitus just leave it in always.) one benefit is that other annoying noises get masked, too. You could... try it, even if it seems kinda "nnnnnnyeh" at first thought.
- If not continuous, the problem becomes remote switching or automation or something. And soft-start - a sudden "on" of masking sound, no matter how it's done, will be a distracting disturbance in itself. As will "off." So volume ramping might increase the quality of the result, in addition to contributing to our primary goal: making life more complicated.
I do love the basic problem. I've mixed live sound my whole life, and decisions about whether two sounds should be blended and making something truly merged, or keeping them separate and clearly distinguishable, is fundamental to the mix. And the applications of the principles are everywhere... neighbors in apartments, trying to get work done, and so on.