What you can't do is stop others copying your idea (as long as they don't copy your implementation), nor stop others implementing the same thing who don't copy you.
Now, a work needs in most jurisdictions to be distinctive, but IMO the length of the work would inter alia be enough to make it distinctive, or perhaps use of a specially designed noise generator. But when you release "10 hours of noise" and I copy your idea and make "9 hours of noise" there will be no tort there.
Refusal letter: https://web.archive.org/web/20160412165054/http://ipmall.inf...
That logo is, within the field of logos, indistinct, boring if you will, if you sat down 100 logo designers I bet you'd get a few that look very like it. I'm not sure I entirely agree with it but having copyright but for a logo RTM is probably a stronger right anyway.
In short I don't think just being white noise is sufficient, but I don't think it excluded a piece of itself.