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I don't think so. The thing is on my laptop bed stand. So it is always in arm's reach and I don't have a controlled environment because I move around within the room. I was thinking more like some kind of specific sound damping material to line the interior of the box for the external HD but the thing is small.
Two things here, the cover could be lined with something thin. Or I could shove it inside the box I made originally for my laptop PSU brick but had to mount the latest one underneath (mess of a cord visible on the other side of the HD). I'd need to do the new PSU plug differently to get the HD all the way into that box. Still all I can do is brute force an empirical damping solution. I can't even ballpark about material properties and sound in this space. It would be a handy abstract skill.
I think my point is being missed.
Imagine illustrating noise cancelling by adding two identical but opposite soundwaves.
Source: https://www.soundguys.com/how-noise-cancelling-headphones-work-12380/
Noise cancellation relies on precisely controlling the distances to listener. If OP were to simply set up a tone generator near the hard drive, the waves would alternately constructively and destructively interfere as OP walked around the room.
That's not entirely true, cancelling sound waves can also be done on a large scale. I've seen it on festivals or other open air venues, where they stack two rows of subwoofers behind each other in a precisely calculated distance to cancel out the bass notes towards the back of the stage, meaning walking in front of the subs you could hear it normally, walking behind them and the bass notes where very faint and clearly coming as a bounce back from walls or the like. So they effectively cancelled out the frequencies towards the back.
But that doesn't help OP since that works way better on lower frequencies, with higher notes you'll need the precise localization you're talking about. So the high pitched noises from a hard drive probably won't work.