Here's a 1950 story about a noise cancelling device, like the one in the story you linked to in the question. Nothing about white noise, though.
1950: "Silence, Please!", a short story by Arthur C. Clarke in his Tales From the White Hart series; first published (as by Charles Willis) in Science-Fantasy, Winter 1950, available at the Internet Archive.
From the Wikipedia summary:
This comic story describes the efforts of a brilliant college student to design a machine that would produce a field of absolute silence. The gadget is then used in a prank, with tragic results. The story touches (albeit in a humorous way) on the popular science fiction theme of an inventor coming to grief at the hands of their invention that is best known from Mary Shelley's novel Frankenstein. The piece also references the composer "Edward England", an obvious parody of the work of Benjamin Britten.
The "Fenton Silencer" described in the story uses the same phase-inversion principle found in modern noise-canceling earphones.
Excerpts from the story:
"Suppose you have a train of waves — a peak here, a trough there, and so on.
Then you take another train of waves and superimpose the two. What would you
"Well, it depends on how you do it, I imagine."
"Precisely. Suppose you arranged it so that the trough of one wave coincided with the peak of the other, and so on all along the train."
"Then you'd get complete cancellation — nothing at all. Good heavens — !"
"Exactly. Now let's say we've got a source of sound. I put a microphone near
it and feed the output to what we'll call an inverting amplifier. That drives a
loudspeaker, and the whole thing is arranged so that the output is kept automatically at the same amplitude as the input, only out of phase with it. What's the net result?"
"It doesn't seem reasonable . . . but in theory it should give complete silence. There must be a catch somewhere."
"Where? It’s only the principle of negative feed-back, which has been used in
radio for years to get rid of things you don't want."
[. . . .]
"I haven't made very extensive tests, but this unit can be adjusted to give
almost complete silence over a radius of twenty feet. Outside that, sounds are
deadened for another thirty feet, and further away everything is normal again.
You could cover any area you liked simply by increasing the power. This unit
has an output of about three watts of 'negative sound', and it couldn't handle very intense noises. But I think I could make a model to blank out the Albert Hall
if I wanted to — though I might draw the line at Wembley Stadium."