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There are lots of strange options besides newtonian fluids. Would be interesting to see how dilatant, peusdoplastic, thixotropic etc react to sounds. Perhaps there is a way to make a material that allows quiet sounds to pass through and blocks all the loud ones. My guess is that dilatant liquids should be a good candidate.
A quick search tells me this have to do with shear forces. Sound would be entirely compressive, so those material properties would have no effect, or at least not change due to sound levels.
That’s unfortunate. Just like OP, I would have really liked the idea of using a non-newtonian fluid to filter out certain types of sounds without using electricity. Well, I guess, we’re back to active noise canceling then.