UPDATE: The seller immediately offered me a 300$ discount after I showed him the videos and told him that used JA-0801 speakers go for around 300$ upward. My guess is he knew of the problem. In any case, I'm OK with this.
Now to repairing it...
I finally had the time to
very carefully disassemble the unit. The rubber gasket gave me quite a bit of trouble. I have no experience with the JA-0801 so I'm not sure, but I think these had been refurbished before. One of the wire contacts had a sticker "yellow" on it, so at the very least, the speaker had been removed before, and it seemed to me that the gasket was held in place very tightly with more glue than normal. I tried using a hair dryer to soften the gasket and glue but that did basically nothing. In fact, the metal mesh itself was glued on to the speaker quite tightly. I finally managed to remove the gasket and mesh with minimal damage by using a credit card, insulating tape to protect the metal, a flat screwdriver and a bit of force.
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After removing the dome assembly and inspecting everything visually, I can't really find any problem. The dome looks very well attached to the coil.
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I could only find a few very minor things:
#1 a tiny piece of copper-colored (but magnetic or at least statically charged) material that sat on the edge of the dome. I guess it's possible that this tiny metal piece vibrated against the dome at certain frequencies but I can't imagine how it could cause that much noise. I removed it very carefully using a small blade that the metal piece jumped to quite readily due to the magnetic field so I could hold it against the knife using a finger and remove it without touching the beryllium dome.
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#2 a bit of a goopy feature on the dome, also visible on the first picture above. To me, it just looks like a little bit of the same glue that attaches the dome to the coil. Maybe during refurbishment some of it got onto the dome where it shouldn't be, but I don't see how this could cause a problem.
#3 there are two or three tiny specs of metal dust in the small gap of the magnet where the coil normally sits. I suppose it's possible that these were rubbing or jumping against the coil. No picture, but it's really just some specks of dust in the magnet. Again I don't see how that could cause so much noise.
Any opinions? Do you see something on the pictures that I don't see? Can it be that these dust particles caused the noise I posted?
Debussy - Clair de Lune:
https://streamable.com/ksugwq
Frequency generator:
https://streamable.com/hk95ie
Otherwise, I think I'll just try to clean up the unit as much as possible and reconnect it for another frequency sweep.