325 BMW with a humming noise coming from the exhaust?

I’ve owned this 1987 BMW since February, and after immense and expensive repairs, it became roadworthy. Recently there has been moderately high pitched hum coming from the Drivers side, approximately under the rear door area of the car. It seems to be coming from a part of the exhaust system(?). This part is white, vibrates, and is cylindrical in appearance.

The car only had one previous owner. He kept all record of the car over the year, so I know of no wrecks, repairs to the exhaust, and/or body work done to that area.

It is a menacing sound but doesn’t seem to be affecting anything, and it only happens occasionally. I have noticed it mostly when idling, but it still occurs some when in first and second gear.

Please help! Can someone identify this part? I’ve looked through a few diagrams, but I am unsure. I need to know the importance of this hum, and whether I should seek a mechanic.

At 27 years old there must have been some work done. I doubt the seller was completely honest there, but that doesn’t mean that the car is a wreck. I hope that at least he did the recomended maintence that the owners manual calls for.

From the discription, I’d say that you are looking at the muffler. It vibrating, may be making the noise that you hear. Sometimes a very small exhaust leak will also sound like a hum. If you can hear it in your driveway, try using a stick…like a broom handle and press on it and see if the noise goes away or gets worse. If it changes, it could just be a heat shield, or a hanger that has rusted thru. Simple fixt any shop.

Yosemite

You’ll want to find a good independent BMW shop - who did the “immense and expensive repairs”? Don’t you want to take to them? The only way to find out the cause is to have somebody look at it. It could be as simple as a loose heat shield, or it could be a leak (which you really need to fix) which’ll require replacing some/all of the exhaust system. At that age it wouldn’t be surprising.