I'm not a plumber. Redwood is.
However, there is a standard dimension for kitchen sink and bathroom sink tail pieces. The outside diameter (OD) for kitchen sink tail pieces is 1 1/2 inch while that for bathroom sinks is 1 1/4 inch. So, regardless of what the tailpiece is made of or what colour it is, the trap adapter will be made to fit snugly around a pipe of 1 1/4 or 1 1/2 inch outside diameter. That means the trap adapter should fit around your tail piece snugly no matter what it's made of.
Now, before you take anything apart, use a mirror and a bright light to confirm that the water is leaking out at the top of the trap adapter. It could be leaking out higher up and just running down the back of the tail piece where you don't see it, and accumulating on the top of the trap adapter.
Once you confirm that the leak is at the top of the trap adapter, then the fix is to repair the seal between the trap adapter and the tail piece. There are two ways this seal is made:
1. The threaded cap at the top of the trap adapter may have a lip molded into it's inside diameter that makes the seal, or
2. there won't be any lip on the inside diameter of the threaded cap. Instead, you'll have a bevelled nylon washer under the threaded cap that makes the seal.
If you see a "lip" molded into the inside diameter of the threaded cap, then it's that lip that seals around the tail piece. You can probably fix the leak by cleaning both the OD of the tail piece and putting a new threaded cap on with the same kind of lip on it. Sometimes they sell the threaded caps separately, but if they don't, you'll just have to buy a new trap adapter and just use the new cap to replace your old one. If you can find the name of the manufacturer on the old trap adapter, it'd be best to buy a new trap adapter made by the same manufacturer. That way the caps will be identical. But, more often than not, the cap from one manufacturer's trap adapter will fit on trap adapters made by other manufacturers, so they're somewhat interchangeable in that respect.
If you have a beveled nylon washer under the cap, then it's easier. You can buy a replacement beveled nylon washer at any hardware store or home center. They come in two standard sizes; 1 1/2 inch for kitchen sinks and 1 1/4 for bathroom sinks. Every hardware store and home center will carry both.
So, you need to confirm the source of the leak. Then determine whether it's the cap on your trap adapter that makes the seal or a beveled nylon washer under that cap, and replace the cap or the washer accordingly.
The leak will only occur when the sink is draining, so you'll probably need to get your wife involved in pulling the drain plug on a full sink to confirm the source of the leak.
Redwood may want to add to this if he can see anything I missed.