Weird howly/high pitched noise

I have a 2004 Elantra, and lately it's been making a horrible noise.

It sounds like when you dip your finger in a glass of water, and rub it
around the rim of a glass. That sort of "howl" vibrating/ringing type
of noise.

At first it did it very infrequently and only quietly.

Lately it does it more and more, and is much louder. It only does it
after driving for a while, and especially when I've been in stop and go
traffic (as often happens living and working in a big city). After I
turn it off for a while, the noise goes away, but will come back. It
does not happen every time I drive, and when it does happen it will
stop and start.

I thought it was a wheel bearing, and took it to my mechanic who
checked and said it was not that. He could not make the car make the
noise, either.

Then I thought perhaps it was to do with the transmission which had
been shifting late, so I took it to my dealer, and he fixed the
transmission problem, and could not make the car make the noise for
him. For a few days after I got the car back, it did not make the
sound, so I thought it had been fixed, whatever was making the noise.
But then it started again. Quiet, at first, but then today it made it
really loud after I got off the highway in stop and go traffic. I
noticed it made it first when I came to a corner and stopped to wait to
turn right. Then it made it quietly while I was driving straight, and
very loud when I turned right into a parking spot.

This is very frustrating that no one can make the noise happen, and no
one can stop it. Any idea what it could be? Power steering? Some
other bearing?

My biggest fear is that the car will spontaneously break down or become
undriveable on my way to or from work one day, stranding me somewhere.
The AAA is under my husband's name only, and we can't afford to add me
to it right now.