I've recently had a problem develop with my 1999 Elantra.  When I get above
60 mph the steering wheel starts to shake.  It doesn't shake continually,
but goes into a sequence of shaking for about 5 to 10 seconds and then
smooths out to normal for 15 to 20 seconds before going back into the
shaking sequence, alternating between shaking and smooth normal operation as
long as I stay above 60 mph.

I first thought the tires were out of balance, so I had them rebalanced, but
the problem remains.  The car has 128,000 miles on it.  Is this a problem
that is common with this model and what do I need to do to fix it?  If this
is not a common problem, what might be the cause and what will I need to
check for repair or replacement?

Thanks in advance for your help.