Shaking steering wheel on 1999 Elantra

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.
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At 128,000 miles, you have to start considering wear in the steering components, struts, wheel bearings. A front end alignment and inspection at a good shop would point out what's wrong. Notice I said 'good shop'. Finding one of those is hit or miss, unfortunately.

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How old are the tires? There my be water in them if they have been refilled several times, and now with the colder weather, condensation could have formed. I would also check your tie rods. A good shop will find the problem. Good luck. Bill Knauer
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That's a great idea about the water!!! I've never seen that mentioned before but it sure makes sense!! I've had a clump of snow wedged in a wheel already that just threw the car all over the place. Now if there were only a way to get the water out short of unmounting it. Someone needs to invent 'dry air in a can' to remove it.
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