maygene, 1st welcome to the site. This is a strange topic most guys feel the wobble is due to tire issues, inflation not proper. I have seen some discussion about bearings & actually some products are being touted to stop the "wobble". Than there is the issue that some guys never get a wobble where as some have it only at low speed, some at deacceleration, and so on and so on.
I'm sure some of the other guys will chime in with their opinions, I've never heard that it is so bad that would cause a safety concern. This is in my mind one of the finest motorcycles ever made.
... HOWDY ... IF THE WOBBLE IS BAD ENOUGH THAT YOU FEEL IT MAY CAUSE YOU TO WRECK ... GET IT INTO A SHOP OR DO A FRONT END INSPECTION YOURSELF ... LEAVE NO NUTS UNTURNED ... I MYSELF HAVE BEEN RIDING WINGS SINCE 78 AND HAVE NEVER HEARD ANYONE SAY ANYTHING ABOUT 4000 MI. WOBBLE ... MY THOUGHTS ARE IF THIS PERSON TOLD YOU @ 4k. A WOBBLE WIL START THAT SOUNDS LIKE HE HAD TIRE ISSUES LIKE FOSSIL HAD MENTIONED ... HAPPY TRAILS ...
My '84 Interstate developed a wobble about halfway through the first OEM front tyre. It was cupped so badly I replaced it immediately. On the third front tyre, I got into a tank slapper at about 60 mph and that scared the bejeebers out of yours truly. Thankfully I didn't have the Mrs. with me at the time. Turned out the stock fork brace was worn at the mounting hole so bad it really didn't offer any bracing at all. Put on a Superbrace for 5o bucks and a set of Progerssive fork springs and viola, no more cupping and no more wobble.
Now the 1500's and 1800's are a different ball game, but just pay attention to your bikes' handling and if it shakes its' head, get 'er looked at. If it's new or new enough to still have the warranty in term, it should be no cost to you if there is something wrong.
Hey I never thought of that William,my 85 LTD has a low speed wobble or "headshake" around 10 MPH or so but only if I take my left hand off the grip,it was scarey the first time because I was'nt expecting it. I run with only 60 to 80 lbs. of air in the front suspension for a nice soft cushy ride, maybe thats the problem. I use the automatic pressure settings with a passenger because of the weight. Maybe stiffening up the front end will help,I'll try it with caution.