Large Advents (1974) woofer pops with deep bass.

The LA of 1974 would have a Masonite ring with a 9” paper cone in a 12” basket, designed that way specifically to prevent bottoming.  So either the inverted foam surround is perforated or there is another air leak somewhere…or you are overdriving them.  They should handle a 125W amp easily, but you may have more.  Are you boosting the low end?  Don’t. People used to stack pairs to increase their output.  They had impressive bass but there are limits!

Bad surround may be the issue & hopefully no permanent  damage has been done., There are a few surround re- foaming kits available on eBay for short $ for the original Large Advents that are worth trying & because there were so many of these speakers sold, there’s probably entire woofers available also. Jensen then distributed & I think made a replacement woofer for it that has no Masonite ring & a black metal frame that is a perfect fit & works very well. I owned a few of the Advents back in 1974 ( one pair & then a stacked pair) & again later on in the 90’s ( left in a home I bought!). 
 

I blew a few woofers playing them at stupid high levels but overall they’re were a great speaker in its day for $210 /pair in Walnut cabinet). For the equivalent & $1200 or so today, there are speakers w/ better high end & image better but few if any w/ better bass. They are warm & musical & their faults are more omission than commission.  Get them fixed & enjoy a true classic!
 

Unfortunately, these speakers, as good as they were in those days, are really not very good.  BIG SURPRISE to me when I opened my shop in 1974 and thought they were AMAZING until I put almost any other speaker next to them 

Box speakers have come a LONG WAY since then, no disrespect to Mr. Kloss, so you might be better off buying (even used) a newer speaker than putting money into fixing those.

Cheers!

I had a pair of Large Advents of the same vintage and the woofers required repair about 20 years ago.. I'm surprised your woofers lasted this long.

Send both woofers in for repair and you will be fine. If you want to keep them until you die the Simply Speakers replacement woofers are the best fix. The cost is reasonable. Even if you decide to upgrade your speakers, those Advents are collectable and people will pay a surprising amount of money for them if the cabinets are in good shape. I sold mine with repaired woofers for $300 but the cabinets and grills were in perfect condition. I see them selling on eBay for as much as $450 a pair. If the cabinets are trashed it's a matter of how much emotional attachment you have to those speakers. There are hundreds (thousands?) of used speakers available for under a thousand dollars a pair that will sound quite a bit better than your Advents.

The woofer is history.