Large Advents (1974) woofer pops with deep bass.

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.

Millersound in PA are top notch and have been in business pleasing customers for 1,000 years.  Call Bill at (215) 412-7700 and you will be all set.

However, yogiboy's link to simply speakers OEM replacements for $130 is worth looking into.

As mentioned above, whatever you do you will want to do it in pairs.

@vcheney 

Whatever elements that define the level of attachment you have to the Advents are valid ones, in my view.  Only you can place a value on them.

You probably have an "unhealthy" woofer there.  I like @dekay process for troubleshooting the woofer.  It's a no cost, zero risk method.  So, I think it's a good place to start.  You can also try gently pushing in on the woofer while in the cabinet and letting go.  The woofer should take a moment, or two, to move back to the starting position.  If not, there's a breach somewhere.

Repairing the woofers is a virtual rebuild where all things that move are replaced.  You'll essentially have a new woofer when you are done.  Magnet strenght may weaken over time.  I believe this has been the subject of debate for some time now, and I don't have a strong opinion either way. And, yes, you should do both woofers.  A) the other one is fatigued, and B) this is the only way they'll sound the same.

When this is completed you'll still be the proud owner of a fully functioning pair of 1971 Large Advents, as close to original as possible.

Good luck!