Active Subwoofer Recommendations

+4 on the Rhythmik L12.  Down 3dB at 18Hz for about 500 bucks -- that's just nuts. 

I use two subs — left and right — powered 8”   Velodynes. The bass is utterly awesome. It is almost as if the room shakes. 😁

Before you can enjoy it all, you have to integrate the subwoofer with rest of your system, and how you connect a subwoofer to the rest of your home theater setup depends on whether it is Passive or Powered. That’s how we do in our management setup. حسين البداد

3rd the Rythmik L12.  They stand out in sound quality at this price.  The only other subs I would consider at the price point are HSU.  

Paradigm Defiance V10 Retail $550 - Has Anthem Room Correction and full App Control from your phone - wireless capable although doing that is a last resort, better to run a wire - check it out at Paradigm.com and look up videos on YouTube about the Defiance Subs.

+1 on the Rythmik. Servo Subs.....yum

Rythmik L12 is <$550. It should be good unless your space is huge.

A pair of small 8" units, sealed, might work quite well. you don't need much, it may seem loud, but it is still just a sound bar.
I've no idea how the B&W is wired for handling a sub, though.

At that stated $500 price-point, there is minimal to nil audio performance differentiations between your contenders and pretenders. It’s largely a “just pick on of ‘em” exercise for you.

Personally, I would strongly recommend that you stretch the budget and go with a pre-owned quality build B&W PV1 home cinema subwoofer, as the ideal synergy match to your sound bar.

google the reviews.