Benefits of a cardioid subwoofer?

Cardioids and dipole sub-woofers couple differently to room modes than monopoles in the same location. This works very well with dipoles and the height mode.

Cardioids implemented with an acoustic delay and dipoles don't benefit or suffer from room gain (a 12dB/octave boost below the room's fundamental resonance assuming an infinitely rigid room) which is one source of boomy bass in small rooms.

The disadvantage to cardioids implemented with an acoustic delay (this is the case for the BASIS 11K sub-woofer, which has one driver plus an output slot on one side) and dipoles is that maximum output decreases 6dB/octave compared to a monopole which is already decreasing 12dB/octave. In a domestically friendly enclosure you get 18dB less output at 20Hz which is the difference between sufficient and inaudible.

You can also tame room modes with multiple conventional sub-woofers without this drawback. You can even eliminate eliminate room gain and fix the height mode problem with a pair of sub-woofers at each room corner wired with opposite polarity.

Complicating the discussion a bit is the fact the Basis sub doesn't appear to use a rearward-facing driver, rather it seems to have some vents on the sides which I assume is part of its pattern control. Also, the picture in the link of the Legacy speaker shows two drivers front-to-back in free air, which will produce very different results from that of i.e. a pair of sealed boxes back-to-back.

The theory behind a cardioid radiator is pretty straightforward, pretty much a cardioid microphone in reverse. In all cases, the pattern control is frequency-dependent . . . in the case of both microphones and loudspeakers, playing with different configurations of diaphragms and vents can affect both the directivity and frequency response.

The limitations of these approaches are peculiar to a domestic application, as the wavelengths of concern are larger than the rooms in which the speaker resides. There's also the issue that a woofer is frequently placed near a wall, in a manner where one vent or diaphragm is several times the distance from one room boundary than another. So while a woofer may produce a far-field cardioid response in a large room, in a domestic-sized room this is completely out the window no matter what the design of the loudspeaker.

Curiously, there currently seems to be very little theoretical consensus as to how to acheive "proper" bass response in a domestic-sized listening room. One of the most thorough and convincing analyses that I've seen is that of Dr. Earl Geddes, which I belive has been highly influential on Duke's approach to the problem. Crudely paraphrased, he maintains that below a critical frequency (the "Schroeder" frequency) sound in a room behaves completely as excitations of modes, and the key to smooth, even bass response is to use many radiators of diverse characteristics placed about the room.

I recently purchased one of these for my HT:

http://www.audiogon.com/cgi-bin/cls.pl?homesubw&1308355063&/Underwood-Hifi-Special-Edition

These are set up for bipolar output, with the supposed benefit being "to cancel cabinet vibration." Well, I can say that yes, there is extraordinarily little vibration. More importantly, the quality (both in 'tightness' and in range) of the output is at least as good as the JL F113 that I had auditioned in the system previously. Whether or not this is at least partially due to the bipolar design I cannot say. Needless to say, I think that these are true bargains if you're in the market.

Duke, I would dearly love to hear 4 of these things set up in a SWARM-type system!