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.
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.