What needs to be correct for a subwoofer to integrate properly with the main speaker? Level, phase, time, and polarity. Here is a picture from the SMAART 8 manual showing the differences of phase, time, and polarity. When integrating a subwoofer, you want the speaker and subwoofer to be in-time, in-phase, and in polarity.
Here are the same things shown using a wavelet. Under the Relationship heading are Time (Mis-aligned or Aligned), Phase (In-Phase or Out-of-Phase), and Polarity (In-Polarity or Reverse-Polarity). Any deviation is audible during playback and is one way to check out a system during calibration.
What happens to the signal in typical music listening with stereo content? The content is decorrelated between the left and right speaker until it gets down 80-60 Hz and then the signal is correlated. In other words, the bass is mono, but sometimes not until below 60 Hz. How can one measure this? There are several tools available. RME has a goniometer as part of its analysis tools. MOTU's Audio Tools has an X-Y plot and Phase Analysis. Both audio devices provide a Correlation Meter. There are also plugins available for JRiver Media Center and most DAW's that let one analyze the music for correlated vs decorrelated signals.
It is
because the bass is mono that you can use subwoofers in a stereo configuration. You don't have to worry about the bass only being in one speaker (or subwoofer). The transition from correlated signals to decorrelated is perfectly smooth. The benefit is that you aren't mixing decorrelated signals in the subwoofer. You are also maintaining the signal's time, phase, and polarity alignment.
There are multiple types of wave patterns generated by a subwoofers. There are cylindrical, beamforming, cartioid, hypercartioid, and a plane wave patterns. A single subwoofer or multiple subwoofers spread throughout the room as typically setup in a home radiate bass in a cylindrical pattern. In a rectangular room, or even a room that starts rectangular, the bass from two speakers or two stereo subwoofers (with essentially mono output) against the front wall creates a plane wave. Using a single subwoofer in a room creates a different wave pattern than using two full range speakers. The benefit of a plane wave is that both width and height modes are attenuated. This means that full range mains or stereo subwoofers have an acoustical benefit for many listening rooms.
My past 7 sets of main speakers have been floor standers. However, three of them were not what I would consider full range. The full range speakers have been flat down to below 20 Hz with output at 20 Hz higher than most subwoofers. For example, my JTR Speakers Noesis 215RT's had
20 dB more output
at 20 Hz than a JL Audio E112 subwoofer or
11 dB more output at
20 Hz than a Paradigm Signature Sub 2. This is just with a single speaker. I had three! I've also had at least dual subwoofers while owning the full range mains and have had from eight (8) 15" woofers to eight (8) 18" woofers for my subwoofers. I've lacked nothing in the ability to play down to 20 Hz with full range mains and I've also not lacked anything with the ability to add in subwoofers.
Floor standing doesn't preclude the necessity for subwoofers. I've calibrated many rooms with floor standing speakers that need the subwoofers to have sufficient distortion free output not only in the bass, but also in the midbass. However, adding subwoofers doesn't automatically somehow make full range floor standing speakers sound better. I always prefer full range speakers vs full range speakers with subwoofers if the subwoofers aren't in a stereo configuration.
When the left or right signal is no longer time and phase aligned with the subwoofer then the bass envelope is widened and the bass sounds muddy. Anytime you place subwoofers throughout the room you widen the bass envelope. It is impossible for the left speaker to be time and phase aligned with the signal from a subwoofer one foot away and another subwoofer 10 ft away. It can't be done. The range in which the speaker and subwoofer both contribute to the sound is usually from 60 to 120 Hz with most receivers (1/2 octave above and below the crossover). Sometimes it is from 40 to 160 Hz (1 octave above and below the crossover).
Full range floor standing speakers have the benefit of generating a plane wave in many listening rooms and they keep the entire signal both time and phase aligned. Can this be done with subwoofers? Absolutely. It can cost more for the dual subwoofer/speaker/dsp combination than it does for a proper full range floor standing speaker. It is extremely difficult to get the subwoofer to align in time and phase throughout the window in which both are contributing to the sound. A frequency response graphs will look the same in which the subwoofer is delayed by either one cycle or two cycles in relation to the main speakers, but will only sound right when the subwoofer time aligned.