Would it be correct to reduce your (mojave/mwillems) posts to "the transient response of a subwoofer is completely described by its frequency response & hence all subwoofers will sound the same if they are equalised to produce the same response in that room & don't produce audible amounts of distortion"?
I wouldn't say that's exactly my position. My position is that the transient response of the subwoofer is completely described by it's impulse response, which is informed by: frequency response, phase,
and decay. I would say that two subwoofers that are equalized/convolved to produce identical FR and phase response (at comparable distortion) will "sound" as close as signal manipulation can make them, but still not necessarily identical, because of decay. If you ring two bells, they might have the same pitch (FR) and start ringing at the same split second (phase), but one might keep ringing for much longer after the strike than another based on what it's made of and mounted to (decay).
For those reasons, there will still be a residual difference in the sound between two subs with identical frequency response and phase response based on mechanical factors: the drivers, box construction, and position in the room. The driver, the sides of the box, or the walls of the room are like the bells, they keep ringing once excited for differing amounts of time. Those differences would appear in a waterfall plot, but not necessarily in a frequency response and phase measurement. The only ways to "correct" those residual differences is mechanical (room treatment, box damping, different drivers, etc.).
The bottom line is that, in my view, two speakers whose outputs have identical FR and phase measurements are as close as signal processing can get them, no matter how you got there, but they won't sound identical in part because of differing time-decay response.
To provide a sub-specific example: imagine two subs that both start the uphill journey of the transient at the same time, but drop off at different rates. One of the subs may "ring" at a given frequency for a millisecond after the time zero (the transient takes a millisecond to fully drop off), and the other sub may ring for two dozen milliseconds after the time zero. Those two subs might show identical frequency response and phase response because they both ramped up to the correct volume (FR) at the correct time/part of the sine wave cycle (i.e. phase), but the time it takes them to stop making noise (decay) isn't shown on a normal FR and phase plot. And that difference will be audible; the sub that's slower to stop ringing will sound more smeared than the other sub; the sub that decays faster will sound tighter.
It's one of the reasons that vented speakers get a bad rap; a port introduces significant phase wrap and also ringing around the port frequency. In that case, I've found that fixing the phase can fix most of the port ringing, and, more generally, phase manipulation can definitely improve decay measurements to some extent. But some kinds of ringing or decay problems (i.e. box vibration, "slow" drivers), can't always be fully "resolved" through signal manipulation.
Does that make sense?
Check out this article for an interesting case study in why time decay can make a big difference in the sound of a speaker:
http://www.soundonsound.com/sos/sep08/articles/yamahans10.htmAnd check out this page over at Zaph audio:
http://www.zaphaudio.com/5.5test/compare.html . He's tested dozens of drivers, and if you compare the Frequency Response graphs with the "CSD" graphs (cumulative spectral decay, aka waterfall plot) you can see how differently different speakers perform on those tests even with fairly similar FR.
I can't speak for mojave's view (although I'm interested to hear his thoughts).