I'm not sure if I can answer the OP question. Perhaps refine it? Here are some measurements that I hope can cast light on the sound field behind a speaker, and the effect of a baffle vs. no baffle.
I have an old ADS tweeter, with ~5cm minimal baffle. I think B&W has reduced the baffle more than this example, but it is OK. The ADS was a competent tweeter, I haven't used these in almost 40 years and am eager to see the measurements.
I made a quick and dirty baffle out of a board. It will introduce a host of issues.
I also have a box that will supposedly simulate a regular cabinet.
Also, Harry Olsen already did this study almost a 75 years ago. But not so focused on off-axis, and not on tweeters radiating from minimum baffle.
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If I consider the tweeter with a small but finite baffle, I expect to see an interference pattern in any rear-radiated sound. The frequency, the angle, and the distance to the microphone will modulate the pattern. Consider the ADS tweeter alone:
We expect a bright spot (peak) in the rear-radiated response at 180 degrees for an obstruction! This bright spot is seen in optics, and acoustic waves. And we expect the nulls and peaks in the interference pattern to be periodic in angle due to the superposition of waves of half-wavelength difference. The shape isn't regular like a sphere or a point-source, so this isn't really a simple textbook example.
Here are several off-axis measurements of ADS tweeter with no baffle, taken at 1 meter distance, angles 120 to 180 degrees:
The 180 degree off-axis measurement (heavy purple trace) has no peaks and nulls across frequency, this is the consequence of the 'bright spot' 180 degrees off axis as mentioned earlier. This also suggests I got the tweeter and mic lined up pretty well, I only see a wiggle at 21kHz so.
I didn't take enough angular measurements to see the entire progression of peaks and valleys, but hopefully this makes the main point: we expect a peak in the 180 degree rear-radiated sound field, with nulls and peaks across angle and frequency. At 10kHz the rear-radiated signal is 20dB down, and 50dB down by 20kHz. Tweeters beam, especially radiating into free space, so this isn't too surprising.
The 180 degree does indeed have higher output at 180 degrees. The difference between no-baffle and the two baffle configurations at 180 degrees is an additional 10 dB or more of attenuation. Not that the no-baffle tweeter has much output at 180 degrees, but the tweeter with baffles produces even less.
Also, I did a crap job of making a good baffle
:
I am pretty sure the jagged artifacts in the response of the no-baffle are the tweeter protruding out of the baffle. It clears up to a certain extent in the box since I adjusted the tweeter to be slightly more flush to the surface of the plywood. I used minimal smoothing, so you see all of the warts.
Going back to the no-baffle example, where is the energy in the 180-degree off axis coming from? In fact, the 180 degree measurement shows the SPL is higher than all of the other off-axis measurements (105 to 165 degrees).
So, yes, in a narrow band right behind a speaker you will find a tweeter has more energy. Go a few degrees off of that 180 degree axis and the energy is actually lower than the 180 degree response! Tweeters with no baffle don't just tend to beam in the forward direction, but also backwards!!! I have a hard time thinking this backwards beaming is strongly audible, perhaps if you had a highly reflective surface right behind the speaker with treated surface just off the 180-degree axis. Would need to reflect back at the main listening position. And would need to be significant relative to all of the another reflections in the room. And if audible, would be very MLP and speaker position dependent. As I said before, integrating over the entire sound field of direct and reflected energy, this is really small, and very room and setup-dependent. While this type of B&W are really fussy in setup and room, I am pretty sure it's not because of this.
Here is the Spin of the
No Baffle compared to
Full Baffle:
I used no smoothing. As stated earlier, the tweeter isn't so flush and well integrated in the baffle, but hopefully this makes the point. Yes, over a narrow angle you get 180 degree rear-beaming from the tweeter without the baffle, but less rearward energy to all other angles, and lots of peaks and valleys due to the superposition of all of those angles' radiation.
The baffle reduces that rear-beaming right on the 180 degree axis, and reduces the periodic peaks and nulls in the rear sound field. It also changes off-axis behavior of the direct sound field by reducing the directivity. Many find this extremely useful and audible since it is tens of dB more significant than the rear-radiated field.
I'm not saying the 180 degree off-axis beaming doesn't happen, it does as clearly seen in the measurements. I think what is going on in front of the B&W speaker of this type is what is dominating the sound. I hope this also puts in context the comment about Spin interpretation, you do need to integrate over the entire sound field, unless you have a reflector set up to beam that narrow band of energy right at the MLP, and even than its still tiny... Also, my comment about the Bose, which sends 90% of the sound backwards, the tweeter with no baffle is much less than 1% in that narrow rear-firing band.
I got through that without mentioning diffraction (Basil Fawlty would be proud
). All this can be deduced without having to mention that though!