I'm trying to get my understanding in order. The monopoles have the LP and HP LR4 applied, so are –6dB at crossover, and sum to 0dB. The power response at crossover is –3dB by virtue of the magnitude responses both being –6dB (i.e., Power = sqrt(0.5^2+0.5^2) = 0.7071 = –3dB). If I'm getting it correct, the directivity is referenced back to the power computed on the basis of the "on-axis" SPL, which is 0dB. The two unfiltered monopoles each have a directivity index DI = 0dB (as they are omnidirectional). So, at the crossover frequency, do we have DI = 0dB or DI = −3dB? I can't see which answer is correct...
Isn't DI = 10log10(Q)? Here Q compares the sound intensity of the loudspeaker to what the intensity would be if the loudspeaker were omnidirectional.
So, in my admittedly simplified theoretical example, the combined source comprised of the two coincident monopoles is itself omnidirectional. Thus, any axis is the same as any other. Hence, the DI = 10log10((Pressureon-axis)^2/Powertotal) = 10log10((1^2/(0.5^2+0.5^2) = 10log10(1/0.5) = +3.0dB. Does that look correct?
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This is one of those times it can help to take a step back because simply put, directivity index is the power response normalised to an axis. Effectively it is the direct level compared to the reverberent.
I could suggest areas of doubt by saying the power remains the same while DI and response changes with reference angle, or that the response pressure has no directivity.
When you see it from the point of view that the power is averaged over nominal omnidirectionality anyway, in other words the power is counted without even considering where it came from, it's actually possible to end up with a negative DI.
Edit.. This was written before your post edit. I'd have to dig out my text books to verify what you have shown, but it looks right.
I could suggest areas of doubt by saying the power remains the same while DI and response changes with reference angle, or that the response pressure has no directivity.
When you see it from the point of view that the power is averaged over nominal omnidirectionality anyway, in other words the power is counted without even considering where it came from, it's actually possible to end up with a negative DI.
Edit.. This was written before your post edit. I'd have to dig out my text books to verify what you have shown, but it looks right.
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In the past, I've noticed a negative DI, albeit quite small, with some loudspeakers. If I recall correctly, these were vented loudspeakers, and had the port located on the rear baffle.
Directivity index is the 10log function of the sound intensity on a certain point at a fixed distance (usually far field) divided by the averaged 4π sound intensity of the source at that same distance. So yes, you can find negative Di’s. Certainly ‘off axis’…
Probably a reflection off the ladders top, which is introduced more as you turn the speaker and the drivers see the surface. It didn't appear to be a problem with this speaker so I just ignored it.
The filter results kind of speak for themselves so I'm really not going to spend any time hunting down the cause.
The images below are for an LR2, LR4 and BW3 filter at 2kHz using ideal drivers in VituixCAD. The Left Middle panels shows Sound Power (SP) in Blue and Sound Power Directivity Index (SPDI) in red.
My description wasn't quite right. The "bump" is above the xo point in the SP curve.
Below shows that the "bump" in SP above the xo point is more significant for LR4 than LR2 and produces what many refer to as "off axis bloom" that is typical of LR4 filters and is easiest to see in the Polar maps (Lower Left). A BW3 has smother PR.
LR2
LR4
BW3