For good cancellation behind inline cardioid subwoofer arrays like end-fire and gradient, it's mission critical that the subwoofers have matched levels. But how does the physical displacement between the subwoofers affect the array's performance over distance?
The criteria that need to be met for cancellation in excess of 15 dB are very stringent (figures 1 & 2):
- Relative phase offsets between the canceling subwoofers have to be 180° ± 10°
- Relative level offsets between the canceling subwoofers have to remain within ± 1.5 dB or less
If these conditions aren't met, the window of opportunity for cancellation in excess of 15 dB will be missed.
The primary goals for cardioid subwoofer arrays are straightforward. Summation in the front, preferably as much as possible since the majority of us are members of the "SPL-preservation-foundation" (pun intended), and cancellation in the back, preferably without compromising what happens in the front in any way.
In terms of efficiency, two subwoofers that are coupling, can yield a maximum increase in efficiency of +6 dB (diffraction not taken into consideration) provided they are equally loud with 0° phase offset.
In terms of cancellation, there's the potential for minus infinity, again provided both subwoofers are equally loud in conjunction with a 180° phase offset. In the real world (unlike, e.g., a DAW) this is very unlikely to happen because loudspeaker manufacturers, despite severe efforts, are typically unable to mass produce loudspeakers within a tolerance of 1 dB or less. Now comes the caveat.
The inverse-square-law states that for point sources, in the direct field where the direct energy dominates over the reverberant energy (direct-to-reverberant ratio), sound levels drop at a rate of 6 dB per doubling distance. This implies that for subwoofers that don't occupy the same position, like inline cardioid subwoofer arrays, it's impossible to preserve matched levels over distance.
In case you wonder, in the reverberant field (indoors) that 6 dB loss rate is typically decelerated by room gain also known as low frequency buildup which helps preserve energy over distance in exchange for reduced impact.
Figure 3 shows two subwoofers in front of each other (inline). To keep the math simple, both subwoofers produce a level of 100 dBSPL at 1 meter distance. The live sound industry's, vented enclosure with direct radiator(s), "benchmark" subwoofer, for all intends and purposes, is virtually omni-directional when measured under the right conditions.
In the front of the array, at 1 meter distance to the grill of subwoofer B, subwoofer B is physically two times closer than subwoofer A and therefor 6 dB louder (inverse-square-law). Both subwoofers will add constructively because they are coupled but, due to this relative level offset at this proximity to the array, not yet yield 6 dB of summation together.
Behind the array, at 1 meter distance to the grill of subwoofer A, subwoofer A is physically two times closer than subwoofer B and therefor 6 dB louder (inverse-square-law). Both subwoofers will add destructively because they are cancelling (regardless of how this is achieved) but, due to this relative level offset at this proximity to the array, not yet achieve a combined level of minus infinity (in theory) together.
In the above example, this close to the array, the total back-2-front difference is a mere -9 dB. Meaning that the combined level in the back is down by 9 dB compared to the combined level in the front.
Notice in figure 3 however that, at greater distances on either side of the array, the levels of each subwoofer individually, start to approach each other. As distance increases, the relative level offset between both subwoofers starts to approach 0 dB by itself when the inverse-square-law renders the physical displacement between the subwoofers meaningless.
With the relative level offset between both subwoofers "maturing" over distance to 0 dB, the combined summation in the front of the array will automatically approach +6 dB and the combined cancellation behind the array will automatically approach minus infinity (in theory).
The overall absolute back-2-front difference increases on its own and the cardioid pattern will become more profound. A phenomena that I've come to call "auto-efficiency". In the interest of the neighbors, this would be the way to go about it. Simply refrain from tampering with levels and the situation will automatically improve.
So what if you don't care about the neighbors and insist on a quiet stage, closely behind the array, where the physical displacement still matters and the relative level offset between both subwoofers is not yet 0 dB (requirement for perfect cancellation)?
You would have to address the relative level offset, uniquely defined by that very particular distance behind the entire array, due to the physically displaced subwoofers. One way of doing that would be, e.g., to attenuate the rear subwoofer (subwoofer A seen from the front) electronically. As tempting as that might sound, you now risk "pattern-implosion".
By lowering the level of the rear subwoofer, the front subwoofer is relatively speaking working harder and will, e.g., trigger the limiters earlier, suffer from thermal power compression earlier or suffer from voltage sagging in the amplifier earlier.
On the onset of compression, the 0 dB level offset (mandatory for perfect cancellation) you carefully achieved by tweaking levels electronically is no longer preserved and the pattern implodes. There are better ways of doing this which I'll save for another article.
Besides, regardless of how you achieved that required attenuation to match levels in the near-field, directly behind the entire array, it's absolute. That level offset will always be there, regardless of distance! "Auto-efficiency" would have normally improved cancellation over distance by reducing the relative level offset to 0 dB! You're exchanging far-field cancellation in favor of near-field cancellation. Choose wisely!
For those that wonder how to setup actual end-fire and gradient arrays please consult the Subwoofer Array Designer manual.