Phase Wavelengths: The End Fire Cardioid Array made visible
The end fire array is one of the most talked about, even if not the most often implemented of the cardioid subwoofer arrays. It can be a challenge to wrap our heads around how we get the speakers to play leap-frog in the forward direction and demolition derby on the back side. Looking at the speakers from a coverage angle point of view is a non-starter. They are omnidirectional. How do you add 360 degrees and 360 degrees? Easy: A 720 degree speaker (only made by Bose).
Driven by Phase
But seriously, the answer to the end fire behavior is not in the amplitude domain. All of the speakers face the same way, they overlap by a factor of 100%. The spatial picture of the level is only a small factor in the upper range of our interest, 125 Hz, where the speaker has become somewhat directional.
The End Fire defined:
a line array of emitters (in our case: speakers) that are spaced and time-sequenced to provide in-phase addition on the forward side and out-of-phase rejection on the rear. The timing is set to compensate for the diplacement between the sources in the forward direction. The most forward element is delay the most, and sequentially less as we approach the last element.
In our example we will use 4 elements (you can use more or less- more makes it more directional) and space them 1 meter apart. The delay required will be multiples of 2.9ms to sync them in front. The physical setup is found in Fig 1.
<<<<<< Note: The pictures here will expand to full-size if you click on it. Much easier to see the fine lines>>>>>>
The next thing to view is the individual radiation character of a single element in our frequency range of interest. The MAPP plots are 1/12 octave, which might seem severe for an omni speaker – but we must use high resolution to see the driving action of phase as we progress. Think about the fact that an octave resolution plot incorporates a 2:1 range of wavelength. In order for us to clearly see the driving effect of phase we can’t have a 2:1 slop factor in the data. What you see in Figs 02,04,& 05 are the decreasing omni nature as we rise in freq. This means that as freq rises we will have both level and phase steering controls. At the bottom only the phase lever will be operational.
The Unfinished Product
Next we look at what could have been. What would the response be if we spaced the elements in a 1m line (facing to the right) without the sequential delay taps. We could call it the End-No-Fire array or the Begin-Fire. You choose. The reason to do this is to see where the amplitude goes. The answer is: it follows the phase. Let’s look now at the 31 Hz response in Fig 06. In Fig 06a we see the phase wavelengths laid on to the empty MAPP plot. If the speakers are 100% omnidirectional, this is all y0u need to know to see where the sound will go. The location where the lines cross is where they are in phase. The fronts of the speakers are pointed to the right but by sleight of phase we have magically moved the main lobe up and down. Fig 6b shows the combined response of the 4 speakers and indeed the strongest sound is heading north and south. The steering is not extreme, however. Why? The answer is in the phase again. The speakers are sequentially only 32 degrees apart (2.9 ms and 31 Hz). The response in the left and right directions don’t fall all the way out of phase – no 180 degree type of differrentials. Therefore the relationship between the elements is more like a lack of cooperation than a serious fight.
As we rise is frequency to 63 Hz (Figs 7a and b) the wavelength is cut in half. The displacement (1 m) is still the same but the pahse shift is now 64 degrees per element. By the 4th element we have reached 192 degrees of phase shift. The 11st and 4th elements are in full conflict. The result can be seen in the squeezing of the sides in favor of up and down. where all 4 elements are 100% in phase. As we move around the circle (from the top) we can see the lines gradually moving apart. This coincides with the gradual loss of level as we move to the sides.
Next up is 125 Hz (Fig 8a and b). Once again the wavelengths shrink in half. Now we find ourselves with the 4 speakers lapping each other on the sides and spreading out evenly in the corners. The full laps create addition on the sides – mixed with the speakers that are NOT in phase – creating a push/pull situation. This is how side-lobes are built. On the diagonals we see the deepest cancellations – due to 4 evenly spread arrivals.
The End Fire (with delay)
Now let’s add delay to the array. What happens is that part of the cycle elapses inside the electronics (the delay) and this means that the cycle completes its 1st turn at a shorter distance from the speaker. From then on it turns again at the normal distance relative to its wavelength. In our first look (Fig 9a and b) we will see 31 Hz. The four speakers all arrive in phase at the right side (in front of the speakers). Each travels a different distance, but each has a different electronic head start. The result is the all finish their first lap at the same spot and then go forward from there.
On the back side the electronic head start still applies – but the physical head start is reversed ( is that a butt-start instead?). The result now is that the phase responses fall more quickly apart – such that speakers A and D are 197 degrees apart – big time cancellation.
The next picture shows 63 Hz (Fig 10). The same thing happens in front but now the back side is spread by more than a full lap. The sides (top and bottom of our screen) gradually fall apart as we move from front to back, creating the incremental steering that concentrated energy forward and rejects it rearward. The meachanism is laid bare here – where the lines converge is where we see the energy – where they spread we see blue.
By the time we reach 125 Hz (Fig 11) we are turning multiple laps on the back side and even on the sides (hence the side lobes). There is also a small component of directionality of the speakers here.
So hopefully this helps clarify some of the mysteries of the end fire array. Comments or questions are welcome , of course.
I have done similar work on several other cardioid sub arrays and will post those when I can.