Polar pattern control of multiple sources  Using directional techniques on Subwoofers can have beneficial advantages due to room coupling, etc. But a full understanding of how they work and what some of the shortcomings are of each system is needed. There’s no “magic-bullet,” but when used correctly these techniques can go a long way to taming the “last-frontier” in audio.
Gradient Arrays Subwoofer Directivity can be achieved using the principles presented by Olson  concerning gradient loudspeakers. Gradient loudspeakers utilize the techniques of microphone polar pattern control, but applied in reverse. These methods (with the exception of zero-order variety) call for two or more spaced sources within the array to achieve the desired directionality. A number of configurations are presented by Olson which provides a useful tool set for low-frequency pattern control. (You can skip-over the math stuff; it’s just there for the math-geeks!)
Zero Order The zero-order gradient source is the building block for all higher-order gradient sources. This consists of a single source which radiates energy equally in all directions, omni-directional* (Fig 1).
Fig. 1: Zero-order gradient source (a) configuration and (b) polar pattern
*NOTE: an assumption of dimensionless point source behavior is assumed. In reality, sources should be modeled as CDPS (complex directional point sources), and boundary effects should be allowed for.
First-order First-order gradient sources combine two zero-order sources one of reverse polarity (Fig 2). This configuration’s polar pattern is highly-dependant on the physical separation of the two sources (Eq. 1).
Fig. 2: First order gradient source configuration (dipole)
Spacing at a quarter-wavelength of the target frequency results in a dipole pattern, but spacing at a full wavelength gives a four-lobed pattern (fig 3a,b).
Fig 3: First-order gradient source (dipole) polar pattern with drive-unit spacing at (a) ¼ wavelength and (b) full wavelength
Cardioid The dipole first-order gradient source can be adjusted to give a cardioid pattern. This involves adding electronic delay to the second source which directly corresponds to driver spacing (Fig.4). Again, the polar pattern is highly-sensitive to source-separation (Eq. 2) as a quarter wavelength spacing gives a cardioid pattern, but full wavelength spacing gives a dipole pattern rotated 90 degrees off-axis (Fig 5).
Fig 4: First-order gradient source configuration (cardioid)
Fig 5: First-order gradient source (cardioid) polar pattern with drive-unit spacing at (a) ¼ wavelength and (b) full wavelength.
Given that the range of wavelengths usually covered by atypical subwoofer systems (20Hz – 100Hz) are in the region of 17m (56ft) to 3.4m (11ft), a ratio of approx 5:1, we can see in order to achieve pattern control over the whole pass-band using this approach is impossible. A compromise is to pick a control frequency somewhere in the middle, say 45Hz, and using ¼ wavelength spacing (approx. 1.91m / 6.27 ft) and a delay time of 5.55ms accepting it will “fall-apart” at the extremes of the pass band.
Second-order Second-order gradient sources are formed using two dipole first-order sources and placing them together with a physical separation and an electronic delay on the second first-order source directly corresponding to the separation distance (Fig 6). The polar pattern is described by Eq. 3 and, like with other gradient configurations, is largely dependent on source spacing (Fig 7).
Fig 6: Second-order gradient source configuration.
Fig 7: Second-order gradient source polar pattern with drive-unit spacing at (a) ¼ wavelength and (b) full wavelength.
Higher order gradient sources are realized by combining zero-, first- and second—order configurations in a similar manner. It is expected that as source order increases, polar pattern becomes increasingly focused. It must be noted, however, that as order increases source efficiency decreases due to destructive interference between drive units. Modifying a subwoofer’s polar pattern not only causes a decrease in efficiency, but can also drastically affecst optimal source placement within an acoustic space.
Smaller Spaces: Cardioid sources have been suggested to be relatively position-independent in terms of room-mode coupling in smaller spaces, Ferekidis, Kempe , but cardioid sources can be fine-tuned to best suit the acoustical space by adjusting their orientation. Cardioid sources have also shown to be unaffected by substantial changes in room absorption and are not greatly influenced by room asymmetry due to its highly directional radiation pattern. Below, the first room-mode cardioid sources show no clear advantage over omnidirectional sources and, in fact, can be a poor choice if system efficiency is important. With this in mind, a subwoofer array with a hybrid, frequency-dependant polar pattern below the lowest room-mode, the subwoofer operates as an omnidirectional source, while above this it operates as a cardioid source. This approach ensures system efficiency at VLF while simultaneously reducing source position sensitivity.
Hybrid Array There are two ways of achieving this; one approach (Fig 8a) involves applying an all-pass filter (APF) to one drive-unit to modify its phase, achieving the desired polar pattern. The alternative approach* (Fig 8b) places a high-pass filter (HPF) before one drive unit so only one of the drivers radiates energy below a defined frequency. The first method is preferable since it has the advantage of two drivers radiating below the lowest mode, resulting in better pressurization of the room.
Fig 8: Hybrid loudspeaker configurations.
*A variation of this second hybrid approach is to utilize drivers of different sizes/pass-bands to assemble the array. Next time we’ll look at end-fire arrays and the pro’s and cons between end-fire and cardioid. REFERENCES:  Olson, H.F. “Gradient Loudspeakers” JAES vol. 21 #2. Pp 86-93. March 1973.  Ferekidis, L; U. Kempe. “Beneficial coupling of cardioid sources to small rooms” AES 116 AES paper 6110. May, 2004 [3} Taken from: Adam J. Hill Doctoral Thesis, University of Essex, Jan 2012. Pgs. 65-68
About Martyn “Ferritt” Rowe Industry veteran and OSA’s Director of Engineering Services Martyn “Ferrit” Rowe brings nearly three decades of real-world experience in live event technical services. Ferrit most recently came from Martin-Audio as the technical training manager for MLA, and uses his vast knowledge and expertise of the multi-cellular technology to support client projects as well as support and train engineers and technicians. Ferrit began his career running cables on a Thin Lizzy “Live and Dangerous Tour,” and then taking on the roles of running monitors, front of house, and system technician for some of the most popular acts in music, such Judas Priest, Ozzy Osborne, Black Sabbath, The Police, KISS, The Who, Elton John, Poison, Bon Jovi and Van Halen. Ferrit’s training career began in 2000 when the first of the line array’s from V-DOSC emerged and became an instructor on the line array theory and continued that path with various systems over the years before joining the Martin-Audio MLA division, and then bringing his knowledge and expertise to OSA International, Inc.