Vertical Room Shape
The vertical room shape is evaluated as the head height coverage line from the front row to the last seat. The shape is evaluated as angular spread (from top seat to bottom seat in degrees) and range ratio (the difference in distance from top seat to bottom seat). If the shape is too complex to be evaluated like this then a single loudspeaker is a bad choice.
The horizontal aim target is defined as the middle/middle (middle seat at the midpoint depth). If the loudspeaker is centered on the shape, then the aim point is obviously along the front/back center line. If the loudspeaker is not centered, then the asymmetry is best balanced by aiming through the midpoint center. For every inch we move off center, we must pan the loudspeaker inward to cross the mid-point depth. This assures equal distribution of the over and/or under coverage.
The vertical aim is found by the range compensated coverage method (Figure 2). Consider the vertical bottom (VBOT) of coverage to be 0 degrees (relative). Our example coverage target is 50 degrees, so the range is from 0 degrees at the vertical bottom (VBOT) to 50 degrees at the vertical top (VTOP), with a midpoint of 25 degrees (ONAX). If the range ratio from VTOP to VBOT is 1:1 (0 dB) then the aim point is the vertical center of the coverage line: 25 degrees.
As range ratio rises, the aim point gradually moves upward toward the farthest point. A range ratio of 1.4 (3 dB) moves the aim point upward by a factor of 1.4 (+40 percent) to 35 degrees (10 degrees above the original aim). Increasing the ratio further raises the aim until the 2:1 limit is reached, and the loudspeaker is aiming at 50 degrees (VTOP).
Horizontal Coverage Angle
There are three logical points to evaluate coverage: start, middle and end (Figure 3). If we use the start, our loudspeaker covers the front fully but is too wide for every row beyond. We may drown in reflections as coverage overflows to the side walls.
The opposite extreme is the rear width as reference. We have just enough coverage across the back (6 dB variance) and not enough anywhere else. Reflections are minimized, but tell that to the half of the audience that has no coverage.
Using the mid-point width as the reference evens out the errors. The result is 6 dB of level variance across the midpoint depth. If the shape is a simple rectangle then the front half will be under-covered and back half over-covered in equal proportion.
Reflection risk rises in the rear while coverage gap risk rises in front. The gaps in the front corners can be reduced if the loudspeaker is raised in the vertical plane (which expands the effective coverage width).
Alternatively we can beg for fill loudspeakers to plug the gaps. Both the underage and overage errors are reduced if the room has expanding splay walls.
Vertical Coverage Angle
The vertical aim was determined above and we will reuse the 50-degree coverage shape example (Figure 4). The minimum variance coverage angle is found by range ratio multiplication of the target angular spread.
If the ratio is 1:1, then the loudspeaker coverage angle equals the target angle: 50 degrees. The ONAX location will be 0 dB and the VTOP and VBOT locations will each be –6 dB. A wider loudspeaker will reduce the level variance, which can be weighed against the potential for increased reflections.
A range ratio of 1.4 (3 dB) moves the minimum coverage angle upward by a factor of 1.4 (+40 percent) to 71 degrees. Recall that the loudspeaker is now aimed above the vertical midpoint; therefore we need a wider loudspeaker to reach the bottom. The loudspeaker coverage angle rises proportionally with range ratio until the limit is reached at 2:1, and we have a 100-degree loudspeaker (a 2:1 ratio of the original 50-degree loudspeaker coverage) aiming at the top row.
ONAX is now at VTOP and both are -6 dB. VBOT is also -6 dB. Level variance is minimized while risk of reflections is maximized. If the reflections are too great then a single loudspeaker approach should be abandoned in favor of an array.
Bob McCarthy has been designing and tuning sound systems for over 30 years. His book Sound Systems: Design and Optimization is available at Focal Press (www.focalpress.com). He lives in NYC and is the director of system optimization for Meyer Sound.