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Radial Coverage

Jul 13, 2012 12:39 PM, By Bob McCarthy

Designing sound for the fan-shaped room.

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Figure 3: (a) 2x90° uncoupled array at 12.5 meters covers 100% by the 25m midpoint of the hall, (b) 2x90° uncoupled array at 6.25 meters covers 100% by the 12.5 meter quarterpoint of the hall. Over-coverage at the rear of the hall. See larger image.

So far we’ve looked at two extremes: covering from the apex (while assuming an infinitely small stage) and covering from the midpoint in the hall (which leaves us with half the room as our stage). Neither of these is the most likely scenario we will encounter, but we have bracketed the extremes. The practical solutions will be in between and follow the same rules of coverage/room relationship.

The next step is to return to the uncoupled pair of 90-degree speakers and make another 50=percent adjustment. The speakers move to 12.5 meters forward—the midpoint between the apex and our previous 25-meter midpoint (see Figure 3a). We now achieve 90 degrees of coverage along the 25-meter radial line. A trend emerges. Once again the 90-degree speaker coverage and 90-degree room shape converge at the point that is 2X the speaker’s forward distance from the apex. The first round was from a 25-meter-forward starting point and created a 50-meter-forward coverage completion point. The second round was 12.5 meters forward with 25-meter completion. The coverage continues forward as we go past the 25-meter line, where the coverage variation is steadily reduced. By the 50-meter line, the speaker’s coverage (0 to -6dB) is effectively double the size of the room’s shape and the variation inside the 90-degree room shape is reduced to practically 0dB. You can pretty well guess what will happen when we reduce the forward position by 50 percent again (to +6.25 meters) as shown in Figure 3b. The coverage reaches maturity (0 to -6dB) at 12.5 meters and carries on for the remaining 37.5 meters. As we reach the rear of the room, we have overflowed the shape by a large amount and must consider the potential wall reflections.

Let’s review the three scenarios where we use a 90-degree speaker in a 90-degree fan-shaped 50-meter room. If the speaker is at the apex, all of the seats in the hall are inside the coverage angle. That’s great, but it’s impractical. If a 2x90-degree uncoupled array is used at the midpoint (25-meter forward), the coverage becomes complete in the last row—too little, too late. If we move the same speakers closer by half (25 percent forward distance), we will cover 50 percent of the room. Move closer by half again (12.5 percent forward), and we get 75 percent of the room and on we go.

What happens if we break from the 1:1 relationship between speaker and room angle? If we cut the speaker coverage angle in half, the distance before we reach coverage completion doubles. The room gets sliced into smaller pieces. If we double the speaker, the coverage completion quickens but the overflow into the walls becomes large. Any angles in between halving and doubling will have a proportional effect in coverage completion distance.

Figure 4: Comparison of path lengths to reach 6.25 meters, 12.5 meters, 25 meters, and 50 meters forward of the origin on the floor. (black) speaker at 50 meters, (red) speaker at 25 meters, (purple) speaker at 12.5 meters, (orange) speaker at 6.25 meters. See larger image.

How many speakers?

The big question in designing a system for a fan-shaped room is, “How many clusters?” Typically we see one to four, but occasionally there are more. Which is right?

So far this looks very straightforward, but there is one more twist: the vertical plane. Our considerations to this point have been in 2D with our speaker and audience on the same plane. In practice this would be a ground-stacked PA in a flat-floored auditorium. In such a case, the level variation between the front and back is the highest possible. We gain 6dB as we move from the back row to the middle row and another 6dB each time we get half again as close. In short, think of every row as 1 meter closer and 1 meter louder. That is so 1970! \

If we raise the speaker up in the air, the relative distance between the front, middle, and back rows declines proportionally. To illustrate this, let’s make an extreme case again and then bracket it in. We’ll return the speaker to the apex and raise it to a height of 50 meters. The previous level difference between the rear and middle seats of 6dB has been reduced to only 2dB. If we move half again closer, we gain less than 1dB (instead of 6dB) and the next round yields around 0.2dB. All told, the three rounds of halvings produce about 3dB of level gain in contrast to the 18dB gain over the same distances from the ground stack. Getting more practical, we can cut the height in half to 25 meters where we see a little more than 6dB variation over the 6.25-meter to 50-meter range. When we lower it in half again to 12.5 meters, the variation rises to around 12dB. If we continue to go lower, we will have a similar response to the ground stack.

How does the height affect the coverage angle? Does the 90-degree speaker still fit the 90-degree room? This is the tricky one. We have to visualize the coverage in meters instead of degrees. Putting the speaker back on the floor and back at the apex, we can measure the coverage width for a given distance. Notice that the coverage stays constant over distance, but the coverage width varies. At our familiar range of 6.25 meters, the width (between the -6dB points) is 8.75 meters by 12.5 meters. The width is 17.5 meters and onward until we reach the 70-meter wide at the 50-meter point. Now let’s lift the speaker back up and see how it fits.

We will use a practical height of 12.5 meters with the coverage beginning 6.25 meters in front of the speaker. The seating area on the floor is 9 meters wide at this point. The path down from the speaker is 13.5 meters, which creates a coverage width on the floor of about 19 meters. Our 90-degree speaker is more than enough for the front row. By the back row the path length from this height is only 10 percent longer than from the floor, so we are nearly a perfect fit back there.

We need three numbers to determine the number of speaker locations for a fan-shaped room: the radial width of the room at the start of coverage, the distance from the speaker to the start of coverage, and the coverage angle of the speaker. It is the same technique we use for front-fill spacing on a radial stage.

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