Radial Coverage

Designing sound for the fan-shaped room. 7/13/2012 8:39 AM Eastern

Radial Coverage

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

Designing sound for the fan-shaped room.

Figure 1: The macro shapes of the room: the rectangle (red), the fan (orange), and the fantangle (black). See a larger image.

The macro shape of the horizontal plane in most listening spaces can be summarized as one of three types: rectangular, fan-shaped, or weird. The rectangle is easily described: length by width. The fan can also be described quite simply: angle by depth. There are plenty of rooms that hybridize the shapes by placing a fan in the front end that squares off in the rear. I’ll leave it up to you to come up with a better name for this than “fantangle.” The third possibility is the “architects gone wild” option: some bizarre shape that reminds us why we overwhelmingly choose rectangles, fans, and fantangles for our performance venues. The fan-shaped room is a particularly popular choice for the modern-day house of worship, chosen for it intimacy and its friendliness to video projection.

At first glance, filling the fan shape looks like the easiest design task conceivable. After all, speakers are specified by coverage angle and so are the fan-shaped rooms. So we can spec a 90-degree speaker for a 90-degree-by-50-meter room and print up the invoice. OK, let’s try it. (See Figure 1.) Oops. Just one small issue: The speaker needs to be placed at the focal point of the radius (which may be the corner behind the stage). We now have “perfect” coverage, but the speaker is behind and facing directly into all the open microphones. Obviously this is totally impractical, and we will never leave the speaker in this location, but before we go on, let’s take a careful look at the coverage we have. This “ideal” aiming point sets the standard for what we want to achieve in coverage for the radial room shape. As we move the speaker forward into the room, we will refer back to this.

The first things to consider are other options for the speaker coverage pattern. Matching 90 to 90 means that the center seats at a given distance will be 6dB louder than the outermost seats. That is the maximum variation we want to see over the arc. We certainly can’t use a narrower speaker than a 1:1 match of the room shape, but we could use a wider one. As the ratio of speaker angle to room angle increases, the variation between the center and outermost seats will decrease. We can weigh this certain benefit against the potential problems from sidewall reflections by factoring in the location and acoustical properties of the side walls. If they are highly reflective, we may stick with the 1:1 ratio. If there is an outer aisle (a buffer zone) and/or the wall is covered with soft goods, we can expand the speaker’s coverage substantially (up to a maximum of a 2:1 ratio) to optimize angular coverage uniformity. A unique aspect of the speaker being located at the focal point is that the center vs. side coverage is uniform all the way from front to back. As soon as we move the speaker forward out of the corner, that will cease. From now on the radial coverage will vary with distance from the speaker(s), ranging from too narrow to too wide, with a sweet spot in the middle.

Figure 2: (a) A single 90° speaker at 25 meters covers only 50% at the rear, (b) a single 180° speaker covers 90% at the rear, (c) 2x90° coupled point source covers 100% at the rear, (d) 2x90° uncoupled array covers 100% of the rear. See larger image.

Going forward

Now it’s time to move the speaker forward. The farther forward we go, the more stage area we get behind us. That’s all good, but now the 90-degree speaker shape no longer lines up with the 90-degree room shape. For every foot we move forward, we will need a foot more speaker to spread out over the room. It is not literally a foot more speaker, but there is a proportional relationship of give and take here. The easiest proportion to visualize is 50 percent. If we cut one parameter in half and double another, then things will even out.

For our example room, we will use a 90-degree radius with a 50-meter depth. In practice, there are many fan-shaped rooms where the apex of the fan is not in the performance space. This area might be backstage or sometimes in the back yard. Even so, our evaluation of the room shape begins from this focal point. Let’s move the speaker forward to the midpoint (25 meters) and observe the response at the rear of the hall (shown in Figure 2a). We now cover only half the width we had before, i.e. 45 degrees out of the 90 degrees. If we double the speaker coverage angle to 180 degrees, we will have coverage restored within 6dB for nearly the full 90 degrees by the rear of the room (see Figure 2b). Closer in, however, the coverage will not reach to the outermost seats. This is because the outer rows are at a much longer distance than the center rows. Even a 360-degree speaker could not solve this because of the differences in path length between the center and sides.

Alternatively, we can get 180 degrees of coverage at this distance by making an array from a pair of 90-degree speakers. There are two ways to do this: coupled and uncoupled. In the coupled version, the speakers are centered, positioned together, and splayed 90 degrees apart. The coupled option (see Figure 2c) provides a better fit over the 50-meter radius than the single 180-degree speaker. However, the closer outside areas still have too much distance to overcome from a central source. The second option (see Figure 2d) separates the room into two 45-degree slices, centers the speakers in the two halves, and splays them at 45 degrees. This option, the uncoupled point source, is the most common method for covering the fan-shaped room.

It doesn’t stop there. From this same distance (half the depth), we can divide the room into three slices and use 3x60-degree speakers at 30 degrees, four slices with 4x45-degree speakers splayed at 22.5 degrees, and on and on like that.

Radial Coverage

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

Designing sound for the fan-shaped room.

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.

Radial Coverage

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

Designing sound for the fan-shaped room.

Figure 5: (a) Coverage width for a 90-degree speaker placed on the floor at 6.25 meters, 12.5 meters, 25 meters, and 50 meters, (b) coverage width for same speaker from a height of 25 meters, (c) same from a height of 12.5 meters. See larger image.

A field example

Our example room is still 90 degrees at 50 meters. Our clusters are located 22 meters forward of the apex (at the stage edge) at a height of 8 meters. Our closest seats are 6 meters forward of the cluster(s), which is 28 meters from the apex. Can we do it with one? In the front row we need 40 meters of radial coverage length. The 90-degree speaker only gives us 14 meters of coverage width there. Bzzt! Thanks for playing. Could a wider speaker do the job? Sorry. We are losing the game by a 3:1 margin here, so it would take a 270-degree speaker. Any takers? Next option: We could raise the speaker up to 24 meters, but now it’s above the roof. Two clusters would expand our coverage to 28 meters and three would give us 42 meters. We needed 40 at minimum. We have a winner. Well, there you have it: It takes three 90-degree clusters to cover this 90-degree fan-shaped room. But wait. The position for the three-cluster scenario blocks the video screen. One option is to move the clusters up by one meter (about 10 percent) and use 80-degree speakers. The extra height allows us to use a narrower speaker in the front, and this will help us to reduce some of the extra coverage at the far end. How about four clusters? Four 90-degree clusters would be too much, but there are other options. The first is four 60-degree clusters and another would be a pair of 90s and a pair of 45-degree clusters. The latter is the one that was actually used on the design.

Let’s summarize things to make sure this all comes together. We have established how to characterize a fan-shaped room, its apex, its length, and angular width. We explored how to fill the shape from the apex or any point forward of that. We saw how the height of the cluster affects the width of the coverage on the floor, making the coverage shape change from front to back. Finally we put it all together to show how to make decisions on how many clusters of what coverage angle.

So that takes care of the fan-shaped room. What about the fantangle? Remember that hybrid of fan and rectangle. In the front it widens with distance, and then it stays the same width in the rear. No speaker is going to do this. Our approach is to first see which is the dominant shape. If there is more of the room in the fan, then go with the fan approach. If more of the room is the rectangle, then go with the rectangle design approach. What’s the rectangle design approach? Well, that’s another story.

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