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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 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.

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