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Running The Numbers: Using “Old School” Math To Help Optimize Loudspeaker Placement

Employing trigonometry to quickly and effectively upgrade coverage; it’s also a good idea to have a backup for your computer in case it goes down during setup.
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Computer software has made the job of audio engineering much easier on the tech of today. From SPL prediction software to room dimension and aspect ratio matching, finding the right loudspeaker to cover an audience or figuring out how to work with the gear that you have efficiently can be a quick process and a lot of fun.

When attempting to deploy loudspeakers in a room, one must realize that the way the sound is projecting from the loudspeaker is somewhat predictable and that it’s worth the effort to optimize that known range of frequencies in order to start with the best chance of a great result.

Most full-range or mid-to-high loudspeakers we work with today can be considered to emit sound energy in the form of an isosceles triangle at high frequencies. The means their usable “pattern” across the listening plane is symmetrical, and if you can find that data (or measure the data yourself), you can use math to place them in an ideal location. The pattern is the area included within the -6 dB edges of the triangle (Figure 1).

Figure 1.

If pre-production is not possible and you’re walking into a job where the equipment specifications are unknown, it may be necessary to use software to quickly find placement and aim strategies for the production. It’s important to understand the underlying math, and it’s also a good idea to have a backup for your computer in case it goes down during setup.

Most devices with an app store will have some sort of “triangle solver” available. This math, called trigonometry, has been around since the first century B.C. and was used by ancient Greeks such as Hipparchus, Menelaus, and Ptolomy for astronomical observations. They used a function called the “Chord” that is closely related to the more modern “Sine” function. A great history paper entitled “The Beginnings of Trigonometry” written by Joseph Hunt, a student at Rutgers University, explains the early development in more detail.

Defining Parameters

For our application, we first need to describe the room and then we’ll understand how the loudspeaker will interact with it. Once we know the room’s width, we will understand what will happen with “x” loudspeaker in it at the very far end of its coverage length, and vice versa.
The software I use for loudspeaker placement can calculate multiple loudspeakers at once and shows how to fill the whole room or only a section. However, using a triangle calculator can produce the same result simply by dividing the room into smaller sections.

The ballroom shown in Figure 2 is one in which I often work. The layout is four loudspeakers that are matched in their coverage angle to the one-quarter (1/4th) area of the room that each needs to cover. To recap on matching goals, the loudspeakers’ -6 dB line should touch the edge of its zone at exactly the “half” or “mid” depth of that section.

Figure 2.

Figure 3 shows the ideal audience location but don’t sweat it if it cannot be arranged; it’s still minimizing the drop to the back of the room by following this goal. Remember, reflections from boundaries are like ears that throw the sound back into the room instead of listening and keeping it to themselves – don’t give them much to throw back at you.

Figure 3.

Now let’s chop up the room into equal sections and see how we can get to engineering with a triangle solver. By the way, this example section (Figure 4) measures 31 feet wide by 52.5 feet deep.

Figure 4.

To begin, open the triangle calculator and find “isosceles” mode. The width of the section (at its mid depth point) should be entered for the bottom side of the triangle.

Now divide the length of the section by two and use that for the other two sides of the triangle. The angle at the top that was calculated represents the correct loudspeaker pattern that fits that ratio of section width to length. Note that all calculator graphics are in units of feet (Figure 5).

Figure 5.

For this particular example, the correct horizontal coverage pattern for the loudspeaker is 72.8 degrees. Now that we have a target, perhaps we can request the perfect unit, or if we’re stuck with something that’s not ideal at least we still have an idea of how it will behave in the room before measuring it with an analyzer.

If there are four 90-degree loudspeakers in this case, we know there will be overlap if they’re placed at exactly one quarter (1/4) section intervals. There would be gaps in coverage with a 60-degree loudspeaker placed in the same spot.

To check exactly how much of the section the coverage pattern will fill, simply enter the known coverage pattern at the top, half the length of the section for each side, and the answer for the bottom side of the triangle is how wide the pattern is across that area (Figure 6).

With a 60-degree loudspeaker at the same position, there will be a gap of just under 2.5 feet at the mid depth point from the -6 dB point of the pattern to the edge of the section: 31.17 minus 26.25 equals 4.92 feet. (Divide 4.92 by 2 since the loss is on both sides). Keep in mind if the next loudspeaker in line is also 60 degrees, there will be the full 4.92-foot gap between each loudspeaker’s pattern at mid depth (Figure 7).

Figure 7.

Correcting The Gap

How about a curveball scenario? Maybe we have three 72.8-degree loudspeakers and a 60-degree loudspeaker because the person who loaded the truck was in a hurry. (Still, give the person credit for finding the 72.8-degree units.)

So far we know what is happening width wise at mid depth. We can partially correct the gap at that point by moving the misfit 60-degree speaker back. Let’s find out by how much.

Put the coverage pattern at the top. Now take 180 and subtract that known pattern. This answer – divided by 2 – goes into the triangle calculator as the other two angles. The answer of the left and right sides of the triangle is how far into the section the “unity” depth of the loudspeaker itself occurs. Normally we always want this unity depth to line up with the mid depth of the room dimensions (Figure 8).

Figure 8.

The answer is the unity depth of a 60-degree loudspeaker placed in the original position in the room will occur at 31.17 feet. (This is all looking very symmetrical because we happened to come across a special kind of isosceles triangle called the “equilateral triangle.” All angles and sides are equal on this one).

Back to our purpose, mid depth for the section is still 26.25 feet for the room: 31.17 minus 26.25 once again equals 4.92 feet. Push that 60-degree loudspeaker back 4.92 feet and we will once again have aligned acoustic crossovers where one source hands off coverage to another from left to right (Figure 9). Note: there will be some change in coverage pattern shape.
If we need to place delay loudspeakers, we can measure out dimensions from where we’d like to place them and use the solver to tell us how they’ll behave once set and aimed.

Figure 9.

It’s always great to have a backup, and I take my phone to every show. However, if you really want to be safe, you could do some research on how to do the underlying math manually. Hmm… maybe I’ll write a part 2 to this article!

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