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Array, or not to array? That is the question

Mar 14, 2012 2:49 PM, By Bob McCarthy

Speaker considerations for best deployment.

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Figure 5: Response comparisons of a single speaker and four, 16, and 40 element arrays in the 63Hz range. (a) Single speaker with 360° coverage (b) 4x20° speaker splayed at 10° creates 200° coverage (c) 16x10° splayed at 5° creates 90° coverage (d) 40x10° speakers at 2° creates 52° coverage. See larger image.


The workhorse of the modern sound system is the coupled point source. The most common modern version consists of a curved line of boxes acting as a single element in the horizontal plane and a coupled point source in the vertical plane. Let’s compare and contrast a single 80-degree speaker with an 80-degree coupled point source. We will create our 80-degree point source seven different ways, varying from two units splayed 40 degrees apart, up 40 units splayed 2 degrees apart. Each of these configurations will meet the 80-degree spec, but they will differ in how much of their operating range holds that number. Notice the simple math behind the example arrays here: Multiply the number of elements by the splay angle and you will get 80 degrees every time. Is it really this simple? Not quite. Is it just coincidence? Not at all.

Let’s start with the easiest one: a pair of 40-degree speakers splayed 40 degrees apart will add up to 80 degrees. This is the 100-percent isolation scenario, with the isolation being achieved by splay angle with each speaker taking on half of the job. How is this different from a single speaker? The principal differences in coverage shape are two-fold: an area of comb-filter interaction around the center and sharper edges on the outside of the pattern. Otherwise the patterns match. Figure 1 shows a comparison of the single 80-degree speaker and three array configurations of 40-degree elements (40 degrees, 20 degrees, and 0 degrees respectively). The plots show the 100-percent isolated array (40-degree splay) duplicates the single coverage, where the 50-percent and 0-percent isolated version cause pattern narrowing.

By contrast, the individual patterns are very wide in the low end. Therefore, even the widest splay angle (40 degrees) leaves the speakers highly overlapped. Therefore, the two-box array is consistently narrower than its single counterpart. This is shown in Figure 2 where the individual 500Hz response (160 degrees) is compared to the two-box version (65 degrees).

Now let’s move on by filling up our 80-degree slice of coverage with three, four, eight, 16, and finally, 40 speakers. Each added speaker brings a new set of relationships to all the other speakers. The isolation/overlap equation plays out in every pairing of boxes. In the case of the 40-box array that is around 780 pairings! Now recall that the individual elements change their response over frequency and you get a feel for just how complex this all is. Fortunately, even though the math is super complex, the observable (and predictable) behavior is straightforward.

As we slice the pizza into smaller and smaller pieces, we wisely choose narrower individual elements. The two-element configuration must have at least a 40-degree element, and for each doubling of quantity we can cut the minimum coverage in half. Recall that individual speakers can’t hold 5-degree patterns beyond a small range of high frequencies. Therefore, isolation percentage rises with frequency and overlap percentage increases as we move down the spectrum. That is how we can start with an element that is 360 degrees wide at the low end and 5 degrees wide at the top end and build an array that ends up around 80 degrees at both ends. Isolation spreads the top, while overlap squeezes the bottom. The extended quantity creates a push-pull between these factors that keeps the coverage angle consistent over frequency.

Figure 3 shows a chart of coverage angle over frequency of our eight versions of 80 degrees. Notice that as quantity increases we are able to extend the control of the array all the way down. In fact, the 40-box scenario is so long that it is actually narrower than 80 degrees. Yes, we can have too much of a good thing! Figures 4 and 5 have some plots of the array responses. A complete set of the data is available on my blog:

So now we know how we can get an array to mimic a single speaker and even to be able to create an idealized speaker response with uniform shape over frequency. In part two, we will explore how to use arrays to bend the shape of sound to get optimal coverage over the space.

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