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Size Matters: Observations On Loudspeaker Directivity
Physics hasn’t changed -- when it comes to pattern control, size still matters...
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Trap boxes and line arrays get all the attention. And that’s no surprise—they’re big and loud, and dare I say it, glamorous.

But the truck rarely rolls without a complement of two-way loudspeakers sporting a 12-inch or 15-inch woofer and a horn. Whether its monitor wedges, drum fill, front fill or just “speakers on sticks,” small 2-way boxes do many of the everyday jobs that make up a typical sound reinforcement day.

We take the performance of these boxes for granted, but they can be used to better effect if we really understand their directivity characteristics and what makes them perform the way they do. They’re often described as a “90 by 60 box” or some other dubious reference. But 90 degrees by 60 degrees at what frequency? Certainly not from DC to light.

There are four principle ingredients that govern the dispersion pattern of these loudspeakers, including the cone driver, horn, crossover and cabinet. Let’s look at these one at a time and assess their contributions. Before we go through our list, though, let’s review some basics.

The amount of directivity any device can exert on a sound wave is directly related to the proportional sizes of the device and the sound wave. To understand this relationship it is important to have a good grasp of how big or small a sine wave is at a given frequency.

Sound at sea level at 72 degrees Fahrenheit travels at approximately 1,130 feet per second. We express frequency or cycles (sine waves) per second as Hertz. So if the frequency of a wave is 1 Hz, the wave is 1,130 feet long. Logically, a 10 Hz wave is 113 feet long, a 100 Hz wave is 11.3 feet long, and a 1,000 Hz wave is 1.13 feet long, etc.

While it’s not overly difficult to do the math to determine the wavelength of any given frequency, there is an old “cheat” called the rule of 5-2-1:

20 Hz = 50 feet
50 Hz = 20 feet
100 Hz = 10 feet
200 Hz = 5 feet
500 Hz = 2 feet
1,000 Hz = 1 foot
2,000 Hz = .5 foot
5,000 Hz = .2 foot
10,000 Hz = .1 foot

While not perfectly accurate, it fills the bill for “quick and dirty” calculations. Physics dictates that a source be physically large in comparison to a wavelength to exert directional control over it.

Figure 1: Horizontal directivity balloon of a 12-inch 2-way loudspeaker at 100 Hz (box facing left)

So let’s look at the low frequency directivity of a 12-inch driver in a 2-way loudspeaker with a 90-degree by 60-degree horn.

Matter Of Control
Remember that the low frequency driver’s only means of controlling the dispersion of the sound wave in a front-loaded loudspeaker are its cone diameter, and to a lesser extent, some boundary effects (we’ll discuss that later).

At 100 Hz, the driver is physically small in comparison to the 10-foot wavelength and provides almost no directivity (Figure 1).

Figure 2: Horizontal directivity balloon of a 12-inch 2-way loudspeaker at 500 Hz (box facing left)

If we increase the frequency gradually, the 12-inch driver does not suddenly exert pattern control over the sound wave when it reaches 1,000 Hz (1 foot), and is the same size as the driver itself.

Rather, it has more and more effect as the frequency gets higher and the wavelengths get shorter (Figures 2 & 3). In this frequency range (800 Hz as shown in Figure 3), the cone driver is actually providing approximately 90-degree horizontal dispersion.

Figure 3: Horizontal directivity balloon of a 12-inch, 2-way loudspeaker at 800 Hz (box facing left)

But also realize that since this pattern is conical (the driver is round), it is not producing the specified 60-degree vertical pattern. As the frequency increases the driver exerts more and more control until it begins to “beam” at higher frequencies. But by the time it narrows that much, it’s above the crossover frequency.

This particular loudspeaker crosses over about a half-octave above the balloon in Figure 3. This has an overriding effect on the polar behavior of the box, especially in the vertical domain, so we will discuss the range from 1,000 Hz to 1,500 Hz when we discuss the crossover.

Now, on to the horn.


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