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Figure 6.12 Directivity Index of Four Point Sources

Sources are arrayed in a line and spaced 2' (oa m) apart.

Figure 6.12 Directivity Index of Four Point Sources

Sources are arrayed in a line and spaced 2' (oa m) apart.

boo Hz

10oo Hz boo Hz

10oo Hz

This has profound architectural consequences because it says that loudspeaker systems must be large in the vertical direction to control directivity in a vertical plane. For example, to limit the beamwidth angle to n/3 radians (60°) at 500 Hz, the array should be about 750 mm (30 inches) high. In cabinet systems the horn, which emits the 500 Hz signal, must be about 30 inches high to achieve a 40° vertical coverage angle. (Note that the coverage angle, which is the angle between —6 dB points, is less than the line array beamwidth.) Architecturally this means that in large rooms such as churches and auditoria, which often require highly directional loudspeaker systems to achieve adequate intelligibility, a space at least 4 feet (1.2 m) high must be provided for a speech reinforcement system. If live music is going to be miked, the directivity should extend down an octave lower to reduce feedback. This requires a line array about 8 feet long, similar to that shown in Fig. 6.12 in a concert venue, or significant loudspeaker displacement or barrier shielding in a permanent installation.

Line source configurations can also be used to control low-frequency directivity in concert systems. When concert loudspeaker systems are arranged by unloading truckloads of multiway cabinets and stacking them up on either side of the stage, there is little control of the low-frequency energy and extremely loud sound levels are generated near the front of the stage and at the performers. If instead, low-frequency cabinets (usually dual 18-inch woofers) are stacked vertically to a height of 20 to 30 feet (6 to 9 m), a line source is constructed that controls bass levels at the stage apron. The front row seats are at an angle of nearly 90° to the midpoint axis of the line source so even coverage is maintained from the front to the back of the seating area.

Continuous Line Arrays

The continuous line array is a convenient mathematical construct for modeling rows of sources that are all radiating in phase. Line arrays have a relatively narrow frequency range over which they maintain a simple directivity pattern. If their length is less than a half-wavelength, they will not provide appreciable directional control. At high frequencies line sources have a very narrow beamwidth, so that off-axis there can be a coloration of the sound.

The directional characteristic of a coherent line source can be obtained (Olson, 1957) by substituting l = n d into Eq. 6.45. This approximation is true for large n.

k where R0 is the directional characteristic of the sound pressure relative to the on-axis sound pressure, and l is the length of the line source.

The directivity plots are shown in Fig. 6.13. Practical considerations limit the size of a loudspeaker array to an overall length of about k to 4 k or so. This two-octave span is adequate for many sound source applications, where horns are used on the high end and line arrays are used for the midrange. Directional control is seldom required below the 250 Hz octave band except in concert venues.

As sources are added to a line array, the beamwidth decreases and the number of lobes increases. To compensate for this effect, a line source can be tapered by decreasing the level of the signal fed to loudspeakers farther from the center. The directional characteristic (Olson, 1957) of a tapered line source, whose signal strength varies linearly from the center to zero at the ends is

As Fig. 6.14 shows, tapering broadens the center lobe of the directivity pattern and decreases the off-axis lobing and the expense of overall sound power.

Curved Arrays

Loudspeakers can be configured in other ways, including convex or concave arcs, twisted line arrays, or helical line sources, which look like a stack of popsicle sticks. For a series of sources arranged in a curve the directivity pattern in the plane of the arc is (Olson, 1957)

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