Pure line array theory is based on pure geometry and the thought experiment of the "free field" where sound is free to propagate free of environmental factors such as room reflections or temperature refraction.
In the free field, sound which has its origin at a point (a point source) will be propagated equally in all directions as a sphere. Since the surface area of a sphere = 4π r² where r is the radius, every doubling of the radius results in a four-fold increase in the sphere's surface area. The result of this is that the sound intensity quarters for every doubling of distance from the point source. Sound intensity is the acoustic power per unit area, and it decreases as the surface area increases since the acoustic power is spread over a greater area. The ratio between two acoustic pressures in deciBels is expressed by the equation dB = 20log(p1/p2), so for every doubling of distance from the point source p1 = 1 and p2 = 2, thus there is an sound pressure loss of approximately 6 dB.
A line source is a hypothetical one-dimensional source of sound, as opposed to the dimensionless point source. As a line source propagates sound equally in all directions in the free field, the sound propagates in the shape of a cylinder rather than a sphere. Since the surface area of the curved surface of a cylinder = 2π r h, where r is the radius and h is the height, every doubling of the radius results in a doubling of the surface area, thus the sound pressure halves with each doubling of distance from the line source. Since p1 = 1 and p2 = 4 for every distance doubled, this results in a sound pressure loss of approximately 3 dB.
In reality, dimensionless point sources and one-dimensional line sources cannot exist, however calculations can be made based on these theoretical models for simplicity. A cone driver, for instance, may have an actual width of 12 inches, but the further a listener is from the driver the more it behaves as a point source as its dimensions become less significant. Thus there is only a certain distance where a line source of a finite length will behave as one - past a certain point, it begins to act more as a point source when its length becomes insignificant. Thus, a true line source has to be infinitely long.
Interference pattern is the term applied to the dispersion pattern of a line array. It means that when you stack a number of loudspeakers vertically, the vertical dispersion angle decreases because the individual drivers are out of phase with each other at listening positions off-axis in the vertical plane. The taller the stack is, the narrower the vertical dispersion will be and the higher the sensitivity will be on-axis. A vertical array of like drivers will have the same horizontal polar pattern as a single driver.
Other than the narrowing vertical coverage, the length of the array also plays a role in what wavelengths will be affected by this narrowing of dispersion. The longer the array, the lower frequency the pattern will control. At frequencies below 100 Hz (wavelength of 11.3 ft) the drivers in a line array will start to become omni-directional, so the system will not conform to line array theory across all frequencies. It is theoretically possible to construct an audio line array that follows the theory at low frequencies. However, the array requires more than 1,000 fifteen-inch drivers, spaced twenty inches center to center, to do it. Above about 400 Hz the low-frequency cones become directional, again violating the theory’s assumptions, and at high frequencies, many practical systems use directional waveguides whose behavior cannot be described using classical line array theory. In short, the geometry of real-world audio line arrays is too complicated to be modeled accurately by ‘pure’ line array theory.
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