Acoustic Geometry & Line Source Theory
1. Point Source vs. Line Source Expansion The fundamental difference between a single loudspeaker and a professional line array lies in the geometry of the wavefront expansion. A point source—theoretically an infinitesimal pulsating sphere—radiates sound energy in an omnidirectional, spherical pattern. According to the Inverse Square Law, the sound pressure level (SPL) decreases by 6 dB for every doubling of distance (r) from the source, as the energy is spread over the surface area of a sphere (4\pi r^2).
A line source, however, produces a cylindrical wavefront. In the "near field" of a line source, the energy is concentrated such that the SPL only drops by 3 dB per doubling of distance. For an array to function as a line source, the drivers must be physically spaced closer than half a wavelength of the highest frequency they are reproducing. If this criteria is not met, the array "breaks" into a series of individual point sources, causing destructive interference and "lobing."
2. Near Field vs. Far Field Transitions Every line array eventually transitions from a 3 dB cylindrical drop (near field) to a 6 dB spherical drop (far field). The distance of this transition—the Fresnel zone—is determined by the length of the array (L) and the frequency (f):
d_{transition} \approx \frac{f \cdot L^2}{2 \cdot c}
Where c is the speed of sound. This math dictates that a longer array will maintain near-field projection (and thus better impact) over a greater distance than a short array. In a stadium environment, the engineer uses this physics to ensure that the "throw" of the PA reaches the back rows without losing the high-frequency "detail" that naturally attenuates faster than low frequencies.
3. The Mathematics of Splay Angles The "splay" is the angle between adjacent cabinets in an array. By adjusting these angles (typically between 0.25° and 10°), we determine the density of acoustic energy across a vertical plane.
Tight Splay (0-1°): This concentrates the energy of multiple boxes into a narrow beam for long-distance projection.
Wide Splay (5-10°): This spreads the energy over a wider area for the "down-fill" or "near-fill" sections. A common mistake in system design is "over-splaying," where the gaps between cabinets become large enough that the wavefront is no longer continuous, leading to "holes" in the frequency response as you walk through the venue.
4. Air Absorption and Atmospheric Interaction Sound does not travel through a vacuum. It travels through air, which is a viscous medium. High frequencies (above 4 kHz) are significantly more susceptible to air absorption than low frequencies. This attenuation is a function of distance, frequency, temperature, and relative humidity. Interestingly, dry air (low humidity) absorbs high-frequency energy more aggressively than humid air. Professional system designers use "Atmospheric Compensation" filters (like L-Acoustics Autocline or d&b ArrayProcessing) to mathematically boost the HF energy in the top boxes of an array to compensate for the predicted loss over distance.
5. Wavefront Sculpture and Summation For a line array to sum correctly at high frequencies, the drivers must use specialized waveguides. These waveguides take the spherical expansion of a compression driver and "fold" the path lengths so that the exit of the horn is a flat, rectangular ribbon. This ensures that the wavefronts of adjacent boxes meet seamlessly. Without this, the high frequencies would arrive at the listener at slightly different times from different boxes, causing "comb filtering"—the jagged peaks and valleys in a frequency response that make a system sound "harsh" or "thin."
Practical Lab: 3D Predictive Modeling
Tool: Professional Modeling Software (Soundvision/ArrayCalc).
Objective: Design a system for a 60-meter deep theater with a balcony.
Tasks:
Create a "Venue" with a floor (40m depth) and a balcony (starting at 30m, rising 5m).
Insert a 12-box array at a 10m trim height.
The Manual Challenge: Set all splay angles to 0°. Observe the "hot spot" on the back wall.
The Optimization: Use splay angles to distribute energy such that the SPL variation from the front row to the balcony back row is less than +/- 3 dB.
Atmospheric Test: Change the humidity from 50% to 20% in the software and observe the change in HF response at 50 meters.
Daily Assessment
Calculation: If a point source is 100 dB at 2 meters, what is the SPL at 16 meters?
4, 8, 16 - the distance doubles 3 times, therefor the SPL reduces by 3x6dB = 82dB
Theory: Why does a long line array "throw" further than a short one? Refer to the Fresnel zone in your answer.
Line array boxes are designed in a way that the speaker makes a cylindrical, concentrated wave form that decreases degredation in the nearfield and loses only 3dB per doubling of distance. These wave forms will eventullay become spherical over longer distances transitionaning from the tight fresnel zone over the transition distance (or the start of the farfield region. this means that a longer array has a much deeper fresnel zone.
Synthesis: You are outdoors. It is noon, and the humidity drops. Which part of your PA system (the top boxes or the bottom boxes) will require an EQ adjustment?
The high end (4K and up) will be most susceptible to the change in frequency and may require a boost, and this will be most apparent across the top boxes that have the furthest distance to trvel.