End-Fire and Cardioid Subwoofer Spacing

An end-fire array puts subwoofers in a line pointed at the audience and delays each one behind the one in front by exactly the time sound takes to travel the spacing. Forward, every cabinet's output arrives together and adds; backward, the electronic delay and the acoustic travel time add rather than cancel, so the rear arrivals spread out and the level collapses. The rejection is deepest where the spacing is a QUARTER wavelength, because that puts the rear arrivals a half wavelength -- a full polarity flip -- apart, which makes the optimum frequency the speed of sound divided by four times the spacing. That quarter-wave relationship is the whole design, and it also sets the band: an array tuned for deep rejection at 90 Hz is progressively less directional as frequency falls, and above roughly twice the tuning frequency the pattern breaks up. Adding elements deepens and broadens the rejection without moving where it is centered. Four cabinets on 3.0 ft centers at 70 F want 2.667 ms per element, with the deepest rejection at 1125 / 12 = 93.75 Hz, right in the kick-drum band, and a 12.0 ft wavelength there of which the spacing is one quarter. Moving the tuning down to 60 Hz opens the spacing to 4.69 ft, which puts over fourteen feet of stage depth behind four cabinets -- and that trade, stage depth against depth of rejection, is the real constraint. The reverse-stack cardioid variant works on the same arithmetic with the spacing set by cabinet depth rather than chosen. A design relation; a measurement system and the room govern the deployed result.

c = 1125 x sqrt((temp_f + 459.67) / 529.67); delay_per_element_ms = spacing_ft / c x 1000; optimum_freq_hz = c / (4 x spacing_ft); wavelength_ft = c / optimum_freq_hz; spacing_for_target_ft = c / (4 x target_freq_hz).

End-fire and reverse-stack cardioid subwoofer arrays from the quarter-wavelength spacing relation, by name; standard live-sound system design practice built on public acoustics. A measurement system and the room govern the deployed result.

The quarter-wave rear-cancellation geometry is public acoustics; the spacing, element count, and temperature are the designer's own values.

Estimate. AHJ and licensed professional govern.

Field names used by the API: spacing_ft, elements, temp_f, target_freq_hz, delay_per_element_ms, optimum_freq_hz, wavelength_ft, spacing_for_target_ft

• Quarter-wave spacing deepest rejection where the spacing is one quarter wavelength, putting rear arrivals a half wavelength apartpublic array acoustics
• Usable band less directional below the tuning frequency; the pattern breaks up above roughly twice itlive-sound design practice
• Element count more elements deepen and broaden the rejection but do not move its center frequencypublic array acoustics