Driver Spacing, Lobing, and Crossover Ceiling

The highest frequency two sources on a given center-to-center spacing can share before the pattern acquires a null, and where that null sits when they are crossed above it. Two sources radiating the same signal are in phase everywhere on their perpendicular bisector and progressively out of phase off it, because the path lengths differ by the spacing times the sine of the off-axis angle. When that path difference reaches half a wavelength they cancel. Below the frequency at which even the WORST case -- the full spacing, at 90 degrees off axis -- is under half a wavelength, no null can exist anywhere and the pair behaves as one source, which puts the crossover ceiling at the speed of sound divided by twice the spacing. Cross two drivers below it and the array is coherent through the crossover region; cross above it and a null sits in the pattern at the crossover, moving with frequency, audible as the audience walks past it. Two 15 in woofers on 18 in centers at 70 F have a ceiling of 1125 / 3 = 375 Hz: crossed at 250 Hz the ratio exceeds one and there is no null anywhere, while crossed at 500 Hz the ratio is 0.75 and the null lands 48.6 degrees off axis. Run it backward and keeping 500 Hz clean would need the spacing in to 1.13 ft, about thirteen and a half inches center to center, which two 15 in drivers physically cannot do -- which is why large-format two-way boxes cross low, and why the spacing constraint is a cabinet design decision long before it is a system tuning one. The same arithmetic answers how far apart two subwoofers can be spread before the center of the room gets a hole. A geometric screen; measured polar data governs a real cabinet.

c = 1125 x sqrt((temp_f + 459.67) / 529.67); crossover_ceiling_hz = c / (2 x spacing_ft); ratio = c / (2 x spacing_ft x test_freq_hz); null_angle = asin(ratio) when ratio <= 1, otherwise no null; max_spacing_ft = c / (2 x test_freq_hz).

Two-source interference geometry -- path difference = spacing x sin(angle), with the first null where that difference reaches half a wavelength -- giving the crossover ceiling c / (2 x spacing), by name; public acoustics, standard in the loudspeaker-design literature. Measured polar data for the real cabinet governs.

Two-source interference is public physics. The spacing, test frequency, and temperature are the user's own values.

Estimate. AHJ and licensed professional govern.

Field names used by the API: spacing_ft, test_freq_hz, temp_f, crossover_ceiling_hz, wavelength_ft, null_angle_deg, max_spacing_ft

• First null where the path difference spacing x sin(angle) reaches half a wavelengthtwo-source interference
• Crossover ceiling c / (2 x spacing); below it no null exists at any angleloudspeaker design literature
• Idealization point sources radiating identically; measured polar data for a real cabinet governsloudspeaker measurement practice