Gradient Cardioid (CSA) vs. End-Fire Acoustic Array Physics

Low-frequency subwoofers radiate omnidirectionally because their cabinet dimensions are small relative to wavelengths below $100\text{ Hz}$ ($\lambda > 3.4\text{ m}$). Left unchecked, omnidirectional bass floods the stage, causing low-end feedback into open vocal and drum microphones and muddying the hall’s reverberant field.

In a Gradient Cardioid Subwoofer Array (CSA), a rear subwoofer is placed at distance $s = \lambda/4$ behind the front cabinet (or reversed inside a stack), delayed electronically by $\Delta t = s/c$, and polarity-inverted ($180^\circ$). Traveling toward the stage ($180^\circ$), the front wave takes $\Delta t$ to reach the rear sub, meeting the rear sub’s delayed and inverted wave at exact $180^\circ$ opposition, creating a deep rear cancellation null! Traveling forward toward the audience ($0^\circ$), the rear wave travels an extra $\lambda/4$ path ($90^\circ$) plus its $\lambda/4$ electronic delay ($90^\circ$) plus its $180^\circ$ polarity flip, totaling $360^\circ$ ($0^\circ$ in-phase summation) at the tuning frequency $f_0$.