Subwoofer Box

A sealed box turns the air trapped behind the cone into an extra spring. That stiffens the driver's suspension, raising both its resonant frequency and its Q. Three Thiele-Small parameters from the driver's data sheet are all you need.

• $F_s$: free-air resonance of the driver.
• $Q_{ts}$: total Q, how lightly damped that resonance is.
• $V_{as}$: the volume of air with the same springiness as the cone's suspension.

Box tuning

A smaller box means a larger $\alpha$, a stiffer air spring, and so higher $Q_{tc}$ and $F_c$. $Q_{tc} = 0.707$ is the Butterworth alignment: the flattest response with the deepest extension before any peak appears.

Frequency response

The system behaves as a second-order high-pass filter, rolling off at 12 dB per octave below resonance. With $x = f / F_c$:

The −3 dB point

Cabin gain

A car interior is small compared with bass wavelengths. Below the frequency where half a wavelength fits the longest cabin dimension, the sub pressurises the whole cabin and output rises at about 12 dB per octave. That almost exactly cancels a sealed box's roll-off, which is why modest sealed subs reach so deep in a car.

References

1. Small, “Direct-Radiator Loudspeaker System Analysis”, JAES 20(5), 1972 — AES E-Librarysealed-box Qtc / Fc analysis used above