Sealed Box — System Parameters

System resonance:

F_c = F_s × √(1 + Vas / Vb)

System Q (Qtc):

Q_tc = Q_ts × √(1 + Vas / Vb)

F3 (−3 dB frequency):

For Qtc ≥ 0.577:

F3 = F_c × √((1/(2 × Q_tc²) − 1) + √((1/(2 × Q_tc²) − 1)² + 1))

Simplified for Q_tc = 0.707 (Butterworth):

F3 = F_c

Optimal volume for Butterworth alignment:

Vb_butterworth = Vas × ((Q_ts / 0.577)² − 1)⁻¹

Worked example:

Driver: Fs = 35 Hz, Qts = 0.65, Vas = 45L

Target Qtc = 0.707:

Q_tc = Q_ts × √(Vas/Vb + 1) = 0.707
√(Vas/Vb + 1) = 0.707 / 0.65 = 1.088
Vas/Vb + 1 = 1.183
Vas/Vb = 0.183
Vb = Vas / 0.183 = 45 / 0.183 = 246L

That's enormous — 8.7 cubic feet. Not practical.

Try smaller box: Vb = 30L:

Q_tc = 0.65 × √(45/30 + 1) = 0.65 × √2.5 = 0.65 × 1.58 = 1.03

Overdamped — tight bass, limited output. More realistic for a car build.

F_c at 30L:

F_c = 35 × √(45/30 + 1) = 35 × 1.58 = 55.4 Hz

F3 ≈ F_c for Qtc near 1.0: ~55 Hz

This is why many car subwoofers have a reduced bass extension compared to home audio — real-world box sizes force higher Qtc.