Ported Box - Helmholtz Resonance

Tuning frequency:

F_b = (c / (2 * pi)) * sqrt(S_p / (V_b * L_eff))

Effective port length:

L_eff = L_physical + k1 * sqrt(S_p) + k2 * sqrt(S_p)

Where:

• k1 = 0.732 for an open outer end
• k2 = 0.732 for a rounded inner end or about 0.850 for a square inner end

Solving for physical port length:

L_physical = (c^2 * S_p) / (4 * pi^2 * F_b^2 * V_b) - end_corrections

Illustration in preparation
Description: three-axis calculator view showing box volume, port area, and tuning frequency, with the resulting port length at their intersection.

Worked example:

Box volume: 2.0 ft^3 = 56.6 L = 0.0566 m^3
Target tuning: 35 Hz
Port: 4-inch diameter round port (area = pi * 2^2 = 12.57 in^2 = 0.00811 m^2)

F_b = (343 / (2 * pi)) * sqrt(0.00811 / (0.0566 * L_eff))

Solving for L_eff:

35 = 54.6 * sqrt(0.1432 / L_eff)
35 / 54.6 = sqrt(0.1432 / L_eff)
0.641 = sqrt(0.1432 / L_eff)
0.411 = 0.1432 / L_eff
L_eff = 0.1432 / 0.411 = 0.348 m = 13.7 in

Subtract end corrections:

L_physical = 13.7 - (0.732 * sqrt(12.57)) - (0.732 * sqrt(12.57))
L_physical = 13.7 - 2.59 - 2.59 = 8.5 inches

Port length: 8.5 inches for a 4-inch round port in a 2 ft^3 box tuned to 35 Hz.

Verify port velocity at the worked-example tuning:

V_port = (S_d * X_max * F_b) / S_p

Using a typical 12-inch subwoofer with S_d = 480 cm^2, X_max = 12 mm, F_b = 35 Hz, and S_p = 12.57 in^2 = 81 cm^2:

V_port = (480 * 0.012 * 35) / 81 = 2.49 m/s

That is well below the rough 30 m/s port-noise guideline, so this example alignment is comfortably safe on port velocity.