Speaker Box Designer

Take a good woofer out of its cabinet, wire it to an amplifier and play a record. The midrange is all there. The bass is gone, and no amount of volume brings it back.

The reason is geometry. A cone pushes air with its front and pulls with its back at the same instant, and at low frequencies the two cancel. Wikipedia’s entry on loudspeaker enclosures says it plainly: the rear of a driver radiates out of phase with the front, and without an enclosure the interference costs the bass. Close the back with a box and the cancellation stops. There is a catch, worked out below: the air trapped in a closed box stiffens the cone’s suspension and pushes its resonance upward. A cabinet is never a neutral container. It is a spring, sometimes a resonator, and it has to be tuned to the driver inside it.

Three numbers on a data sheet

For decades that tuning was done by trial and error. The vented enclosure goes back to a patent application filed by A. L. Thuras in 1930, yet the first complete quantitative treatment, by A. N. Thiele, was published in Australia only in 1961, and by Richard Small’s account it was barely read for ten years. Small’s own papers on closed and vented boxes, in the Journal of the Audio Engineering Society in 1972 and 1973, turned the theory into a design method. Their message was blunt. Trial-and-error design techniques, Small wrote, are not only wasteful but unnecessary.

What made that possible was the finding that a few easily measured parameters describe everything a driver does at low frequencies. Fs is the resonance frequency of the driver, where the cone, left to itself, moves most readily. Qts is the total Q at that resonance, one figure for how firmly the motion is damped; it combines an electrical part, Qes, and a mechanical part, Qms, which counts every other loss. Vas is the odd one: the volume of air that has the same acoustic compliance as the driver’s suspension. It states the softness of a spring as a quantity of air, which is what makes it useful.

Read together, they say where the driver resonates, how tightly that resonance is held, and how much air its suspension is worth.

The sealed box is a spring you can size

Mount the driver in a closed box and the trapped air stiffens the suspension. Small reduces the whole effect to one ratio, Vas divided by the box volume, which he calls alpha. The resonance of the system and its Q both rise above the driver’s own values by the same factor, the square root of alpha plus one. A box as large as Vas raises both by 41 percent. A box one third of Vas doubles them. Below resonance the output falls at 12 dB per octave, the mark of a second-order filter.

So choosing a sealed box means choosing a final Q, written Qtc. At 0.50 the system is critically damped. At 0.71 it has the maximally flat Butterworth response, and the 3 dB point sits exactly at the box resonance. Go higher and a peak appears above cutoff in exchange for a little more extension: at a Qtc of 1.0 the 3 dB point drops to 0.786 of the resonance frequency and the response peaks by 1.25 dB. When Small measured commercial closed boxes in late 1969 he found total Q anywhere from 0.4 to 2.0, with the smaller cabinets showing definite peaking.

Stuffing changes the sum only a little. Small credits filling material with a theoretical 40 percent gain in the effective compliance of the box and says real materials probably never deliver more than about 25 percent. What it does reliably is damp standing waves inside the cabinet. An encyclopedia entry on enclosures gives the same ceiling of 40 percent and the reason for it: sound travels more slowly through the filling than through air. Ported boxes get less. A Wikibooks text on bass reflex design says the acoustic material in them is usually much less than in a sealed box, often none at all, so that air can flow freely through the port.

The ported box borrows the back wave

A port turns the cabinet into a Helmholtz resonator, the same device as a bottle that hums when you blow across its neck. The air in the neck vibrates like a mass on a spring, and the spring is the air in the cavity behind it. Tuned correctly, the port projects sound from the back of the cone in phase with the front, at least near the resonant frequency. At that frequency almost all of the output comes from the port, and the cone hardly moves.

The reward is real. Small calculates that a vented box has a maximum theoretical efficiency constant 2.9 dB above that of a closed box, and about 3 dB in practical designs, to be spent on higher efficiency, a smaller enclosure or a lower cutoff. Because the cone moves so little near tuning, the displacement-limited power rating on average program material is higher too, by a factor he puts at 3.5.

The bill arrives lower down. A ported box is a fourth-order filter and rolls off at 24 dB per octave, twice the slope of a sealed one. Well below cutoff the excursion advantage reverses: Small warns that the vented system then has a higher relative displacement sensitivity than a closed box and is more vulnerable to turntable rumble and other subsonic signals.

There is a timing cost too. Researchers at the University of Kaiserslautern note that vented boxes pay for their bass with increased group delay at low frequencies, which can be perceived as muddy, boomy or slow. For a Butterworth alignment they put the delay at the box resonance at about 0.59 divided by that frequency, roughly 15 milliseconds at 40 Hz. Small grants the closed box nominally better transient response, then cites Thiele’s view that the differences between correctly adjusted systems are likely to be inaudible.

Which response a ported box gives depends on how its volume and tuning are matched to the driver, and Thiele’s alignments are the classic recipes. The Butterworth alignment, flat right to the corner, suits exactly one driver: in a lossless box it needs a Qts of 0.383, a volume of Vas divided by 1.414, and tuning at Fs. Drivers with lower Qts take the quasi-Butterworth alignments, the usual choice, as the Wikibooks text puts it, when Qts is not large enough for the rigid requirements of the Butterworth; they call for smaller boxes tuned higher. Drivers with higher Qts take Chebyshev alignments, in larger boxes tuned lower, with some ripple.

Real cabinets leak, and that shifts every figure. Small measured leakage Q values between 5 and 20 and calls 7 a very satisfactory starting point. Above a Qts of about 0.6 he rules a vented box out.

Small’s own worked example shows the method, and the calculator above reproduces it. He measured an 8 inch driver: Fs 33 Hz, Qes 0.45, Qms 2.0 and Vas 57 liters, hence a Qts of 0.37. Expecting a loss Q of about 10, he read a quasi-Butterworth alignment off his chart: 37 L at 35 Hz, with the response 3 dB down at 38 Hz. Enter 33, 0.37 and 57, set the box loss to 10 and press the classic alignment button, and the tool returns 37.4 L tuned to 35.2 Hz, 3 dB down at 37.4 Hz. The gap is what chart reading and a rounded Qts would produce.

Now seal the same driver. For a Qtc of 0.71 the tool asks for 21.3 L, with the 3 dB point at 63.1 Hz. The ported box is about 1.75 times larger and reaches three quarters of an octave lower.

Air has a speed limit

The port itself is a design problem. Its length follows from the Helmholtz relation, less an end correction, because air just outside each end moves along with the air inside. For an end set flush in a flat panel, the textbook air load on a piston in a wall works out to an extra 0.85 of the tube radius. For an end in free air, Levine and Schwinger calculated 0.6133, often rounded to 0.6. For his example Small chose a 65 mm tube and read a length of 175 mm from his nomogram. Set the tool to his rounded 37 L and 35 Hz, with one port end flush and one free, and it says 173 mm.

Why 65 mm? Because air driven too fast through a tube turns noisy and stops behaving like a simple mass. Small gives an experimentally determined ceiling of about 5 percent of the speed of sound, provided the vent is smooth inside and its edges are rounded, and he turns it into a formula that sets the smallest acceptable vent from the tuning frequency and the volume of air the cone can displace. He calls the formula a general guide only, not a rigid rule.

Engineers at JBL and Infinity documented what lies beyond the limit: broadband chuffing, compression, and at very high levels a port whose output falls out of step with the cone until, in their words, the port is simply a leak in the box. The older studies they cite put the safe speed for a plain straight tube lower still, below about 10 m/s.

Here the arithmetic bites. Give the tool the example driver’s 6 mm of travel and 201 square centimeters of cone, and for a steady tone at full excursion it reports 22.5 m/s in that 65 mm port, 6.5 percent of the speed of sound. Small sized his vents for program material, a gentler test than a sustained tone. Widen the tube to 75 mm and the speed falls to 16.9 m/s, but the tube must grow to 239 mm, longer than the 206 mm inside depth of the tool’s default cabinet. A quiet port wants to be wide, and a wide port wants to be long. A long, thin tube, Small warns, tends to act as a resonant pipe.

What the wood adds and what the model leaves out

A box of the right volume can still sound wrong, because its walls move. Siegfried Linkwitz observed that panel vibration modes are always resonant, store energy and can radiate sound, and his rule of thumb for three-quarter-inch wood allows no unbraced panel area larger than four inches squared. An encyclopedia entry on enclosures lists the general remedies for a bothersome resonance: more mass or rigidity in the cabinet, more damping in its walls, stiff cross bracing, or absorbent material inside.

Shape and symmetry matter too. A Wikibooks chapter on sealed boxes says the internal height, width and depth should never stand in whole-number ratios such as 2:1, which would encourage standing waves, and reports that some designers use the golden ratio. The tool’s default proportion, 1 : 1.618 : 0.618, follows that suggestion. Braces obey the same logic. A United States patent of 1997 observes that box resonance is most noticeable when all sides resonate alike, and it reinforces each side differently so that the peak is spread over a range of frequencies.

Air leaks count as well. The Wikibooks text names leakage as the larger loss in a ported box. In Small’s model they are the leakage loss already described, so a joint that does not seal moves the finished box away from the alignment it was cut to.

All of the numbers above come from a small-signal model, and that model has edges. Small’s circuits hold only where the cone moves as a piston, and they neglect voice-coil inductance. They treat the driver’s parameters as constants, whereas Klippel’s measurement notes show the force factor, the suspension stiffness and the inductance all varying with cone position, the changing stiffness alone generating significant distortion below resonance. Nor does the model know the room: its results assume radiation into a free half space, which Small calls only an approximation to average listening-room conditions. Enclosure losses, he concedes, cannot be known exactly in advance.

So the method cannot tell you how loud the system will play cleanly, how the panels will color the midrange, or what the room will do. Treat a calculated box as a first prototype that is already close. Build it, measure it, and compare. Small asked for exactly that step, checking the completed system against the specification it was designed to. The theory tells you which box to build. The finished cabinet tells you whether you built it.