When going through vented box loudspeaker box design (at least two or three simulators I've tried), they accept the speaker parameters as input, and I get the volume of the box, and diameter and length of the port as output.

It is my understanding that what's happening in a vented box, by analogy with an electric circuit, is that the air cavity (the box) acts as a capacitor (it accumulates pressure), and the pipe acts as an inductor (it accumulates momentum / air movement).

I would like to know the formula for this "inductance" as a function of diameter and length, so that if I change the diameter, I can determine the required length to maintain the inductance.



1 Answer 1


Yes it does, but it's counterintuitive, so watch closely now:

We need to solve for the inertance of a duct, this takes into account both the quantity of mass contained in the duct and its geometry, which determines how a force gets distributed on that mass.

The formula for the inertance of a duct is I = ((rho) x L)/A where rho is the density of air, L is the length of the duct, and A is its cross-sectional area.

The counterintuitive part is what's the area doing in the denominator? This is the geometry part. A long and skinny duct contains a long and skinny-shaped mass of air and if we wish to set it in motion and establish a specified flow rate through the duct, it takes more work to do so than if the same mass were distributed in a duct of large area and short length. This is because in the skinny duct, the mass has to be set into faster motion to establish that flow rate than in the case of the large diameter, short duct.

  • $\begingroup$ One of the simulators I have gives me several options for diameter/length combination, and they are close, but not an exact match for this formula (D=4.7 → L=3.62; D=6.8 → L=9.8; D=8.2 → L=15.5; D=10.2 → L=25.8; D=14 → L=52.4). The ratio L-to-D² is not constant (although for larger values of D, it gets closer to a match). Are there other practical factors that affect these parameters? Is it possible that this software is running a numeric solution of the "complete" model including non-linearities? $\endgroup$
    – Cal-linux
    Jun 13, 2020 at 12:27
  • 1
    $\begingroup$ yes there is. a certain amount of air in the immediate vicinity of the ends of the duct is coupled to the air inside it; this adds to the inertance of the air in the duct and is called the end effect and DIY builders take this into account by cutting the duct long on purpose and then trimming it to get the correct response curve for the speaker. In addition, small diameter ducts get detuned because of the flow resistance of air in them; this effect goes away for large diameter, short ducts and is significant for long, small diameter ducts. $\endgroup$ Jun 13, 2020 at 16:40

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