Bass reflex theory and practice

An article by an Italian acoustician, reproduced here with the author’s blessing, was originally titled Teoria e pratica del condotto di accordo. That is, in a literal translation – “Theory and Practice of Bass Reflex”. From the editorial board of Avtozvuk magazine:

This title, in our opinion, corresponded to the content of the article only formally. Indeed, we are talking about the ratio of the simplest theoretical model of a phase inverter and those surprises that practice prepares. But this is – if formal and superficial. And in essence, the article contains the answer to the questions that arise, judging by the editorial mail, all the time when calculating and manufacturing a subwoofer-phase inverter.

The first question: “If you calculate the phase inverter according to the formula that has been known for a long time, will the finished phase inverter get the design frequency?” Our Italian colleague, who has eaten about a dozen of dogs on phase inverters in his lifetime, answers: “No, it won’t work.” And then he explains why and, what is most valuable, how much it will not work.

Question two: “I calculated the tunnel, but it is so long that it does not fit anywhere. How to be? ” And here the signor proposes such original solutions that it is precisely this side of his works that we put in the title. So the key word in the new title should be understood not in New Russian (otherwise we would have written: “in short – phase inverter”), but quite literally. Geometrically. And now Signor Matarazzo has the floor to speak.

About the author: Jean-Piero Matarazzo was born in 1953 in Avellino, Italy. Since the beginning of the 70s he has been working in the field of professional acoustics. For many years he was responsible for testing acoustic systems for the magazine “Suono” (“Sound”). In the 90s, he developed a number of new mathematical models of the process of sound emission by loudspeaker diffusers and several projects of acoustic systems for industry, including the Opera model, which was popular in Italy.

Since the late 90s, he has been actively cooperating with the magazines “Audio Review”, “Digital Video” and, what is most important for us, “ACS” (“Audio Car Stereo”). In all three, he is in charge of measuring parameters and testing acoustics. What else? .. Married. Two little sons are growing up, 7 years old and 10.

Magic formulas!

One of the most common wishes in the author’s e-mail is to give a “magic formula” by which the ACS reader could calculate the phase inverter himself. This is, in principle, not difficult. The phase inverter is one of the cases of the implementation of the device called “Helmholtz resonator”. The formula for calculating it is not much more complicated than the most common and available model of such a resonator. An empty Coca-Cola bottle (only a bottle, not an aluminum can) is just such a resonator, tuned to a frequency of 185 Hz, it has been verified.

However, the Helmholtz resonator is much older than even this, gradually falling out of use, packaging of a popular drink. However, the classical scheme of a Helmholtz resonator is similar to a bottle (Fig. 1). In order for such a resonator to work, it is important that it has a volume V and a tunnel with a cross-sectional area S and a length L. Knowing this, the tuning frequency of the Helmholtz resonator (or phase inverter, which is the same) can now be calculated by the formula:

where F_b – tuning frequency in Hz, s – sound speed equal to 344 m / s, S – tunnel area in m², L – tunnel length in m, V – box volume in m³, π = 3.14, it goes without saying.

This formula is truly magical, in the sense that the bass reflex setting does not depend on the parameters of the speaker that will be installed in it. The volume of the box and the dimensions of the tunnel determine the frequency of tuning once and for all. Everything, it would seem, is done. Let’s get started. Suppose we have a box with a volume of 50 liters. We want to turn it into a bass reflex box with a 50 Hz setting. It was decided to make the diameter of the tunnel 8 cm. According to the formula just given, the tuning frequency of 50 Hz will be obtained if the length of the tunnel is equal to 12.05 cm.

We carefully manufacture all the parts, assemble them into a structure, as in Fig. 2, and for verification we measure the actual resonant frequency of the phase inverter. And we see, to our surprise, that it is not equal to 50 Hz, as it would be expected by the formula, but 41 Hz. What’s the matter and where did we go wrong? Yes, nowhere.

Our freshly built phase inverter would be tuned to a frequency close to that obtained by the Helmholtz formula if it were made, as shown in Fig. 3. This case is closest to the ideal model described by the formula: here both ends of the tunnel “hang in the air”, relatively far from any obstacles. In our design, one of the ends of the tunnel mates with the wall of the box. For the air vibrating in the tunnel, it is not indifferent, because of the influence of the “flange” at the end of the tunnel, its virtual lengthening occurs. The phase inverter will be tuned as if the length of the tunnel was 18 cm, and not 12, as in reality.

Note that the same will happen if the tunnel is completely placed outside the box, again aligning one end of it with the wall (Fig. 4). There is an empirical dependence of the “virtual lengthening” of the tunnel depending on its size. For a round tunnel, one cut of which is located far enough from the walls of the box (or other obstacles), and the other is in the plane of the wall, this elongation is approximately 0.85D.

Now, if we substitute all the constants in the Helmholtz formula, introduce a correction for the “virtual elongation”, and express all dimensions in the usual units, the final formula for the length of a tunnel with a diameter D, which ensures the tuning of a box of volume V to a frequency F_bwill look like this:

Here the frequency is in hertz, the volume is in liters, and the length and diameter of the tunnel are in millimeters, as we are used to.

The obtained result is valuable not only because it allows at the stage of calculation to obtain a length value close to the final one, which gives the required value of the tuning frequency, but also because it opens up certain reserves for shortening the tunnel. We have already won almost one diameter. It is possible to shorten the tunnel even further while maintaining the same tuning frequency by making flanges at both ends, as shown in fig. 5.

Now, it seems, everything has been taken into account, and, armed with this formula, we seem to be omnipotent. This is where difficulties await us.

First difficulties

The first (and main) difficulty is as follows: if a relatively small box needs to be tuned to a fairly low frequency, then, substituting a large diameter in the formula for the length of the tunnel, we will get a large length. Let’s try to substitute …