# #018: Slow Subwoofers?

"Is your sub constantly trying to catch up?" Umm... what?! Today's topic is slow subwoofers. It seems as if some subwoofers have this magical property of being 'fast', whereas the rest are just these sloths, that are perfectly content running at a lower pace than the rest of the audio chain. We will investigate if there is any merit to this from a mathematical and physical standpoint.

I think the topic of "slow subwoofers" is similar to that of "time-alignment in loudspeakers": It is complex enough that the explanation cannot be formulated in a few words, but simple enough that laymen and hobby enthusiasts want to chime in. This is true for many aspects of HiFi in general, and loudspeaker design in particular. Anyway... Let's get on with this. A subwoofer is a loudspeaker intended only for low-frequency output, typically below 80-ish Hz. It is pretty much always an electrodynamic driver in either a closed enclosure, a ported enclosure, or a more intricate internal build. Now, among audio enthusiasts there seems to be this idea that some subwoofers are 'fast', while others are 'slow', with the latter being a negative trade. This has always baffled me quite a bit, since when dealing with the modelling of loudspeakers, the words 'fast' and 'slow' really don't enter anyway. It is not like we talk about certain electrical components being slow, such as a slow capacitor for example, and yet a loudspeaker can be described via an analog electrical circuit. So what is going on here. Well, first let's see how we could possibly describe 'slow' in the first place:

1) Always lagging behind

2) Late to start/stop

3) Low velocity of diaphragm

Regarding the second type of slow, the inability to start or stop quickly, this is determined by the system's transfer function. A sub with a closed enclosure will have a second-order high-pass transfer function, and so all you need to know (at least before placing the sub in an actual room) will be available in this function. Take a look e.g. at the Linkwitz Thor sub below:

This is a closed box with a twelve inch driver. Linkwitz applies an equalization filter to extend the low-frequency capabilities, but let us look at the pure driver/box combo. There are two poles in the transfer funtion; one at s=2Pi*30 rad/s and one at s=2Pi*46 rad/s. This gives a cross-over frequency of 37 Hz, and a Q of 0.49. In this case, we have two real poles, and a relatively low Q, and for most subwoofers, a higher Q with complex poles will probably be more common. But either way, knowing the zeros and poles will allow you to calculate the subwoofer's step response, impulse response, or whatever else response, you are interested in. You just have to either do the necessary inverse Laplace transformation analytically [1], or use a mathematics or electronics tool to do it for you. This way you can investigate the temporal properties of your subwoofer. I did the inverse Laplace transformation analytically, mainly to refresh the technique, and if everything went well, this should e.g. be the step response:

The above analysis shows that you can find the temporal behavior, if you simply model the transfer function, apply the impulse/step/whatever and map the response to the time domain. For a ported sub, the responses may look more irratic, and the Q may be higher, to give you more of an 'ooomph', at the expense of the sub sounding 'less precise', 'sluggish', 'slow', or whatever subjective measure you feel makes sense. And you could argue that there is something to it, but it would show up in the analysis, so there still isn't anything magical, that engineers have yet to discover. By the way, I have this sub also, and without the EQ I think most would describe its sound as 'dry', 'fast', or something similar, and I would agree, even though it is not moving any faster than other subs. It is most likely tied to the low Q of the system, although placement in the room, and matching with the front speakers are also important points.

Once you put your sub in your room, you can of course run into issues, as the sub operates at frequencies where the modal behavior of the room itself becomes important. You probably have to play with the placement of the sub, and if you have multiple subs, you need to know what you are doing. I suspect that a lot of problems, and probably subjective evaluations with the sub sounding 'slow', are due to the room-sub interaction.

You could probably also experience issues if you have a small leak in the enclosure, even if it is ported, but then you are looking at an imperfection that you could still model, and again there is nothing inherently slow about the sub. The same would apply if the enclosure is not rigid enough; you could possible hear a less accurate sound reproduction, but it would make sense, since the vibration of the enclosure would add to the total SPL in a deterministic way.

The final way to look at a sub being slow is that its membrane could possible have a low velocity. But that is really not much of a topic for discussion: The area of the sub driver and the acceleration of the driver, along with the port ditto, determines the SPL. If the acceleration is low, the velocity is low, but if the area is large, you still get the same output as a 'faster' driver with a smaller area. One thing to consider here, however, is that smaller drivers have to have larger excursion for a given SPL, and so non-linear effects will become more important. Also, as the driver plays, it will heat up, and this will affect both the magnetic system and the structural materials, and so in practice there are quite a few things to consider. Again, if you have an experience that your sub is 'slow', it could be that some of these effects are influencing the subwoofer output in an adverse way.

So in conclusion: The signal being send to any subwoofer is slowly varying by nature, in the sense that for a given time, a low-frequency signal has fewer oscillations than a high-frequency signal. A well designed subwoofer (reasonable Q, rigid built, ...) will be fast enough to reproduce this signal in a satisfactory way. Poor sound may still be present, if e.g. the sub is placed in the room in such a way that the frequency response is adversely effected, if the cross-over to the front speakers is chosen poorly, if the front speakers are to small to ever bridge the gap between sub and themselves and so on. Faults in the sub, such as leaks or subpar enclosures, could of course be affecting the output, just as design choices regarding e.g. Q of the sub or low power amps, could equally make your sub sound in way, that you feel is 'slow' compared to better constructions. But overall, I do not believe that the term 'slow' makes much sense, when talking about subwoofers, and it most likely covers over a myriad of effects, and a lack of physics and mathematics knowledge of those using the term.

[1]: Active All-Pass Crossover Networks with Equal Resistors and Equal Capacitors, René Christensen, J. Audio Eng. Soc., Vol. 54, No. 1/2, 2006 January/February

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