Bandwidth applied to amplifiers and speakers is a familiar term. This page is taking a look at the low-end of the system bandwidth, and tries to add it all together. Speakers perform better for bass in large vented cabinets, but are more convenient desirable in small cabinets, which affects the frequency where the gain starts to diminish. Amplifiers can more easily be designed with a "flat" bandwidth, until we start adding capacitors in the signal path to block DC currents, but this can be optimised for minimal effect on bandwidth. Preamps ditto. Cables driven badly roll off badly, so we tend toward low impedance outputs driving high impedance inputs. The double bass instrument, as we have noted, is just like the speaker, a compromise of size and sound. As the violin evolved into a family of ranged instruments, it required twists and turns in the dimensions to keep the design intact. The cello, for example, is notorious for commonly suffering from wolf-tones and uneven response. The bass too, while it has to be big, also has to be playable. The bass roll-off here is not measurable without some Thiele-Small style equations thrown at it. It could be guessed at, but the figure would be irrelevant. We are already accustomed, if not programmed, to accept the bass with its limitations and enjoy its character when in expert hands.
I could digress here, on the difference between my years of listening pleasure on radios and mono record players (I exaggerate for effect), and those of someone brought up on pristine cd recordings on headphones, a practice I find uncomfortable. We are, to a degree, programmed with the repertoire of our upbringing, which gives rise to a diverse range of opinion, also immeasurable.
Sparing much detail which is already well addressed elsewhere, the roll-off of bottom-end bandwidth, or bass response, can be described as high-pass.
This simple term is misleading, implying that there is an electronic gate which will allow through it, say, any frequency over 123.4Hz, for example. The term is often backed up by misleading graphs, which show bandwidth as a set of straight lines. I have already given a similarly misleading statement on this very page, when referring to a "flat" bandwidth. So let's get to the nitty-gritty and iron out the terms. Firstly, I'll deal with the straight line bandwidth diagram. It is just that, a simplified representation of an ideal, knowing that it is not achievable. Playing with my op amp filter calculators, I can see that the only straight line graph available to me in a real situation with a negative feedback loop amplifier is by the use of a single component. A resistor provides all-pass gain, (horizontal line), a capacitor provides a slope of 6dB/octave (Circuit 1). If two components are combined, the slope can be made to change to horizontal at either end, but there are no sharp corners where horizontal meets slope. It occurs over a frequency range, actually a very large frequency range; the slope never truly becomes horizontal. I hope it becomes horizontal "enough", within the frequency band of interest to save having to make it more complex, or more expensive. Certainly, looking at a single opamp stage, the graphs of Circuit 3 or Circuit 4 can be "tuned" to a degree, to reach a point of some satisfaction. If not satisfied, more stages are required, and more fine tuning, and this usually happens when the join between slope and horizontal is "not good enough".
When I hooked up a piezo element to my prototype preamp, I had to minimize the length of cable between the two, knowing that the piezo crystal's impedance is high, a bad situation, causing roll-off. Yet the input of the preamp demanded a capacitor, to protect the op amp from very low frequencies and d.c. Already I had two threats to my super-sounding future which I could only partially address, with a shorter cable and a larger capacitor. I knew that the "slope" from these two elements would never become truly horizontal, but hoped it would be good "enough". In practice, I know now that the piezo's position on the bass is quite isolated from the danger of being thumped, so I might consider removing the capacitor from the preamp.
And so, on goes the signal, through the amplifier and to the speaker, where it meets another slope, the bass response of my "too small" cabinet. Let's imagine that in such a system there are three elements, like the capacitor, which might be there for good reason. They each have a high-pass effect on the signal, and for the sake of this example, we find that the cut-off is at 10Hz for each one, that's the point where the signal has dropped by 3dB.
The cumulative effect of three such high-pass filters will be found by looking at the point where the signal has dropped by 1dB
Perhaps the speaker cabinet should be more specialized, focus on the exact area of concern, and let another speaker do the rest, the easy stuff. There is such an animal, the bandpass cabinet, which tunes a volume of air behind and in front of the speaker, addressing a frequency range of perhaps only an octave or two, but with a 'flattish' response in that band.
Chebyshev and Butterworth
According to my Thiele-Small parameter machine, the slope is considerably steeper than 6dB/octave, and the cabinet is tunable to make it horizontal with a very sharp corner frequency. The parameters and the plots hint at "Butterworth" and "Chebyshev" tunings