The woofer or subwoofer can be chosen first either by how loud you want to play or how much volume you have available to install it. The next parameter is how large is the vehicle - what is the interior volume?
Knowing this you can make a selection.
The vehicle size sets the installed box resonance. It is also known the "Q" of this resonance should be between 0.7 to 0.8. 0.71 is ideal. A typical size vehicle has a volume turnover frequency of 66Hz. 48Hz for a Van and 87Hz for a compact are the outer limits.
Take for example is a midsize car with a 66Hz "resonant volume" with space for 1 cubic foot of enclosure.
Now we need a woofer whose Fs and Vas and Qt parameters allow it to be mounted in the enclosure such that Fb is 66, and Qb is about 0.7. A woofer Qt of 0.5, Fs = 47 and Vas = 1 cubic foot fits.
After mounting in the enclosure the Q is raised to .7 and the resonance to 66.

It is possible to work out the numbers in your head by trying many combinations. A computer program can do this quickly of course.
Remember Qt and Fs will rise with the square root of the total compliance. The total compliance can be referenced to Vas. So if the Qt was 0.35 it should be mounted in a box with a Vb of Vas/3. The parallel of these real and virtual volumes is Vas/4 so Fs and Qt will double after mounting. Now we can note that this woofer needs an Fs of 33. For a given type of vehicle there is an Fs and Qt relationship we can see developing. This is roughly Fs = 95*Qt for a typical 66 Hz size car.
Note that any Qt over 0.7 has no simple ideal solution in a mid size car. The same is true for any Fs over 66. If Qt was a low 0.2, the Fs would need to be a low 19Hz. The box size would be very small compared to Vas. The total virtual box size would be 12.6 times less than the Vas. This means the box size is Vas/11.6.
So for the midsize vehicle: Enclosure Volume = Vas*((Fs/66)*(Fs/66)/(1-(Fs/66)*(Fs/66)))
Change the 66 number up and down for vehicle type to 48 for a van or 87 for a compact.
Fs = 95*Qt for a midsize car 70*Qt  for a Van and 124*Qt for a compact.

A parametric equalizer centered on Fb can correct errors in Qb so then just getting Fb right is required. Note that a parametric band pass plus high-pass/lowpass equalizer can make any subwoofer in any cabinet work in any vehicle.

The louder you want to play the more woofer diameter and stroke you need.
Given all this you can select the amount of amplifier power you are ready to deliver based on what power the subwoofer can handle.
The larger the enclosure the less power that will be needed.
Take the power handing in mind when you turn up the wick. This is totally separate from getting the "sound" correct.
Systems built to pound and wail naturally will be set for that effect - the system described here is theoretically accurate not stylized for effect. Ported boxes are better for effects. That is another subject.

 

CDT engineering