The design of audio subwoofers has, for many years, been predicated on a theory known as "Hoffman's Iron Law" which provides:
Loudspeaker effiency = volume of enclosure / ( cube of cutoff_frequency)
(F3) is the desired low frequency cutoff or limit for the subwoofer
Unfortunately, if one wishes to reduce the low frequency cutoff of a subwoofer from, for example, 50 Hz to 18 Hz while retaining the same efficiency, the volume of the enclosure must be significantly increased. Or, if one wishes to decrease box volume from, for example, 1 ft3 to 0.4 ft3 and, at the same time, decrease the low frequency cutoff from, for example, 50 Hz to 18 Hz, efficiency drops by a factor of approximately 53.
Consequently, a woofer designer finds that where a 50 watt or 100 watt amplifier might have operated a 1 ft3 woofer at a 50 Hz low frequency cutoff, a 0.4 ft3 box at 18 Hz low frequency cutoff will require an amplifier that is approximately 53 times larger than conventional... in other words, an amplifier somewhere between 2500 watts and 5300 watts. They would also require an subwoofer driver able to handle this amount of power without burning up and without running out of linear excursion (don't want the voicecoil bottoming out or being launched into the room) This explains why you will not find any small subwoofers that can play very deep with any respectible sound-pressure-level. Basic physics get in the way.
Today, there are many DVDs with sound-tracks with low-frequency effects down to 10Hz and lower. Many commercial subwoofers have low frequency cutoffs somewhere 50 and 100 Hertz and amplifiers around 100 to 200 watts, these subs don't even come close to reproducing the low bass on these sound-tracks, but might be fine for music where low-frequency bass seldom goes below 50Hz. Other subs sold for use in a home theater may have a response down to 20 Hz. These are usually a bit larger, with larger amaplifiers, trading larger size and efficiency for lower frequency response.
There are many commercial subwoofers that employ amplifiers that are capable of over 1000 watts. The "Sunfire" brand is one example. They use a powerful 2500 watt amplifier in a small enclosure to produce good quality low frequency bass. They trraded efficiency for small enclosure size.
Other brands of subwoofers have much larger enclosures and much smaller amplifiers (Most subwoofer "plate-amplifiers" produce from 75 to 250 watts)
Hoffman's Iron Law says that the efficiency of a woofer is proportional to cabinet volume multiplied by the cube of the F3. In other words, even thouogh you want a small box, high efficiency and deep bass, you only get to pick 2... the third is decided by physics. (Dispite what the marketing departments want you to believe)
If you want a small box, you can have deep bass or high efficiency, not both. Likewise, if you want deep bass, you get a small box or high efficiency, not both. Last, if you want high efficiency, you get deep bass or a small box, but not both.
There is no way to cheat this law until you find a way around the laws of physics, so it's something that you need to be aware of when evaluating your needs, and also when evaluating manufacturer's sometimes outlandish claims.
How many "watts" do you need??? I'll say you probably need somoewhere between 50 and 50,000 watts.
(I have two 18" Ascendent Avalanche drivers, each in a sealed 12.5 cubic foot enclosure, each powered by a 500 watt amplifier, in a room roughly 20x14x8 feet.
As a result, my low frequency cutoff is somewhere near 10Hz. Note... these are LARGE enclosures compared to most subwoofers, but I traded size for extended low frequency response... and I needed 500 watts per driver)