Like any speaker system, you have to isolate the front from the back. The out-of-phase "wave"from the back of the speaker comes-around to the front at the speed of sound, and it will tend to cancel the sound (or wave/impulse) from the front. A large sealed box can be used, or maybe you can use a "tuned" ported box, or a long pipe.
A bigger speaker will move more air, as will a speaker with a larger X-max.
Power (Watts) is calculated as Voltage squared/Impedance. 15 VDC will give you slightly more than 50W. But, speaker ratings can be tricky. A 50W speaker should be able to handle the bass from a 50W signal (which is most of the 50W), but a 50W amplifier playing undistorted music at full volume is only putting-out an average of around 5W. You speaker can probably stand short 50W pulses, but it will probably burn-up with 50W constantly applied for several seconds.
A square wave (or DC suddenly applied) has a high-frequency component and a low-frequency component. A woofer can't produce high audio frequencies, but it can produce mid-frequencies. So with DC, you'll get a more of a "click
than a "thud". If you wan't a particular sound, you'll need to use an audio amplifer and probably a few cycles of a low frequency sine wave.