Wave Physics - can a dynamic waveform be constrained to a specific geometry by signal processing alone? Suppose that you have a signal source, a set of point-transducers, and a handful of moderately powerful DSPs. Is it possible to construct an arrangement of the transducers such that the original signal propagates within a given region, but is absent or significantly attenuated outside of that region?
The obvious application would be sound systems that don't leak loud sound into neighboring rooms. But this would also be useful for limiting RF propagation to a physical space. In the audio example, you can shield the room with sound-absorbing materials. You could also build a faraday cage for the RF case, but either physical solution is likely to be prohibitively expensive. Now that powerful DSPs are cheap, can this be done entirely in the signal domain?
 A: For the simple problem where you have a set of point sources and you want them to cancel at infinity in all directions (i.e. there is very little signal except in a bounded region), the answer is yes. You just have to arrange for the sources at the surface to absorb the radiation going outwards.
An estimate of how many signal sources this will take (in the audio case): The highest frequencies are the worst. At 20KHz, the wavelength is about 1/20 feet so that is about how close your sources will have to be. For a 8 foot cube sized room this will be something like 200,000 sources (all 6 walls and a half inch apart) and will be pretty expensive.
On the other hand, you know that it's a lot cheaper to insulate high frequency sound than low frequency. So you hear your neighbors bass but not his treble. So if you are interested in sounds only below 300 Hz, now you need elements separated by about 3 feet and you only need something like 42 sources.
Given that these are audio frequencies, I'm thinking that the cost of the DSPs will be quite small compared to the cost of the speakers.
