Someone tried to explain the process of beam forming to me, and how beam forming is far more efficient than an omnidirectional signal of similar amplitude.
This seemed obvious at the time: the power of the signal is proportional to the integral over time, so since the magnitude is only large in the desired direction, you only need to supply the power for that part.
But from what I understood, beam forming is using interference patterns to make sure that the signal strength in one direction (the 'beam') is much larger than others. I assume that the maximum amplitude of the signal cannot become larger than the sum of the amplitudes of the point sources.
So let's say that I have 2 point sources to form my (admittedly pretty terrible) beam. I model them as both sending out sine waves with amplitude A omnidirectionally. The maximum amplitude of the interference is now 2*A. And the power consumed scales with twice the integral of the sine over time.
Now compare this to a single omnidirectional point source of equivalent amplitude (2*A). This also scales with twice the integral of the sine over time.
So where do the power savings come in? Is it because I ignored the reduction of the signal strength with distance? But that should scale inversely with distance^2 either way, right? Are my cows too spherical?