> an omnidirectional antenna that transmits 1W forward, but also 1W backward, and up, and down, etc
By definition, any antenna with positive gain must concentrate power in one or more directions. So in practice, omnidirectional antennas are only omnidirectional in the horizontal plain, they have greatly reduced power in the up and down directions (which makes sense as generally there is no need to transmit into the sky or ground).
> Overall the omnidirectional antenna may transmit much more power total
This doesnt make sense - an antenna is a passive device, not an amplifier. Regardless of gain, you cant get more energy out of an antenna than you put into it.
By definition, any antenna with positive gain must concentrate power in one or more directions. So in practice, omnidirectional antennas are only omnidirectional in the horizontal plain, they have greatly reduced power in the up and down directions (which makes sense as generally there is no need to transmit into the sky or ground).
> Overall the omnidirectional antenna may transmit much more power total
This doesnt make sense - an antenna is a passive device, not an amplifier. Regardless of gain, you cant get more energy out of an antenna than you put into it.