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For plane waves, the wavefunction is exp(ik.r) and multiplying it with its complex conjugate gives a constant amplitude and thus a constant power (I think). But who says that all 2-D waves are plane waves!

In special cases, power can be constant, but I guess, in general, it must have a dependence on 'r'. I read in AP French's book 'Vibrations and Waves', somewhere in the last chapter where he discusses the solutions of wave equation in two and three dimensions, that far from the source the amplitude falls off as 1/sqrt(r). So Amplitude squared would be 1/r and thus power will fall as 1/r.

Is that correct?