If you stabilize the current at I0 I think that the question of average is solved.
you have U(t)=U0+R(t)I0 P(t)=U(t)I0 what is proposed here is that R(t) is hugely unpredictable and changing quickly. in fact if you measure U(t) average over the time period, you have simply the power injected. no need to panic with RMS, because I is constant. Pavg=Uavg.I0 if I is not constant, as if you drive by voltage, you have to measure the RMS of the current. I think that that is what McKubre found, probably faster than me, and maybe simply reading electrochemistry literature. the question is however to have a good current source with a high bandwidth, wider than the one of the bubble instabilities, but there is power supplies sold for that... note that on the opposite of what was said here, adding capacitors would not help. adding inductors in serie may however help the current source to overcome huge transients, but with modern electronics it is useless.
