OK, I wont lose sleep, but back to units. I get a buzz out of digging into fundamentals, I dont like taking things at face value much. Regarding the ringing frquency, β, for now, I whipped up a series cct using a 68mh inductor having an inherent 190Ω series resistance, and a 10nF cap. I switched it from an oscillator using a low resistance MOSFET as the swithing element. I got a nice sine wave with little decay and measured the period as T = 170uS, giving a frequency of 1/T = f = 5,880 Hz. The radian frequency is therefore w = 2.pi.f = 36,900 rad/sec. Now β = √(1/LC - R²/4L) = 38,300 (units of something) ≈ w so β must be in units of rad/sec (accounting for errors measuring T and component tolerances). OK, the proof there is in the pudding (as well as being stated in most texts), but this leads me to ask further. Why does pi seem to apear out of knowhere, for no particular reason? There must be a deeper physical (or mathematical) reason for this, which I can in no way elude to. I will get back to the decay period issue at a later date.
PS A pi that looks like one with wiggely legs, not π, seems to have been monumentally dropped from our list of symbols.