Lots of these photos on the internet today, for obvious reasons (this one from
The Atlantic). Am I correct in assuming that the lengths of the icicles on the lower power line would form a normal distribution (a bell-shaped curve without any skew?) Eyeballing it seems to suggest this, but is it fair to assume the shape of the distribution? I obviously don't have time to make the measurements...
Normal distribution? Probably not. The cable is going to be hanging in a catenary, which means there will be a tendency for water to flow towards the middle, so you'd get slightly shorter icicles on the ends, longer in the middle. My hunch is that you'd get a better match with a catenary plus a noise function. And the noise function would give an average length plus a random variation, but adjacent icicles would probably also 'feed' off one another, smoothing things out. There's an academic paper's worth of analysis lurking here, for sure.
ReplyDeleteI see no evidence in the photo of a catenary curve. Those typically develop in power lines in hot weather when the metal expands. In winter cables typically tighten up, as shown here. My impression of a normal distribution stands.
DeleteI also had the feeling there was a papers worth of analysis available in this. Although I certainly lacked the insight about the curve of the cable.
ReplyDeleteAll suspended cables, chains etc. follow a catenary curve. Even a metal rod would. Shallowness of the curve varies depending on stiffness of the cable and the tension. And once the surface it's one is wetted, water will flow down even the shallowest of curves, so even though it's basicaly not perceptible I'd still expect a slight bias towards the centre. Icicle formation isn't fully random - even in the picture, while the lengths vary, the spacing between icicles is very uniform - so I'm dubious that a normal distribution is a good baseline model to use. I think it you measured the icicle lengths on the bottom wire and compared them to a normal distribution, the match would be poor. They don't look random enough to me. A lot of natural processes (esp. watery ones, such as, famously, wave sizes) tend to follow power laws, which give a different curve.
ReplyDeleteI suppose another approach would be to ask, if you wanted a bunch of icicles whose heights did follow a normal distribution, what conditions should you try to arrange?
I have considered actually measuring the icicles with my calipers. There are 114 of them. But life is too short. I'm moving on...
ReplyDelete