While driving along today I notice that the antena of the car was vibrating, which got me thinking about invariants and dimensionless numbers. I'm not sure if there's anything to this, but I'll think out loud a bit about it.
The argument for dimensional analysis is a bit weird. The idea is that in any physical system there are bound to be some invariants, somethings that are independent of the scale of the system. This is basically the idea that the universe is simple and linear, which empirical observation shows is often true.
Another way of looking at it is to say that a lot of things depend on the ratio of various forces, sizes or speeds, but not on their absolute magnitude. An airplane flying depends on the ratio of lift to weight being greater than one, for instance.
But on this skimpy foundation we can then build a pretty substantial castle. The most famous dimensionless number is the Reynolds number, which is the ratio of viscous to inertial forces in fluid flow. Viscosity is a measure of how well a fluid supports shear stress, which is roughly how "runny" a fluid is. Honey is viscous, air is (relatively) invisid. It turns out that the dynamics of flow are dominated by the Reynolds number. At very small Reynold's number (less than one) viscous forces are practically zero and the fluid is described by the equations of potential flow. At Reynold's numbers up to about 2000 the fluid is described by the equations of laminar flow. And at Reynold's numbers greater than 2000 the flow becomes turbulent, and can hardly be described at all.
Heisenberg is reported on his deathbed to have said that he had two questions he wanted to ask God: why relativity, and why turbulence? He further remarked that he was pretty sure God would be able to answer the first one.
Given the complexity of turbulent flow, it is quite remarkable that there is this simple numerical value that can predict when it will occur, but that's the way it is.
The frequency of vibration of a cylinder like my car antenna in a flow is governed by something called, if memory serves, the Stroul number, which is a dimensionless constant. The quantities we have to work with are:
velocity (dimensions of distance/time)
frequency (dimensions of 1/time)
size (dimensions of distance)
The obvious combination of these is:
v/(f*l) = constant
where v=velocity, f=frequency, l=size. I haven't specified which size, and there are two to choose from: the diameter and the length of the antenna. There were also two modes of vibration: a fast mode, in which the antenna vibrated with a maximum at the center and nodes at both ends with a frequency of about 10 Hz, and a pulsation in that mode with a frequency of about 0.5 Hz, or a period of 2 seconds. The diameter of the antenna is about 4 mm and the length about 1 m, so the ratio of the lengths is about 250 to 1, while the ratio of the frequencies is only about 50 to 1, so this suggests that either there is a different constant for the length, or something else is going on.
A quick check with D. J. Triton's Physical Fluid Mechanics strongly suggests something else is going on. The Stouhal number is defined as the reciprocal of the quantity above:
St = f*d/v
where d=diameter of the cylinder, and has a roughly constant value of 0.2 for Reynold's numbers between 40 and 2000. Above 2000, where turbulence sets in, the flow is so chaotic that no single frequency really dominates. The Strouhal number actually describes the frequency of vortices being shed from the cylinder, not the vibration of the cylinder itself, which is dominated by the mechanics of the cylinder, which I should have thought of.
The car was going about 90 km/hr = 25 m/s, and the diameter of the antenna is 0.004 m, so a St=0.2 predicts a vibration frequency of 1250 Hz, whereas the antenna was vibrating at perhaps 10 Hz. This suggests that the 10 Hz frequency is a natural frequency of the antenna, which is just been driven by the much higher frequency of the vortex shedding.
Does any of this mean anything? Probably not, but I rarely get to do this kind of physics these days, so it's fun to look into. I haven't had a fluids book open for (as you can probably tell) five or six years.
Stalled about midway through the Iliad, but still slogging. There's only so much senseless death a fellow can take in a week.
I've started reading Melody Beatie's Codependent No More and am getting lots of insight from it. Christianity has a lot to answer for, and teaching us (me, specifically) that you're only worthy if you're helping others is one of the worst. The inductive approach to codependence (not currently a term much favored by therapists, apparently) is valuable and good, and the relentlessly positive message that you can change, you can impove, you are worth more than your ability to help others, you are not responsible for anyone but yourself...is really good.
But I've got a few of criticisms.
The first is the emphasis on God, which is no small part of what got me into this mess in the first place. Long after I fled the church, I still thought there was some value in the sentiment expressed by "We do not presume to come to this thy table Oh Merciful Lord trusting in our own righteousness, but in Thy manifold of great mercy." Even after I knew what a jerk Bishop Laud was, that little snippet from the Book of Common Prayer stayed lodged in some dark corner of my mind, sapping my will like a spiritual hookworm.
The second is that simply telling people "Smarten up! Love yourself!" isn't enough. I don't count this as a criticism, really, because one book can't do the whole job, but apart from a few exercises it leaves unanswered the question of "What do I do to change?" More concreteness would be valuable. A friend once summed up his "life lessons" in two aphorisms, which I repeat here for your benefit: 1) Don't be an idiot, and 2) Don't do anything stupid. The second is an immeadiate corollary of the first, but more valuable because it is more concrete: it is often easier to determine when we are about to do something stupid than when we are about to be an idiot.
For the question of "What do I do to change?" I recommend people have a look at Branden's The Six Pillars of Self-Esteem. Walter Foddis has recently written an essay for Enlightenment on the value of sentence-completion as a technology of self-discovery.
"Know thyself, presume not God to scan,
The proper study of Mankind is Man."
Pope's imprecation, stripped of the sexist language, is as true today as ever, and I think that to live "the examined life" we should be happy to pick up any technology that lends itself to self-discovery, which sentence-completion surely does.