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Fri, 21 Apr 2006
When I was in high school, I'd see water running out of a faucet growing narrower, and wonder if I could figure out what determines that curve. I found it was rather easy to do.I puzzled over that one for years; I didn't know how to start. I kept supposing that it had something to do with surface tension and the tendency of the water surface to seek a minimal configuration, and I couldn't understand how it could be "rather easy to do". That stuff all sounded really hard! But one day I realized in a flash that it really is easy. The water accelerates as it falls. It's moving faster farther down, so the stream must be narrower, because the rate at which the water is passing a given point must be constant over the entire stream. (If water is passing a higher-up point faster than it passes a low point, then the water is piling up in between—which we know it doesn't do. And vice versa.) It's easy to calculate the speed of the water at each point in the stream. Conservation of mass gets us the rest. So here's the calculation. Let's adopt a coordinate system that puts position 0 at the faucet, with increasing position as we move downward. Let R(p) be the radius of the stream at distance p meters below the faucet. We assume that the water is falling smoothly, so that its horizontal cross-section is a circle. Let's suppose that the initial velocity of the water leaving the faucet is 0. Anything that falls accelerates at a rate g, which happens to be !!9.8 m/s^2!!, but we'll just call it !!g!! and leave it at that. The velocity of the water, after it has fallen for time !!t!!, is !!v = gt!!. Its position !!p!! is !!\frac12 gt^2!!. Thus !!v = (2gp)^{1/2}!!.
Here's the key step: imagine a very thin horizontal disk of water, at distance p below the faucet. Say the disk has height h. The water in this disk is falling at a velocity of !!(2gp)^{1/2}!!, and the disk itself contains volume !!\pi (R(p))^2h!! of water. The rate at which water is passing position !!p!! is therefore !!\pi (R(p))^2h \cdot (2gp)^{1/2}!! gallons per minute, or liters per fortnight, or whatever you prefer. Because of the law of conservation of water, this quantity must be independent of !!p!!, so we have: $$\pi(R(p))^2h\cdot (2gp)^{1/2} = A$$Where !!A!! is the rate of flow from the faucet. Solving for !!R(p)!!, which is what we really want: $$R(p) = \left[\frac A{\pi h(2gp)^{1/2}}\right]^{1/2}$$Or, collecting all the constants (!!A, \pi , h,!! and !!g!!) into one big constant !!k!!: $$R(p) = kp^{-1/4}$$ There's a picture of that over there on the left side of the blog. Looks just about right, doesn't it? Amazing.So here's the weird thing about the flash of insight. I am not a brilliant-flash-of-insight kind of guy. I'm more of a slow-gradual-dawning-of-comprehension kind of guy. This was one of maybe half a dozen brilliant flashes of insight in my entire life. I got this one at a funny time. It was fairly late at night, and I was in a bar on Ninth Avenue in New York, and I was really, really drunk. I had four straight bourbons that night, which may not sound like much to you, but is a lot for me. I was drunker than I have been at any other time in the past ten years. I was so drunk that night that on the way back to where I was staying, I stopped in the middle of Broadway and puked on my shoes, and then later that night I wet the bed. But on the way to puking on my shoes and pissing in the bed, I got this inspiration about what shape a stream of water is, and I grabbed a bunch of bar napkins and figured out that the width is proportional to !!p^{-1/4}!! as you see there to the left. This isn't only time this has happened. I can remember at least one other occasion. When I was in college, I was freelancing some piece of software for someone. I alternated between writing a bit of code and drinking a bit of whisky. (At that time, I hadn't yet switched from Irish whisky to bourbon.) Write write, drink, write write, drink... then I encountered some rather tricky design problem, and, after another timely pull at the bottle, a brilliant flash of inspiration for how to solve it. "Oho!" I said to myself, taking another swig out of the bottle. "This is a really clever idea! I am so clever! Ho ho ho! Oh, boy, is this clever!" And then I implemented the clever idea, took one last drink, and crawled off to bed. The next morning I remembered nothing but that I had had a "clever" inspiration while guzzling whisky from the bottle. "Oh, no," I muttered, "What did I do?" And I went to the computer to see what damage I had wrought. I called up the problematic part of the program, and regarded my alcohol-inspired solution. There was a clear and detailed comment explaining the solution, and as I read the code, my surprise grew. "Hey," I said, astonished, "it really was clever." And then I saw the comment at the very end of the clever section: "Told you so." I don't know what to conclude from this, except perhaps that I should have spent more of my life drinking whiskey. I did try bringing a flask with me to work every day for a while, about fifteen years ago, but I don't remember any noteworthy outcome. But it certainly wasn't a disaster. Still, a lot of people report major problems with this strategy, so it's hard to know what to make of my experience.
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