Sunday, 28 February 2016

Slowness limits

A stretch of road that I regularly drive along recently had its speed limit reduced from 60 miles per hour to 50. No complaints from me, but I would like to know how much extra time to allow for my journey. A glance at the odometer tells me that the section with the new limit is two miles long so that means it'll take an additional ... err ... well ... an amount of time that I found rather tricky to work out in my head while driving.

Perhaps I'm just lousy at mental maths? For a week or so I annoyed my friends with this puzzle. Some worked it out sooner than others but none got it straight away and quite a few couldn't be persuaded/shamed into working it out without using pen and paper, or in some cases at all. Why is such a simple question so hard to answer?

The problem is speed, which is distance per unit time. If you're travelling at a speed of x 'distance measures per time measure' then, if you keep doing so for a whole time unit you'll have travelled x distance units. Hardly clear, I realise, and it doesn't help that I'm using 'unit' in two senses:in 'per unit time' it means with a value of one; in 'a whole time unit' it means the amount we use to measure time, be it hours, seconds or whatever.

The size of those measurement units determines the numerical value of the quantity - the Young's modulus of a material is an enormous number in SI units because it is the stress necessary to cause unit axial strain (i.e. doubling of axial length) and stress is force per unit area (in SI units a square metre). So the Young's modulus of steel is the force that would have to be applied to opposite faces of a one metre steel cube to stretch it till it was two metres long. No wonder it's a big number.

The measurement units might determine the numerical value, but the definition determines the meaning. With stress and strain there's a clear cause-and-effect: the stress causes the strain and the higher the material's modulus the more stress will be needed to cause the same strain. Is that the case with speed?

How often do you get in a car knowing how much time you intend to drive for but with an open mind about how far you're going to travel? In that strange situation knowing your speed is useful, because it tells you how many more miles each of the remaining seconds will add to your total distance. But what we almost always do is travel a set distance and want to know how long it will take. You might have a time constraint as well as a distance constraint but the distance constraint always wins: "Sorry, I couldn't drive fast enough so I got to the meeting five minutes late" might annoy your colleagues, but "Sorry, I couldn't drive fast enough so I stopped a mile away when it was time for the meeting to start" will annoy them more.

So we adjust speed in order to control arrival time, hopefully subject to speed limits. When you're running late it's tempting to go faster in order to get there sooner, but do you have a clear idea of how much journey time you save for a given increase in speed? I'd say no for two related reasons: as we've seen above it's a hard calculation to do, and the effect on journey time depends on how fast you're already going. As a result I suspect that some drivers who break speed limits do so because they overestimate the time it will save them. That's a testable proposition and I'd be interested in knowing if anyone's tested it, but for now I'm just going to assume it's true.

We could avoid the whole situation if we switched from speed to slowness, the reciprocal of speed. A speed of 60 miles per hour is a slowness of one sixtieth of an hour per mile, or 60 seconds per mile. If I tell you that a two-mile stretch of road has had its slowness increased from 60 seconds per mile to 72 seconds per mile the additional journey time will be obvious, and it'll also be clear that it's not a big deal; the additional 24 seconds is down in the noise compared to the uncertainties in my journey time.

The slownesses corresponding to our current speed limits mostly come out to whole numbers of seconds per mile because the number of seconds in an hour, 3600, is highly composite (thanks, Babylonians). So 10, 20, 30, 40, 50, 60 and 70 mph correspond to 360, 180, 120, 90, 72, 60 and 51.43 spm (oh well Babylonians, you did your best). Let's round that last one up up 52 spm, or 69.23 mph, rather than down to 51 spm, or 70.59 mph.

If we're going to argue that using slowness could reduce the perceived advantages of going faster when you're already going fast then we should consider what the effect will be when you're going slowly. Schools on main roads, which already have 30 mph speed limits, quite rightly have signs saying "Twenty is Plenty" - might motorists be less willing to increase their slowness from 120 spm to 180 spm than they are to reduce their speed from 30 mph to 20 mph? They might be, but the remedy is to address what they care about, journey time, with a sign saying "Take an extra 10 seconds [say] to go past our school at 180". OK it's not as snappy as "Twenty is plenty", nor does it rhyme, but someone can work on that.

You might be wondering what this has to do with acoustics: when we use rays to find approximate solutions to short-wavelength wave problems it's very common to calculate the slowness along a ray, when can be integrated over the ray's path to give its total travel time. This usage started in seismology (i.e. low-frequency short-wavelength solid acoustics) but shows up in plenty of other areas of acoustics.

Rays, however, have an important difference from cars: they don't stop at traffic lights. If we're calculating journey time by integrating slowness with respect to distance and the car stops we'll have a singularity, where slowness is infinite for an infinitely short section of the journey. Fortunately most cars have clocks and time proceeds untroubled when the car stops, so the time spent stopped can just be added to the journey time. As a bonus your slow-ometer (the old speedometer with a new dial) would have an infinity symbol on it which would a) look cool and b) encourage mathematical literacy.

Unlikely though this is to happen there's a more serious point here: the fact that speed and slowness are mathematically equivalent does not mean that they are psychologically, sociologically or political equivalent and there are plenty of similar choices to be made. Sticking with cars do you measure its mileage or its fuel consumption? Which map projection should you use when you plan an international trip? There are arguments to be made in each such case; I don't intend to make them here but my overarching point is that engineering is, or should be, a person-centred discipline and engineers have to think about these things - getting the maths right is necessary but not sufficient.

Postscript: There's an important case in acoustics where reciprocal quantities are less equivalent than they seem. When I first learnt that admittance was the reciprocal of impedance my reaction was to wonder why I was being asked to, effectively, remember a new name for something I already knew, and the fact that the real and imaginary parts of each had a special name didn't change my mind. It was much later that I encountered the multi-channel case where the admittance matrix is the inverse of the impedance matrix that I appreciated the value in having both concepts, so I try to explain that when introducing the concepts to my students.