## Fun with the Airy Equation

Today being a Maynooth Thursday, it has, as usual, has been dominated by computational physics teaching. We’re currently doing methods for solving ordinary differential equations. At the last minute before this afternoon’s lab session I decided to include an exercise that involved solving the following harmless-looking equation:

This is usually known as Airy’s equation and it comes up quite frequently in problems connected with optics. It was first investigated by a former Astronomer Royal George Airy, after whom the function is named; incidentally, he was born in Alnwick (Northumberland, i.e. not the Midlands).

Despite its apparent simplicity, the Airy equation describes some very interesting phenomena. Indeed it is the simplest ODE (that I know of) that has the property that there’s a point at which the behaviour of the solution turns from oscillatory to exponential. Here’s a result of a numerical integration of the equation: obtained using a simple Python script:

(I stopped the integration at as the magnitude of the solution grows very quickly beyond that value for the particular initial conditions chosen).

One of the reasons for including this example (apart from the fact that Airy was a Geordie) is that the students were so surprised by the behaviour of the solution and most of them assumed that there was some problem with the numerical stability of their results. Some integration methods do struggle with such simple equations as the simple harmonic oscillator, but just sometimes weird numerical results are not mere numerical artifacts!

Anyway, my point is not about this particular equation or even about computational physics, but a general pedagogical one: finding interesting results for yourself is much more likely to motivate you to think about what they mean than if they’re just described to you by someone else. I think that goes for numerical experiments in a computer lab just as much as it does for any other kind of practical experiment in a science laboratory.

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April 12, 2018 at 5:10 pm

Hang on, you called Durham the Midlands some posts ago…

April 12, 2018 at 5:12 pm

Durham is in the Midlands, Alnwick is not.

April 14, 2018 at 2:43 pm

the Airy equation describes some very interesting phenomena. Indeed it is the simplest ODE (that I know of) that has the property that there’s a point at which the behaviour of the solution turns from oscillatory to exponentialThat’s because it is of the form y” = f(x)y; f(x) changes sign between the oscillatory and exponential regimes; and you can’t get a simpler function which does that than f(x)=x. So you won’t find a simpler one.