Season 7 Episode 2

Part of the power of mathematics is that one equation can turn up in many different places. In this episode; the wave equation.

Further Reading

Waves on squares and circles

Perhaps you don't care about waves on a string, you want to see waves in higher dimensions. Good news! The basic solutions for waves on a square drum, pinned at the edges, are fairly nice. They're

$$y=\sin(kx)\sin(my)\sin(\sqrt{k^2+m^2}ct)$$

where $k$ and $m$ are whole numbers.

The energy associated with such a wave is proportional to $k^2+m^2$, so this is a physical problem which requires us to think about which numbers are the sum of two squares. We're doing Maths and Physics at the same time!

Perhaps you don't care about square drums, because who has a square drum? Luckily for you, mathematicians can also solve for what I’m calling the basic solutions (really, eigenfunctions) of a circular drum. In fact, this is in the Oxford Mathematics course. The solutions are great; see Wikipedia for animations.

For an interactive solver, see the VisualPDE website. This is exciting because it actually solves the partial differential equation numerically in your browser. They've got other partial differential equations on their website for you to explore, including the heat equation (which someone mentioned in chat) and ones for mathematical biology. We're doing Maths and Biology at the same time!

Physics

In case anyone actually needs to revise A-level Physics standing waves, I like Isaac Physics for that sort of thing. If you're looking for a fun webapp that demonstrates waves on a string, see this page. As a challenge, set it to manual, set damping to none, and try to oscillate the wrench at a frequency that causes a standing wave. Now go back to VisualPDE and try to tap the screen at a frequency that sets up standing waves. If you've got it right you should be able to stop tapping and let the waves continue to oscillate.

Quantum mechanics

I mentioned quantum mechanics in passing, referring to atomic orbitals.

This is probably too advanced for now, but the upshot of getting through all the mathematics to rigorously solve for the energy levels of the Hydrogen atom is that you get to understand why the periodic table is the shape that it is (why does the row starting with Lithium have exactly eight elements in it?). We're doing Maths and Chemistry at the same time!

Music

Something I didn't have time to discuss is the effect of the length of the string (I mentioned the importance of tension and density of the string). I said in passing that I'd used a string of length $\pi$ to make things prettier. Let's generalise that and consider a string of length $L$. We'll put the ends at $x=0$ and $x=L$.

With just a minor tweak to the calculation we did on the livestream, we now want $y=\sin(kx)\sin(kct)$ to be zero at $x=L$ for all time (rather than $x=\pi$). That means that we want $kL=n\pi$ for some positive integer $n$. A quick bit of dividing and we find that we want $k=\frac{\pi}{L}n$ for $n=1,2,3,\dots$. Perhaps you can see why I picked $L=\pi$ for the livestream.

The frequency of these waves is $kc=\frac{\pi c }{L}n$ which gets smaller for larger values of $L$. So shorter strings give higher musical notes. Where have we seen this? Perhaps on the fretboard of a guitar; putting your finger on the string to hold it still effectively reduces the length of the string, and putting your finger nearer the bridge gives a higher frequency note. Ever wondered why the frets get closer to each other as you get nearer the bridge? If you want the note to be one octave up from the base harmonic of the string, then you need a string of half the length, then one more octave up is half the length again, and so on. The same sort of thing is true for smaller intervals than an octave (but the proportions are different of course, depending on the interval).

Beats

My Desmos graph is available here.

You might have wondered why I didn't use the hear graph feature in Desmos to demonstrate the beats effect. (if you haven't seen it before, click the keyboard icon in the lower left, click the speaker icon, and click "Hear Graph" for an audio interpretation of your graph).

There's a good reason! The hear graph feature does something slightly different from what I want here. I would like to send the waveform on screen directly to your speakers, to make them oscillate in precisely that pattern. The Hear Graph feature is designed to instead convert a constant function into a sine wave, so that you can "hear the value of the function". It's a bit like Hear Graph is treating your graph as sheet music, whereas I want to treat the graph as the literal sound wave. Desmos keeps the amplitude of the noise constant, and it cannot play two notes at once, so there's no way for it to play a beats interference pattern.

Wikipedia has some sound files if you want to experience this effect, including a nine-minute binaural version which is, in my opinion, too much. 

If you want to get in touch with us about any of the mathematics in the video or the further reading, feel free to email us on oomc [at] maths.ox.ac.uk.