Bessel – Bean Thinking

Happy New Year!

Nicaragua, direct trade, Amoret, Java beans, — Sometime this week I’ll brew this with a V60 and adapt an ‘examen’ to help me review 2019. I was thrilled to be able to meet the farmers, Dania and Desiree, at Amoret coffee earlier this year. One of the things I’m sure will feature in my ‘gratitude’ examen.

Each New Year is an opportunity to look back at the previous year, anticipate the future year and perhaps make resolutions to improve our lives, or even of those of people around us. Maybe this is even more true this year which is not just the start of a year, but of a decade.

This year I have been lucky to meet, or to continue friendship with, many people who have taught me all sorts of things about life, physics and coffee. There have also been some great finds of some fantastic cafes, trying to make a difference to their local community while serving excellent coffee.

And yet, as the year or the decade turns and we resolve to get fitter, pay more attention to sustainability or whatever seems important to us right now, we will inevitably take our existing selves into the new day and our resolutions will meet the reality of who we are: a bell rings at certain frequencies owing to the resonances of the vibration on the surface of the bell. The resonances of the bell depend on its exact shape and size, it is not easy to change the sound of the bell unless you change its temperature or even the interior to a different gas or muffle. (You can see images of how a violin vibrates at resonance here). The surface of a coffee resonates similarly, if we put it on a vibrating surface with a frequency that matches the fundamental vibrations of the surface. Nonetheless, thinking about these resonances can take us in surprising directions. The mathematics that describes them was developed by Friedrich Bessel (1784-1846) but, Bessel was not thinking about resonances when he formulated what is now known as Bessel functions. And it is possible, his life may have taken a very different direction were it not what happened from 1799.

In the new year of January 1799, when he was just 14, Bessel was apprenticed to an imports and exports company with the hope that he would become an accountant. And maybe we would have heard no more about him had he not got interested in the problem of longitude and solving the navigational issues of the time, important for the company for which he was working. This issue got him thinking about astronomy and he caught the attention of the authorities of an observatory who gave him a job there and encouraged his observations and interest. But it was while thinking about “many body problems” or how multiple massive objects interact with each other via gravity that he came up with the mathematical description that we now know as Bessel functions. It is these Bessel functions that also describe the resonances on a bell and in a coffee cup.

Sun, heat, nuclear fusion — What links coffee to the Sun? So many things! But for the purpose of this post, we can find clues as to the interior of stars by watching the way they vibrate, analogously to a bell. What would Bessel think? Image © NSO/AURA/NSF

What does this leave us with in our thoughts for 2020? That what we are interested by may lead us to discoveries in various tangential and scarcely believable connections? That what we plan for our lives may not be how they have to end up? That it benefits us to stop for 5, 10 minutes, even half an hour and just contemplate our world in our coffee? (ok, that last one did not come from Bessel). Where-ever your paths lead and your interests lie, happy new year! May the 2020s be a decade where we can all slow down, notice, contemplate and appreciate the beauty of this strangely connected world which is our home.

A lawyer, an accountant and and emperor walk into a cafe…

Strata, geology — This is not a resonance in a coffee cup but the concentric circle pattern is similar to a resonance that you could frequently see.

Have you ever noticed concentric rings on the surface of your coffee, forming as the table under the coffee cup vibrates slightly? Perhaps you have seen more complicated patterns. You may have observed, as you have played with your coffee, that some patterns are more stable than others. The one that is formed from concentric circles is fairly easy to form and to see. A more complex one looks like a chequer board, you may perhaps of seen others. These patterns are what are known as ‘resonances’ on the surface of the coffee and they are the consequence of standing waves being set up on the coffee surface. Many people who have gone through an undergraduate physics degree will immediately be reminded of Chladni figures and there is a good reason for this. Ernst Chladni (1756 – 1827) was a pioneer in investigating such resonances, one of the reasons that he has been described as “the father of experimental acoustics”.

And yet Chladni was not a physicist in the way that we now think of the term. In fact, by training he was a lawyer, a consequence of following his father’s rather insistent ‘advice’. Obediently, Chladni had trained in law and had started working as a lawyer in 1782 when his father died. Chladni appears to have taken this event as an opportunity to start to investigate the scientific problems that he was actually interested in and so re-invented himself as an acoustician testing the theories of music developed by people like Bernoulli and Euler¹.

transmission lines, electrical noise — Like strings on a guitar. Resonances on a string can be used to make musical notes.

Did Chladni drink coffee in eighteenth century coffee houses while admiring the resonances in the cup? Sadly what comes down to us in history is not his coffee habit but his experiments with sand covered metal plates secured onto wooden rods. Chladni caused resonances on these plates by rubbing them with a violin bow. By exciting resonances similar to those you can see on the surface of your coffee, Chladni was able to test theories about the sounds made by curved metal surfaces (e.g. bells). Indeed, these experiments became so important to understanding acoustic theory that Chladni started a European tour demonstrating his plates and their relevance to designing musical instruments. It was presumably through one of these tours that he met an Emperor of the time, Napoleon Bonaparte.

But despite this great experimental progress, the mathematics used to understand these resonance patterns, was developed by another physicist with a non-typical career path, Friedrich Bessel (1784-1846). Bessel had trained as an accountant but with the good fortune of timing, he had apprenticed into an exports company. At this time, such companies would have been interested in the problem of longitude and so Bessel gained an opportunity to indulge his interest in astronomy. As a consequence of this work, particularly his work on the orbit of Halley’s comet, Bessel secured a job in an astronomical observatory and it was there that he started the work that would eventually lead us to be able to describe, mathematically, the resonances on the surface of your coffee.

Did Bessel drink coffee? Had he seen Chladni demonstrate his plates? We don’t know the answer to those questions and in many ways it is not relevant because Bessel’s mathematics did not concern such resonances at all. Instead, almost to underline the idea that everything is connected, particularly with physics and coffee, Bessel was working on the problem of how to calculate the gravitational attraction between multiple objects.

Kettle drum at Amoret — The note made by a drum is a function of the size and shape (therefore resonance pattern) of the drum and also the gas filling the drum. Would this drum-table sound the same if banged on Venus as on Earth?

Perhaps you remember from school Newton’s famous description of the gravitational attraction between two bodies as being F = GMm/r² (where F is the force, G the gravitational constant, M and m the masses of the two bodies and r the distance between them). That’s all very good but what if there were three bodies, or four, or…

It was this problem that Bessel was working on and by so doing he solved the problem of Chladni’s patterns. The maths that describes the many body problem also describes the way that these resonances form. Those patterns in your coffee are described by the same maths as allows us to calculate complex gravitational problems.

And so perhaps it is not quite correct to title this post as a lawyer, an accountant and an emperor walk into a café, but it would be fair to say that each time you catch those resonances in your coffee cup, the influence and interests of these investigators of nature are infused within the brew.

You can find a sketch of Chladni entertaining Bonaparte with his metal plates here.

¹Harmonius Triads, Physicists, musicians and instrument makers in nineteenth century Germany, MIT Press, 2006