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.
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.
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