If you are tired of London

Jonathan's coffee house plaque
Jonathan’s Coffee House was the site of the first London Stock Exchange. A link between coffee and the financial markets can be made via the maths describing Brownian Motion.

London has a long history for tea and coffee drinkers. The first London coffee house opened in around 1652 in St Michael’s Alley*, followed quickly by a large number of coffee houses all over London. Isaac Newton, Edmond Halley and Abraham de Moivre were all known to drink coffee in the various establishments dotted around the city (de Moivre even charged a small fee for mathematical advice at Old Slaughters on St Martin’s Lane). Famous institutions such as the London Stock Exchange and even the RSPCA were first formed at London Coffee Houses. On the engineering front, Thomas Telford established the Institute of Civil Engineers at the Salopian Coffee House just off Trafalgar Square. Yet the story in this post comes not through the coffee houses but from historic tea drinkers and, in particular via a “prodigious drinker of tea”, Dr Samuel Johnson**.

The story is detailed in Boswell’s “Life of Samuel Johnson” and concerns a friend of Johnson: Mrs. Anna Williams (1706-83). Williams had gone blind in the 1740s and, in the later part of her life lived with Johnson in his various dwellings around Fleet Street. You can now visit the Samuel Johnson museum in one of those properties at Gough Square, just off Fleet St. Between 1759 and 1765 Williams was living a short distance from Johnson but Johnson regularly popped around for tea in the evening. It was on one such occasion that Boswell joined them. Williams would pour the tea for all the guests but Boswell wondered how she knew when to stop pouring given that she couldn’t see. Boswell thought it was possible that she had subtly dipped her finger into the tea to feel when it was about to flow over the edge of the cup. As he later found out, this was not true. Apparently she had “acquired such a niceness of touch as to know by the feeling on the outside of the cup how near it was to being full.”

The question then arises, how could she have known this just by touch?

a heat sensitive coffee mug
This coffee mug changes colour as it gets warm. The constellations slowly appear as the heat is transferred from the hot coffee inside to the outside of the mug.

A first solution would be that she was feeling the heat of the tea through the porcelain of the cup. Is this possible? It is certainly true that, once a cup is full, it feels warmer to the touch than an empty cup, but heat does not travel through a substance, such as a cup, instantaneously. Would Williams have had enough time between feeling that the cup had become warm to stopping pouring to prevent the cup from overflowing? A quick experiment with my colour changing cup (pictured) suggests that this may be unlikely. Even if she had the sensitivity in her fingertips to quickly discern a change in temperature, it took 20 seconds between boiling water being introduced to the cup before the top started to change colour. Although porcelain is thinner than my mug, and the change is probably quicker to feel than to see with the cup, it is likely that she would have carried on pouring the tea beyond the rim of the cup before realising that the cup was becoming full. Although it is possible that she could compensate for this, for example by touching the cup further towards the base, perhaps there are other ways that you can discern a cup is almost full without seeing it?

A second solution could be that her discernment occurred through a mix of touch and sound. There are a few ways in which audio signals will suggest that a cup is almost full. Coupled with the feeling of increasing warmth on the side of the cup, this may give the tea pourer increased confidence that the cup is comfortably full. One such clue is familiar to those of us who fill water bottles up from the tap, the change in pitch as the tea is poured. Similar to a Helmholtz resonator, a pourer would know when to stop as the musical note from the pouring tea changed. But the tea cup is open at the top so this effect is weak. Perhaps the sound clue instead comes from the sound that the poured tea makes as it impacts the surface of the cup of tea. When we hear a dripping tap, what we are actually hearing is the bursting of an air bubble just below the water surface. We would expect the sound of that to depend, for shallow containers such as the cup, on the depth of the tea. Consequently, if we poured the tea slowly and heard the drops of tea as they entered the cup, we may expect to gain an idea as to how full the cup is. As Boswell assumed a slight of hand (and of hygiene) on the part of Mrs Williams, it is probable that she did not pour the tea slowly enough for this to have been her primary route for knowing that the tea cup was nearly full. Perhaps Mrs Williams did not use sound after all, but relied on something else?

Bubble tea? There is a lot of physics to be found in tea as well as coffee. Take time out with your brew. What do you see, hear, feel?

There is one more clue that you could get about the relative fullness of a cup of tea using touch. If you very gently push against the cup with your finger, you can feel the resistance of the cup to movement. As the cup became heavier with tea, the resistance to its being pushed would increase (Newton’s second law). After pushing at the top of a few mugs with my finger, this seems to be possible. Unless the person watching the tea being poured was very observant, it is not clear that they would notice this. Can we gather anything of Boswell’s skills in observation? Throughout his book he describes people and social situations well and yet, he was unsure whether Williams had dipped her finger into the tea cup to know when it was full enough. I think we can probably gather that on this particular point, Boswell may well not have noticed had Williams been pushing gently at the tops of the cup to see how easily they moved. Although there is a risk of pushing the cup over, this does seem to be a very feasible way of filling a cup without using your sight.

There is however one last, maybe more boring possibility. Generally the speed at which liquid comes out of a tea pot is quite reproducible. Through the fact that she was pouring tea every evening for Johnson and his guests, it is quite possible that she knew how long to pour for each fixed angle of pour before the tea cup was properly filled. We’d still have to ask how she measured the time given that she wouldn’t have looked at her watch, but perhaps here we would have a clue from Galileo. He is said to have sung songs with a known rhythm in order to measure time (the use of the pendulum to measure time came later, partly as the result of Galileo’s work). The idea was that the rhythm is a surprisingly reproducible method for comparing time intervals. Maybe Mrs Williams sang to herself while pouring the tea for the guests.

What do you think? Perhaps you can think of another effect that could be used to determine when your tea cup is full, without any visual clues. Or maybe you just disagree with my deductions. Whatever your thoughts, do let me know in the comments below or on the various social media sites (FB, Twitter, Mastodon), I look forward to learning more and maybe, being able to pour my coffee with my eyes closed.

*Information on the London Coffee Houses can be found in the excellent “London Coffee Houses” by Bryant Lillywhite published in 1963

**The story of Mrs Williams, Dr Johnson, his tea drinking and the notes of Boswell is described in “Life of Samuel Johnson” by James Boswell

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


Notes on a cup

Ritzenhoff Mugs

Experimental apparatus

An opportunity for an experiment with a cup of coffee. Sadly though, for the experiment itself, it would probably help if the mug were empty, so there are two choices: Either grab a coffee and drink it so that you have the empty cup next to you, or get an empty cup and wait for your coffee until later. There is though, perhaps a third choice, get two cups, one with coffee in it, one empty, that sounds a much better idea.

Now, get a pen or pencil and start to tap the rim of the cup, make note of the sound that the cup makes as you tap at a point next to the handle, moving around to 45º from the handle, 90º from the handle etc. Perhaps compare the sound of different mugs but, on going around any particular cup, what do you hear? The note that you will hear when you tap the mug just next to the handle, or at 90º intervals from the handle should be lower than the note that you hear at 45º angles to the handle. Why is that?

wobbly bridge, Millennium Bridge

“Couple at St Pauls”, photograph © Artemisworks Photography. The ‘wobbly bridge’ is in the background.

Before answering that question, and to give you some time to think about it, it may be time to consider a (related) anecdote. Back at the turn of the millennium, a new ‘shard of light’ was built across the Thames. The Millennium Bridge takes pedestrians from the Tate Modern on the South bank towards St Paul’s on the North bank (or vice versa). It opened on 10th June 2000 and then closed, two days later, owing to problems that left it labelled the ‘wobbly bridge’. Along with many people, I had been taken in by the newspaper headlines of the time saying that we had built a terrible and wobbly bridge. It wasn’t until I was researching St Katherine’s Docks for the White Mulberries cafe-physics review that I found David Blockley’s book, ‘Bridges, the science and art of the world’s most inspiring structures’ and learned the true story. It turns out that the reason the bridge wobbled was because of a previously unknown phenomenon. Dubbed ‘synchronous lateral excitation’, it is a human crowd response to a platform swaying under their feet. Apparently in response to a swaying platform, people will widen their gait slightly to compensate for the wobble, only this acts to increase the sideways force on the platform itself and so can amplify the wobble. This bit had been known, what had not been appreciated was how the ‘wobble’ would grow if a crowd were present. The reason that the wobbly bridge surprised everyone was that never before had the critical mass of pedestrians been walking on a susceptible bridge. According to Blockley, 156 people walking along a particular section of the (original) Millennium Bridge did not cause a problem, but 166 walking in a group along the bridge caused the wobble to quickly become very appreciable.

hitting Zorro

Poor Zorro being experimented upon.

The solution, of course, was to damp the structure, to add shock absorbers and weights to the bridge so that the oscillation decreased. The cup is behaving similarly. Each time you tap the cup, you are exciting a standing wave around the rim of the mug, this is what is exciting the sound. This vibration has four points of maximum oscillation (called anti-nodes) and four stationary points (nodes) around the mug spaced at equal intervals. If the cup is hit so that the handle (which adds a relative weight to one side of the cup) is at a point of maximum oscillation, the mass that is being moved is greater than if there is a node at the handle so it does not have to move. This change of mass shifts the frequency of the oscillation and so the note is lower than when the handle is at a point of zero movement. For more information on the standing waves in your cup click here.

So it’s not just science in your coffee cup, a world of engineering is mirrored in your brew too.

Bridges – the science and art of the world’s most inspiring structures, by David Blockley was published by Oxford University Press in 2010, it is well worth a read as it is a very accessible and informative guide to bridges as well as being entertaining.

If you notice any engineering in your coffee cup, why not let me know via the comments section below or by contacting me via email.