A three coffee puzzle

Second shot coffee and cake
How would you describe the gravitational attraction between a Long black, a hot chocolate and a piece of cake?

Not a question of how many coffees are acceptable before lunch, but an astronomical conundrum with consequences for your cup.

It starts with gravity. Perhaps you remember that Newton came up with a set of equations describing the laws of gravity. You may even remember the essence of those equations, that the force between two masses is proportional to their product and inversely proportional to the square of the distance between them. If we wanted to phrase it mathematically, the force, F, is given by:

F = GMm/(r x r)

Where G is a constant and r the distance between the masses M and m.

Which is all very well, but suppose we have three masses, or four? M, m and M’, m” for example. If we happened to drop an apple (mass = m) between the moon (mass = M*) and the Earth (mass = M), how exactly, and where exactly, would it fall? How do we add an extra mass into the equation?

It is one of those problems that can seem far removed from your coffee cup, but in fact, the connection is quite close.

The Orion Nebula, M42, can just be seen with the naked eye in the sword of Orion, it is known as a birth place for stars. This image was obtained using the Hubble Space telescope. A separate dust cloud also in Orion was observed for 11 years as a possible host for planetary formation. Credit:
ESA, M. Robberto ( Space Telescope Science Institute/ESA) and the Hubble Space Telescope Orion Treasury Project Team

But although you may not often drop an apple somewhere between the Earth and the Moon, the question became relevant recently when astronomers observed a dusty disc, the sort of environment that is capable of planet formation, surrounding a three star system. The stars are found in the constellation Orion, which is visible in the evening at this time of year (autumn/winter) from the Northern Hemisphere.

Although these dusty discs are thought to be a host to planetary formation, astronomers have yet to observe any planets actually forming out of the dust. It is thought that in some cases, the gravitational perturbations caused by multiple stars at the heart of the dust clouds could lead to the formation of planets. And so the system in Orion, with three stars in the centre of the dust cloud was perfect to observe the effect of the three stars on the integrity of the disc. Over 11 years, the astronomers recorded the system and then included modelling into understanding how the planetary disc was breaking up. But of course, to do this, they would have needed to understand how the gravitational force is affected by having 3 or more interacting masses.

To solve the problem requires mathematical functions known as a “Bessel functions”. These functions were first described by the astronomer Friedrich Wilhelm Bessel in 1817 who used them for exactly this sort of problem. But they don’t just apply to describing the gravity between three or more objects. They can be used amongst other things to understand heat transfer, to model the microwave fields in a microwave oven and to understand vibrations on your coffee.

The beat of a drum or the resonance on our coffee – the mathematical description of the resonance patterns on coffee is shared with the mathematical description of the gravitational force between three or more objects.

Because when you see a series of concentric circles on the surface of your coffee where the table underneath the cup is vibrating, or when you see more complex patterns as you drive a take away cup over a rough table surface, these patterns can be described using exactly the same Bessel functions as would have been used to model the star system in Orion.

And so there is a direct link between the maths describing the planetary formation in a star system visible in our night sky and the patterns of your coffee cup. But if you want to drink your coffee while gazing at Orion, you may want to stick to decaff, or wake before dawn.

Ringing in the New Decade

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.

Resonating coffee.

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.

Good vibrations at Rosslyn, Mansion House

Coffee at Rosslyn, Mansion House, EC4N, coffee clock, base
Coffee time at Rosslyn, EC4N. Why is it that base 60 was used as a counting system in Mesopotamia? And why is it that the echoes of this are still seen in our clocks and the angles of a circle (unless you use the radian system) but not in our everyday counting system?

It’s always “coffee time” at Rosslyn apparently, at least according to the clock above the door. In front of you as you enter the cafe is the counter and, as you move down to collect your coffee (for take-away) the day’s edition of the Financial Times is stuck to the notice board where you wait. An interesting touch, somehow making a resonant connection with the City coffee houses of old such as Jonathan’s, just around the corner, where the stock market was originally located.

There are not many stools or tables in Rosslyn, which appears to be designed as more of a take away space. Nonetheless, we found a perch by the window overlooking the bench seats outside. It is a perfect place to watch the world go by. The massive junction of Poultry providing plenty to see.

Coffee is roasted by Modern Standard and there are bags of roasted coffee on sale (together with some of the mugs) at the other end of the counter to the FT. The occasional (welcome) plant reminds us that life is not just concrete, glass and cars/buses. Although it was sunny, it was not yet hot and so we had a soy hot chocolate and a long black, went back to take our seats and waited for the drinks to arrive.

The wooden spoon that came with the coffee was an interesting touch, reminding me of Barn the Spoon and his work in Hackney. While the clock got me thinking about our use of base 10 as a counting system and the older systems that used base 60.

coffee, hot chocolate, plant, mugs, wooden spoon.
A quiet moment with a coffee and a hot chocolate at Rosslyn. Notice the spoon.

Contemplating these things we noticed a strange effect in my coffee. Or rather, I noticed it and brought attention to it by taking repeated photographs of the coffee while tapping the bench just to try to capture what I was seeing: a resonance pattern on the coffee surface. At this point, your mind may connect to several different things. There’s the resonance effects involved in the Whispering Gallery in St Pauls close by to Rosslyn. There are the resonance patterns caused in bells, drums and violins and the relation between these, air movement and music. There’s the fact that these movements initially revealed the excellence of the table as a movement sensor: the ripples on the coffee revealing footsteps behind us rather like we detect earthquakes in the earth. (My later attempts at photographs were in that sense “faked” as I was tapping the table beside the cup to try to reproduce the effect so that it was visible on my camera).

Or there was the fact that this movement in the coffee cup is exactly the same phenomenon as something in our lab. But whereas in the cup it is an interesting, almost aesthetic feature, in the lab it can be a major pain to deal with.

The problem comes in that the coffee cup was in the middle of the bench. This had been an accident in terms of where we were seated but it had large effects. Because the bench table has its legs at each end, but nothing in the middle, the table itself acts as if it is a massive drum. And one of the more fundamental resonances of a drum has the maximum movement at the centre of the drum: the edges don’t move much but that bit in the middle oscillates wildly. In the coffee cup this manifests as a ripple pattern on the coffee surface, reflecting the street outside in slightly distorted fashion. In the lab this means that some of our instruments become incredibly difficult to use.

ripple pattern coffee Rosslyn
Can you see it? The ripple pattern caused by the coffee being on the drum of the table at Rosslyn. An interesting effect to watch in coffee but what if this sort of thing happens in a physics lab?

Consider for example the Atomic Force Microscope (AFM). This microscope is able to resolve the structure of films down to an almost atomic resolution. It does this by monitoring the resonance of a small silicon cantilever as it approaches the surface of the material being studied. Just for a moment, put a wooden sugar stirring stick (or a lollipop stick) on the edge of a table and ‘twang’ it. It vibrates just as the silicon cantilever does in the AFM. Then think, what if you put the stick in honey and ‘twanged’ it – or put a magnet on the end of it and ‘twanged’ it over a bit of iron, how would the oscillation change? This is what the AFM does but with the atomic forces that are present when you get very close to the surface of a sample. But the phrase “very close” is key. Typically, the cantilever will be nanometers from the surface of the sample and, as it is very sensitive to the forces at the surface of the sample, if that sample moves because the instrument is vibrating up and down on the floor, the image will be at best blurry and unusable and at worst, you are going to be damaging your cantilevers.

And so, it is important to ensure that the AFM is placed in a suitable area of the lab: not in the middle of a floor in a high level building because that will just act as a drum in exactly the same way as the coffee cup was being vibrated at Rosslyn. If you’re not fortunate enough to have the AFM in a basement lab, you could place the AFM (and other vibration sensitive instruments) at the corner of the room, so the vibration amplitude of the floor-drum is minimised. You could also try to place the instrument on concrete blocks to ‘damp’ the vibration. An extreme example of this sort of damping is the ‘quiet labs’ of Lancaster University just next to the M6 motorway. These labs have been designed to minimise vibration noise and the team there routinely achieve atomic level resolution with their atomic force microscopes.

The silence of an area next to the M6 contrasting with the noise of the City. The directions that contemplating a cup of coffee takes you are always surprising.

Rosslyn is at 78 Queen Victoria Street, EC4N 4SJ

In the loop at Coffee is my cup of tea

exterior coffee is my cup of tea, cimcot, coffee Hackney

Coffee is my cup of tea on Dalston Lane. The colour of the exterior matches the crockery used inside.

There is a lot of truth in the name of this café. “Coffee is my cup of tea” in Hackney is a lovely retreat, a place where you can take time to enjoy whatever drink is your cup of tea. Walking through the door, you are presented with a few wooden tables and a cocktail menu on the wall. A breath of calm on an otherwise busy road. Together with the bench just outside, there is plenty of seating inside. There’s even a long table along the window where you can sit if you would like to enjoy your coffee while gazing at the passers-by. There were the usual range of coffees on offer along with fresh juices, other drinks together with a range of food. When we went in the late afternoon, there didn’t seem to be many cakes on offer but maybe we were just unlucky. Coffee only this time. The coffee is roasted by Assembly and there is of course tap water available at the end of the bar.

Facing the bar, glued to the wall, were a circle of stiletto shoes. Forming what seemed to  be a “shoe star”, they were one of a number of art works around the shop. The café is also quite spacious, the window at the front providing plenty of light and contributing to the relaxed space. When my long black arrived, the light coming in from the windows produced great interference patterns on the bubbles of the coffee, an irresistible piece of coffee physics. The cocktail menu provided quite a distraction, again making the point that it was a shame we visited on an afternoon: an evening of coffee and cocktails would make a lovely night. However, a sunny afternoon was a great time to sip and enjoy a long black. While the long black started off very fruity, the taste changed (matured?) as the temperature of the coffee decreased. In the background to this all though, something so subtle as to be almost un-noticeable caught my attention. Completely surrounding the window was a very thin piece of copper wire. Were there tiny little lights on it to make the café more attractive (romantic even?) in the evening? I couldn’t see any. From our table, it seemed as if it was just a thin, closed loop of copper wire forming a loop around the window.

coffee cimcot

Fantastic interference patterns on the bubbles of the coffee at Coffee is my cup of tea

Such a loop could be used as a radio antenna, a “loop antenna”. Indeed, when Heinrich Hertz (1857-1894) first discovered radio waves in 1887-8, he used a (gapped) loop antenna as the receiver. Hertz had been trying to test James Clerk Maxwell’s theory that visible light was part of a much broader spectrum of electromagnetic waves, particularly, that there should exist very low frequency waves far beyond the visible region of the spectrum, waves that we now know as “radio waves”. Radio, TV, wifi, all things that seem so obvious now but were really only predicted and discovered relatively recently. Working in his laboratory in Karlsruhe, Hertz set up a radio generator which consisted of two brass balls that were charged until a spark flashed between them. Sitting a few metres away, a gapped loop of wire, the ‘loop antenna’ suddenly showed a spark over the gap. The spark that Hertz had generated in one part of the room had been mysteriously transmitted, as if through an aether, to be picked up by the antenna a few metres away. Clearly it was consistent with Maxwell’s predictions. The electric spark had generated a low frequency electromagnetic wave that had been picked up with the loop antenna. With further experiments, Hertz showed that this wave was indeed reflected and refracted in the same way as ordinary, visible, light and even determined its wavelength (which for Hertz’s experiment was about 66cm)¹.

loop antenna at cimcot, Dalston Lane

It is probably easiest if you visit the cafe but look very very closely at the frame of the window. There is a copper wire surrounding it.

Although Hertz did not immediately see any practical application of his result (beyond the fact that it was a test of Maxwell’s theory of light), ‘radio’ soon started to be developed. Marconi and others worked with wavelengths of 200-600 m to transmit radio waves across the Atlantic Ocean¹. As amateur radio enthusiasts got hold of radio sets in the 1920s they started working with wavelengths that were initially considered impractical for applications (much shorter than the hundreds of metres used by Marconi). These enthusiasts soon realised that they could communicate with other enthusiasts in distant countries through the reflection of the radio waves off of the (until then unknown) ionosphere¹. Gradually our understanding of radio waves and antennae design developed, leading to further, unexpected applications. Depending on the design of the antennae, radio waves (and microwaves, which have a slightly shorter wavelength of the order of 0.1-100cm*) could be made to be directional. So antennae could be made that transmitted waves only in set directions (or conversely could detect the direction from which radio/microwaves originated). This understanding of antennae design would lead to advances in Radar technology.

Which brings us back to the loop antenna at Coffee is my cup of tea. Loop antennae are grouped into two types, “small” and “large”. It is fair to say that it is a large window at Coffee is my cup of tea and so the loop antenna there would fit into the “large” category. These antenna are ‘resonant’ (meaning that they respond most) to wavelengths equal in length to the circumference of the antenna. From memory, I’d guess that the window was roughly 2m high and 3 m across, meaning it had a circumference of 10 m. We can calculate the frequency of the radio waves that would be resonant with this by using the fact that the frequency (f) is just the speed of light (c) divided by the wavelength (λ) (ie. f=c/λ). The speed of light is 3×10^8 m/s, so the frequency would be (3×10^8)/10 = 30 MHz. There are two last things to notice about this result. First, the name of Hertz lives on in the unit of frequency (Hz). Secondly, the loop antenna around the window at Coffee is my cup of tea is resonant with approximately the frequency of Citizens Band radio (CB radio operates at ~27 MHz). Which may make us question once more what this loop of wire is doing at this friendly little café on Dalston Lane.

Coffee is my cup of tea can be found at 103B Dalston Lane, E8 1NH

¹Britain’s shield radar and the defeat of the Luftwaffe, David Zimmerman, Amberley publishers (2001, 2010)

*Technically Hertz discovered microwaves rather than radio waves. However, given neither were named at the time and they are both of longer wavelength than visible light, it is perhaps too pedantic a point.


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.