# The Actual Mathematics of Popping Champagne Corks

I have a bottle of Champagne-like sparkling white wine
and I'm here at the UCL Department of Engineering
because in there they have both a bubble physicist
and a high-speed camera which we're going to use
to find out the actual mathematics
of popping Champagne corks.
Alright, so I'm here with Helen Czerski, bubble physicist and explosives expert.
Background in explosives? Helen: Yes, I moved from explosives to bubbles.
Matt: Wow Helen: Makes perfect sense!
Matt: So this is the intersection of your career, in a bottle!
Helen: Basically, if only there'd been more of it along the way, life would have been a lot nicer.
Matt: Now, we were talking a little while ago about some media coverage which you were very upset about.
Helen:Well I get sent it, whenever anyone writes anything about Champagne
all my friends go "oh, Helen does bubbles, she wants to read it" so it all arrives in my inbox
and there was one piece recently about someone who was talking about
the frequency of Champagne, the pop that you hear Matt: Right
Helen: Because there's two things everyone knows about Champagne
One is that when the cork goes it makes a loud pop, and the other thing is the bubbles in the thing.
So this is the bit no-one thinks about, which is the pop, and they were talking about the frequency of the pop.
and the relationship of that to the temperature,
and the whole thing set off a few alarm bells.
Matt: Because I saw the same media stuff, people sent it to me going "oh, there's some maths"
and it was saying the pop would ring
at 8 to 12 thousand hertz.
Helen: Which is off the end of the piano. If you imagine a grand piano
and you imagine a few extra keys because it could get... Matt: It's out there somewhere
Helen: it's out there somewhere, and that is definitely not the sound of a boop!
Matt: It's not the doomph, exactly, exactly, right.
So you'd actually been doing research into... like this is your slow motion footage?
Helen: Yeah, so the other reason the whole thing rang alarm bells is that just the week before I had been
in my lab with a bottle of Champagne and a high-speed camera and this is what I was doing
Matt: You don't have to justify it, I accept this is your normal job, It's ok!
Helen: No, no, no, no, my head of department might be watching
Matt: Oh, good point, this is important physics research
So what this is, there's a Champagne bottle down there,
and this is obviously the cork just on its way out, and this is being filmed at 6000 frames a second.
Matt: Wow
Helen: So six of these frames are one millisecond.
Matt: That's insane Helen: That's a lot going on
So this is what the video looks like, so just at the end it's just about to go,
see its beautiful pop, you get a bit of a whoosh, and I expected to see that,
Matt: Right Helen: and what I didn't expect was this bit here
Matt: This Helen: This is the fog forming, and this is sound
so I'd actually seen the sound when that video came along
and you can watch the whole thing oscillate
Matt: So that up-and-down is the sound wave... Helen: Yeah Matt...resonating.
Helen: That's what you hear when you hear the pop.
Matt: So, Helen actually promised me some working out, from first principles, what that frequency should be
using my all-time friend, Helmholtz.
Helen: Yes, because it's not physics unless it's got maths in it, so you're not
interested in physics unless it's got maths in it. Matt: Correct! Frankly I wouldn't be here.
The Champagne is good, but let's do some working out.
Helen: So what's going on here, is that there is
the type of oscillation, let's draw out the type of oscillation here first
which is that here's the top of the bottle
going down like that, shoulders going out,
and up to about here, this is full with Champagne.
And then you open up the top and the cork is.... Matt: Long gone this way...
Helen: off, gone that way, so then you've got this
long cavity here with gas in it and it's just, you know,
it's high pressure, lots of gas in here, so it whooshes out
but then the pressure outside is higher because all the gas is gone, so it pushes back in.
Matt: Right. Helen: So you've got this oscillation which is the gas... Matt: in, out, in, out
Helen: until it gets back to the inside being the same as the outside, so that's what you can see.
And that's the main oscillation,
there are a few other things that can happen
you can have it sort of... it can slosh a bit basically so you can get some higher modes.
Matt: But that main oscillation, the gas up and down, that's subtly a bit different to our normal
standing wave-in-a-pipe scenario Helen: Yeah
Matt: which is why we're not just using wavelength and length. Helen: That's right.
Because it's to do with the compression of the gas
and it being pushed out on the compression and being pushed down.
So it's more to do with that, yeah.
So there is an equation you can actually write down
for this dominant mode, an approximation, which is that the frequency of oscillation
is equal to the speed of sound in air,
divided by 2 pi, Matt: So you're using "c" for...
Helen: Oh, well... Matt: That's fine, that's fine! No, no, no, it's good!
Helen: I'll put "air" down there! Matt: It's great
Helen: and then up here we've got "A", which is the
surface area of the thing
divided by the volume and then this term here
which is... Matt: a delta, Helen: It is a delta, so this is...
So this equation is sort of designed for a
an oscillator that looks a bit more like that. Matt: Oh!
So there's a length "l"
Matt: So not a cylinder, just a volume of gas.
Helen: So the "v" is the volume, whatever you've got Matt: The reservoir
Helen: The idea here is that if you know the internal radius
and the total volume here
and this little thing is a correction factor just because
this is all a nice tube, but it hits the air and the air is a bit messy
so you need a little correction factor in there. Matt: so this is just correcting for the messy interface
Helen: And it's a very small number.
Matt: And it's because of this weird shape, is that why they haven't cancelled down the area with the volume?
Helen: Yes, that's right.
So there we go. So what you can see here is that
the most important thing here is the shape of the cavity,
how big the entrance is and how big the insides are
and then the speed of sound in air matters a little bit
and that's the only place where a temperature difference could affect this whole thing.
Matt: Because everything else is not temperature dependent.
Helen: Everything else is the same, but the speed of sound in air does vary a tiny, tiny bit with temperature
Matt: And this will change temperature Helen: Yes Matt: That makes sense.
Can we... ooh! That's a pi! Helen: Oh, yeah, should I have written tau?
Matt: No, no! I'm pro-pi all the way! My goodness!
I was just thinking, so if you knew all these values, and you measured the sound,
you could calculate pi. Helen: You could calculate pi,
it would be one of the more... one of your more...
Matt: Don't, don't, don't phrase it like a challenge. I've just noted for later.
Can we... Lets, let's, let's move on quickly
and can we put some numbers into this to see. Helen: Yes
Matt: Okay, to get some actual numbers to plug into this
I'm going to do the most careful Champagne opening in the world
right next to... I'm not very good at this...
Helen: Right next to my computer and all my kit over there.
So, be good. So it actually turns that if you shake the
bottle, you lower the pressure so that is not
Matt: That sounds like the beginning of a YouTube prank
Helen: No, it is true, but the problem is... Matt: and if you look right into it...
[POP]
Helen: So that is why I've got a towel over here Matt: [giggling] You weren't kidding
Helen: No, I told you it was violent Matt: Oh my goodness, that is violent!
Well, I've ruined our, um... Helen: Well you won''t be able to measure how much was in it now!
Matt: I've ruined our measurement of the height. Sorry.
Helen: You've ruined quite a lot of it!
Helen: It was loud, though, wasn't it? Matt: I'd like to apologise...
by giving you this bottle of Champagne
I've got it all over the maths... Helen: Oh, God...
Matt: Right, so anyway, now that we're...
Helen: I'm just looking to see if there's a mark on the ceiling
Matt: There's someone three floors up going "where did this cork come from, I dunno"
Helen: So the point is, so this is... Is this chilled?
Matt: You're such a professional! Alright, you're like "Right, I'm carrying on".
Helen: Did this come out of the fridge? Matt: Eh, no
Helen: Where's this been, it's been sitting here, hasn't it? Matt: No, it's been sitting in your office
Helen: So here is the reason, it turns out that warm Champagne,
and warm counts as 22 degrees, is extremely dangerous because is does that, and actually I've seen bottles
that are that warm go with a much bigger fountain.
Matt: Really? That was my cat-like reflexes.
Helen: Well, once you take... If you're trying this at home
do not under any circumstances take the
cage off the top and then
do anything other than stand back. Matt: Stand back.
Helen: Keep your pets out of the way, because that cork...
Matt: I don't... I wasn't even... Wow. Helen: 10 metres per second around then
depending on how fast it is. So that will take your eye out.
Matt: So if people filmed that happening at home
with a normal camera there'd be a cork, and then Helen: No cork.
Matt: That'd be so good. Helen: There'd be a blur.
So even on here, even on my images here Matt: How many frames do you get before it's...
Helen: It's maybe four or five.
But... where am I going... oh we don't want the sound
we want this one, the one with all the stuff.
So even here it's blurred. Even with an inter-frame time of
So... Matt: Oh! Helen: Let's go back here. So even there
the image is blurred as it accelerates. Matt: You're right, it's not even...
That's insane.
Helen: So it's going with a hell of a bang, basically.
Matt: That's amazing.
Helen: And the reason the temperature makes a difference is that when the
wine is warmer the equilibrium state is more gas,
more carbon dioxide in the gas phase
and so there's more gas up here so the pressure is really really high.
Matt: And you're arguing that if I'd shaken that it would have dissolved some of the gas.
Helen: A tiny bit, yep, but would have happened is the gas would have distributed itself through the bottle
so then when you release the pressure the gas is escaping
and brings the liquid with it. Matt: Got it.
Matt: Now we're here, let's get some values
and we'll try to compensate... Helen: For the fact that we've just...
Matt: I'm actually just going to squeeze the towel back into the bottle, so it'll be fine.
546, 550-ish.
Helen: Okay, 546 hertz, which is pretty close
Matt: Wow! That's pretty close to 600 hertz.
Helen: I know, it's an approximation
and there will be harmonics that give you the shape of the sound
but they'll all be at multiples of that, so that'll be about 500
they'll be around 1000, 1500, so they're very much lower than ten kilohertz
First of all, the useful thing should you wish to be rude at parties
is that not only, if you don't want to weigh the bottle as you arrive
you can record the pop,
measure the bottle and see whether it was full or not. Matt: That's good.
Helen: Well it might be good, depending on who your friends are
and whether you still want them to be your friends afterwards.
Matt: But it means if you don't know any one of these... Helen: Yeah...
Matt: you can work it out once you've got the frequency.
Helen: Yeah, as long as you can measure the inner radius in there
Matt: Well you can certainly do that at a party
Helen: So the lovely thing is you get the number, it works, you can see it, and... yeah!
Matt: I can't tell you how happy I am that your
actual experimental result with the high-speed camera
matches the Helmholtz approximation
of the gas resonating, well oscillating, in the tube.
But the other thing that the press was covering with this press release "research" that we're...
I mean, to be fair it was commissioned by a cork association so it must be entirely impartial
Helen: Yeah, I'm sure that it would pass all unbiased...
Matt: But, it said that the frequency changes depending on the temperature.
Helen, Yeah, so now we can actually make an estimation of that quite straightforwardly here
because you can see that the speed of sound comes into this here,
the speed of sound at, let's say, 5°C
is around 334 metres a second Matt: Right
Helen: The speed of sound at around 20°C is around 343 metres a second
Matt: Which was all we used here Helen: Now if you're drinking Champagne outside that range
something has gone wrong.
So we've got a difference of maybe a few percent... Matt: That's a percent!
Helen: And here's the thing
They're saying that the speed of sound, that the temperature would make a difference to the frequency,
is correct if you have exactly the same volume in the bottle every time
However, implying that you could hear the difference is not correct because
the way that the musical scale works is it's logarithmic.
So a single octave is a doubling in frequency, so one percent... nothing.
Matt: Absolutely nothing, but, to be absolutely certain, Helen has, HAS, we're going to use your...
I mean I know you've done some legitimate research on it,
but we're going to take down, we've got more bottles of Champagne
and we're going to see if the temperature changes
because in the press they said put a bottle in a bucket
of ice for forty minutes to get the optimal noise. Helen: Yes
Matt: We've done that! And we're going to see if it sounds any different.
So now we're down in Helen's lab
several floors below the streets of London, I believe,
and in eleven seconds we're going to take
the refrigerated, well, refrigerated and then put in a bucket of ice Champagne out.
So this has been there for exactly...
...as of now... [Phone alarm sounds]
...stop...
40 minutes. That's coming out,
and I'm going to put it here
so Helen's rigged up a microphone which then just goes into the...
Helen: That goes into a pre-amp, and then into... this is an oscilloscope
It looks like a computer, but it actually is an oscilloscope.
Matt: Of course, of course. Helen: Obviously.
Matt: Right so, now, the trick is not to let this go off too early,
so if I...
Okay, here go. Ready? Helen: Yes.
Matt: That is the least tidy way anyone's ever opened...
Helen: You would never get a job in a posh restaurant with that kind of technique
Matt: There's a long list of reasons why I wouldn't.
Okay, are you ready? Helen: Yeah.
Matt: So I'm going to release the cork, and you're going to start. Okay ready?
Okay. It's not moving
Helen: Because it's cold, you're going to have to give it some help.
Matt: Right, so I'm going to start. Ready? Helen: Yeah.
Matt: I'm going to loosen it first... Helen: As soon as you feel anything. Matt: It's going , it's going, it's going.
Ready? Helen: Yeah. Matt: Do you want to count me down and I'll let go?
Helen: Three, two, one
[POP]
Yes! Matt: We got it! Helen: Right at the end of the recording!
Matt: And, this is going everywhere, that's fine, that's fine, that's fine.
Helen: That's what the towel's for. Matt: That's why we have a towel. Shall I just mop this up with your
Matt: Okay, okay, so that was the, allegedly, freezing cold one. Should we get a temperature reading on that?
Helen: Thermometer there, yeah.
Matt: I have just put a lab thermometer in a bottle of wine so it is no longer safe to drink.
We didn't think that through, did we?
Helen: I swear it's seen worse. Anyway, were you planning on drinking it?
Matt: I was not planning on drinking it.
Helen: 12:21, and lets just have a look at the shape of it.
[POP]
Helen: Yes! Got it. Matt: Got it, right in the middle!
Helen: So, don't be looking over the top of it! Matt: A lot of pressure, that's like eight atmospheres
Helen: Eight atmospheres of pressure, so just treat it with respect.
As you would any other loaded weapon.
Matt: For hilarity purposes, if the cork can ricochet and hit the camera, that would be...
Helen: Perfect. Are you listening, Champagne bottle? Matt: Let's do it, let's do it.
Helen: There's two reasons that posh Champagne people don't do this.
Only one of them is that the Champagne doesn't taste as good.
[POP]
[unintelligble]
Helen: So it's topped-out there, and you can see there's this beautiful oscillation here, dying away
Matt: That is amazing Helen: That is what we're after
Matt: And that's our frequency Helen: That's our frequency
Matt: Okay, so this worked?
Helen: Beautiful little signal there, it looks different to the warmer temperature, so we've got what we need.
We can go and have a look. Matt: Excellent, back upstairs.
Data time. So this is the waveform of the 3.5° pop.
You can see where the cork came blasting out,
and this little bit here is where the gas coming out of the bottle has hit the microphone
but pretty quickly it settles down to this nice oscillation pattern, that's the frequency we're after.
You see pretty much the same thing on the 8° pop
and the 19° pop looks very similar as well.
Now we want to work out what the main frequencies are for these pops.
The lazy way would just be to count the number of peaks and divide it by the amount of time.
What Helen has done is a Fourier analysis,
so she's taken that waveform and worked out
how much of the power is at different frequencies
and you can see for the 3.5° pop
there is a major spike at 648 hertz.
If we compare it to the 8° pop
it's pretty much the same, if anything it's slightly lower
that's 637 hertz,
and that's within the error that we would expect
from that fact that all the bottles have slightly different amounts of Champagne in them,
and so the length is slightly different which is why that's moved ever so slightly down.
However the 19° Champagne you can see is a little bit higher now at 695 hertz.
It's pretty much exactly what
was predicted by our working out.
However the press coverage was looking at things up at 8-12,000 hertz,
this only goes to 2,000 hertz.
So what we can do is take the 3.5° one
and scale it down and then
extend the plot into the much higher frequencies
and you can see there is definitely happening over there on the left,
and there is nothing happening over there on the right.
There's no meaningful data,
it's just noise at those frequencies.
If we add on the 8° plot, there's still nothing over there,
and if we put on the third one, the 19° one,
you can see now with three different plots
out there it's a complete nothing burger.
If anything, the 19° one
has slightly more movement, which is the opposite
of what was stated in the media.
So there you are. The experimental data perfectly matches up
with what we predicted with the maths.
So I think we can all agree that Champagne does not ring at 8 to 12 thousand hertz,
absolutely not, and cooling it down doesn't give you a higher frequency,
and the maths works.
Helen: Yup, and it's a nice simple system and no-one ever thinks about the pop
but once you know it's there, you'll be listening out on new year's eve, seeing what you can learn.
Matt: I have to admit, I did try, because I've got a spectral analysing app on my phone.
I had a go, and I could see a peak at about 600 hertz.
So, I'll put a link in the description below
to a previous video where you can go and get one of those apps
and give it a try, do look into it.
And you've got a book! Helen: Oh, yeah Matt: Which you never bring up!
Helen's got an amazing book about teacups,
and so I'll put a link Helen: It's all about the physics of drinks.
Matt: Basically, yeah! I've got a link to that in the description below.
Thank you very much!