Estimate the speed of the waves of the coffee

[vinny380]

Question: When you walk with a cup of coffee (diameter 8cm) at just the right pace of about 1 step per second, the coffee sloshes more and more until eventually it starts to spill over the top. Estimate the speed of the waves of the coffee.

Relevant Formulae:

f= 1/T
v= f * wavelength

Solution:
I am not really sure if I am doing this right... but I said since the cup has a diameter of 8cm, the wavelength of the coffee is also 8cm (not sure it this is true)...
If the above is true, all i would need to do is find the frequency. Though I am kind of confused when it says that I am walking at a pace of 1 step per second. How can a frequency be found from this?

After finding these two quanities, and multiplying, a velocity could be found...

 

[Kurdt]

Well frequency is simply 1/T as you have stated. You can assume that the T is the time between the steps. I'd say the wavelength was double the mug diameter aswell.

 

[vinny380]

why double?

 

[Kurdt]

If you think of coffee sloshing back and forth in a cupyou will notice as one side moves up the other will naturally go down. One of these is a node and the other an anti-node which means the wavelength of the moving coffee would actually be 4 times the coffee cup diameter since a node and anti-node defines 1/4 of a wavelength. Perhaps I'm going a bit over the top with the assumptions. Choose whichever you see fit as long as you can justify the choice.

 

[OlderDan]

If you think of coffee sloshing back and forth in a cupyou will notice as one side moves up the other will naturally go down. One of these is a node and the other an anti-node which means the wavelength of the moving coffee would actually be 4 times the coffee cup diameter since a node and anti-node defines 1/4 of a wavelength. Perhaps I'm going a bit over the top with the assumptions. Choose whichever you see fit as long as you can justify the choice.

I think this is close, but since both sides move up and down I think they are both antinodes with a node in the middle of the cup.

 

[BobG]

I think Kurdt's right on the wavelength - it's four times the cup diameter. The sloshing is from the braking action of your front foot hitting the ground and the push from your back foot. All the coffee in the cup is exposed to the same forces, so there's no reason the coffee would slosh two different directions simultaneously.

Plus I tried this out and the coffee sure looked like it was all sloshing the same direction ... at least until I ran into the edge of the door frame and dropped the cup.

 

[OlderDan]

I think Kurdt's right on the wavelength - it's four times the cup diameter. The sloshing is from the braking action of your front foot hitting the ground and the push from your back foot. All the coffee in the cup is exposed to the same forces, so there's no reason the coffee would slosh two different directions simultaneously.

Plus I tried this out and the coffee sure looked like it was all sloshing the same direction ... at least until I ran into the edge of the door frame and dropped the cup.

But sloshing in the same direction just means the liquid on one side is up when the liquid on the other side is down and vice versa. This is exactly what happens with two adjacent antinodes. They are always 180º out of phase. I'm still favoring the diameter being half a wavelength.

Of course the real solution to this problem is more likely a Bessel function for the radial part with a periodic azimuthal function like the vibrations of a circular membrane. I can't find any place that solves the equation using a free perimeter, but there are numerous references for the clamped perimeter. This site shows several animations for the clamped circular membrane.

http://www.arts.uAlberta.ca/~michaelf/Acoustics-demos/Vibrating%20Circular%20Membranes.htm

My guess is the sloshing cup is very much like the m=1, n=1 mode figure, but with the cup radius being about half the membrane radius so the peaks form at the cup perimter. I know I have also seen modes similar to the m=0, n=2 mode in my cup, and even had drops separate from the surface at the middle when the amplitude gets big enough.

 

[Kurdt]

Looking at that animation you're probably right with the centre of the liquid serving as a node. Of course the fact that both sides are moving means they have to be anti-nodes anyway. I've been terribly off lately since most of my time has been devoted to studying languages for jobs in europe.