Why is the bottom of the other shoreline cut off?

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My dad keeps asking me about a claim I made in my post about the curvature of the Earth: That looking at sea level across a 500 m wide part of a lake, we should be missing the bottom 20 cm of ships moored on the other side. So to shut him up, here are the calculations.

First: This is what we assume:

  • The Earth is round.
  • Its radius is 6.371 km.
  • We can actually see in a straight line and the light isn’t bent by weird things in the air or other processes.

This gives us this situation:

We are situated at position x right at water level. We look out tangentially along b, so there is a right angle between the radius of the Earth, a, and b. Side c of the triangle we are looking at consists of c2 (which is equal to a, the radius of the Earth) and c1, which we are looking for: The height below which we cannot see from position x.

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Calculating how much we cannot see at a given distance looking at water level.

We know a to be 6,371 km and b is 0.5 km. Now we just need to put everything into Pythagoras’ theorem, solve for c1 and we are done!

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Calculating how much we cannot see at a given distance looking at water level. Are you sure you really want to look at this?

Turns out we find c1 as 19.6 cm. Which is pretty close to the 20 cm I claimed last time, right? Everybody happy now? If you want to look at a more realistic and less simplified calculation – go do it yourself! :-)

Wave train

When you look at waves, do you sometimes notice the train of smaller waves being pushed forward by the “main” wave? That has always fascinated me. Kind of like in the center of this picture:

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wave train

When we were sailing in Ratzeburg earlier this year, one day there were hardly any wind waves on the lake, so putting a foot in the water from a sailing boat resulted in exactly the phenomenon that had puzzled me for so long.

In the movie below you see it “occurring naturally” and then afterwards “created” like in the picture above. I’m pretty sure it’s the “group velocity is only half the phase velocity” thing, with small waves passing forwards through the group and vanishing, only to be replaced by waves coming from the back of the group. Is this what is happening here? Anyone?

Amplifying waves

A pier full of sail boats, perpendicular to the prevailing wave direction, and voila: Instant demonstration of an amplified wave!

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Ratzeburg 2015

Look at how the masts show a transversal wave:

Maybe consider bringing this video as a backup when you are not sure whether your class will be able to do this visualization of transversal waves and calm back down again :-)

Waves on a slope

Earlier this year at Forscherfreizeit Ratzeburg – the summer camp at which Conny, Siska, Martin, a bunch of teenagers and myself spent a week sailing, exploring and playing with water – I spent a good amount of time staring at waves hitting the wooden boards that form the slip in the port. They create a nice slope with a very interesting structure, especially at the joints where the angle of the slope isn’t exactly the same.

Watch what happens when the wave approaches the shore (and focus on the left part of the picture, where it is clearer):

At first, it arrives pretty much as an ordinary wave.

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As it is running up the slip, you start seeing the structure of the boards below.

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As the wave becomes steeper and steeper, the front one is being slowed down more than the second one, because it is in shallower water (and we all know that the phase velocity of shallow water waves depends on the water depth, right?).Screen shot 2015-09-26 at 4.41.34 PM

Eventually, they form one steep wave and break.
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Watch the movie to see it happen:

For more waves on a slope, check out these posts (Norway, Hawaii).

Interference of waves

The advantage of wind-less days on a sailing holiday is that you can play with water as much as you like. For example in order to look at the interference of waves, which is what I did.

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Check out the movie below!

Sitting on the pier, playing, over time the interference pattern get more and more creative:

On buoyancy

This is an experiment that Martin brought to Ratzeburg and that he let me use on my blog: Using a beam balance to talk about buoyancy.

So at first we have two objects hanging on the beam balance, a heavy one with a large volume, and a lighter one with a smaller volume.


As we lower the beam balance towards the water, the large object starts floating! Whereas the other one does not.IMG_2526

And in fact, the small object sinks and the larger one keeps floating.IMG_2525

What a great experiment to talk about density and buoyancy!

Sink or swim – experiments using tin foil

A pet peeve of mine are books on handcrafts or experiments or any kind of activity that come with drawings instead of pictures, because I always suspect that it was easier to draw whatever the author wanted to show than to take a photo of it. Which, to me, suggests that it isn’t really all that easy to conduct the experiment or build the wicker basket or whatever it is you are attempting to do.

So here is an experiment that I had seen drawings of and that Martin and I went to try: on swimming and sinking.

Step 1: Take two identical pieces of tin foil.

Two identical pieces of tin foil

Step 2: Build a boat out of one of the pieces, and a ball out of the other one.

Two identical pieces of tin foil made into a boat and a ball.

Step 3: Place the boat and the ball on the water surface.

Step 4 to step 9: (And these are the steps that the nicely drawn instructions always omit) Watch the ball float on the surface. With growing annoyance, try to make the ball as compact as possible in order to make it sink.

Step 10: This is what we wanted to see after step 3 already. Even though the boat and the ball are made of identical pieces of tin foil and their mass is the same, the boat floats while the ball sinks.

A boat and a ball made of identical pieces of tin foil. Boat floats, ball sinks. Nice demonstration!

What do we learn from this? Always test experiments before using them as a demonstration, especially those that look extremely simple!

Creating eddies

I always love watching paddles in water, or ships in water, or ducks in water, or anything water, really, but on a wind-less day in Ratzeburg, Siska managed to create the beautiful eddies you see in the movie below, that survived well over half a minute out in the open lake! Beautiful.

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Playing with a model steam boat

I’ve shown this steam boat before, but I thought you might appreciate it in its full glory on the lake in Ratzeburg.


Plus this time round, in the movie below, you can really see the wave field that is caused by the rhythmic thrust of the boat.

Curvature of the Earth

In Ratzeburg, we very much enjoyed our daily early morning swims. One thing that is really nice to observe when you are swimming in a calm lake is how things vanish behind the horizon. Of course, you see the same effect when sitting on the beach or on a boat, but somehow it impresses me most when I’m either in the water or lying flat on my belly on a surf board.

This is what a buoy and boats look like that are fairly close. Even though your eyes (or the camera) are very close to the water level, you still see the water line on the buoy and the boats.IMG_2441

For buoys or boats a little further away it is very difficult to see the water line.IMG_2428

And in this picture you still see boats in the background, but clearly the lowest part above the water line is missing.IMG_2426

The boats in the picture above are maybe about half a kilometer away. Ignoring all effects of refraction of light and just looking at the geometry, the bottom 20 cm of the boats should be missing – which fits well with what we observe. Nice!