# Phase and group velocities in deep and shallow water

When Tor came to visit me in GFI’s basement lab a couple of days ago, he told me about an experiment he had seen in Gothenburg in the seventies. So Elin and I obviously had to recreate it on the spot. Therefore today, we are comparing phase- and group velocities in deep and shallow water!

Waves are excited by means of an oscillating, hand-helt beer can, curtesy of the beer brewing club at GFI. The experiments are filmed and wave lengths and phase velocities are determined from the videos, which is a lot easier than measuring them directly while the experiment is being run.

# Shallow water waves

For shallow water, we are using a water depth of 10 cm. Waves are very easy to see and phase velocities are equally easy to measure.

There is another experiment on (standing) shallow water waves being run at GFI the year before students attend GEOF213, which I described back in 2013.

# Deep water waves

For deep water waves, we use a water depth of 42.5 cm (the exact number only matters when the tank filling is also used to fiddle with the dead water experiment, as I had been when the idea for this experiment came up).

Typical wave lengths that are easy to do are between 10 and 25 cm (wave lengths obviously have to be short enough that the water is still “deep”, i.e. H>>wave length) — Elin’s instruction to me for the kind of waves she wanted was “Allegro!” :-D Elin, you are really the coolest and most fun person to play with tanks with!

In deep water, we now have the added difficulty that the phase speed is twice as fast as the group speed. This makes observing the whole thing a lot more difficult. Also amplitudes are a lot smaller now, since the tank was so full and we wanted to keep the water inside…

Here is t0 — Elin has just dipped the beer can into the water for the first time

t1 — can you see the wave signal has propagated up to where the red arrow is pointing to?

t2 — the signal has reached my thumb at the left edge of the picture.

From timing this, we can calculate the group speed. We can also measure the wave length on the video and then calculate a theoretical phase speed from that. For the experiments Elin and I did, the results were pretty good, as in phase speed was usually about twice as fast as group speed. And I am curious to hear how well this works out when the students run the experiment!

# Sunday #wavewatching. Shallow water waves, deep water waves, #wakewatching, reflections, refractions, and more.

This is the blogpost I wanted to write today before I got distracted by all the adventures on MS Stadt Kiel with sightseeing and swing dancing and all the good stuff that wasn’t wave watching. I had even uploaded all the pictures of this morning’s wave watching walk on my blog already, but then things just got too exciting… But here we go now!

Here we see a really nice example of shallow water waves: The wavelength is a lot longer than the water depth and the shape of the waves looks very different from the almost sinusoidal waves in deep water: The troughs are very long and the crests short and steep.

And here they show the influence of water depth on phase velocity: Where the water gets shallower and eventually the land falls dry, the waves get slowed down. This means the part of the wave crest that is still in deeper water is faster, overtaking the shallower part, and thus the waves get bent around, towards the shallower areas in the water.

For comparison here the shape of waves in deeper water. Do you see how they are much more sinus-shaped?

Above we also see fairly well how gusts of wind change the water’s surface roughness: where the water surface looks darker, it is covered in small capillary waves caused by gusts of wind.

Btw, if you were wondering where the longer wave length waves came from: They are the wake of a boat accompanying the training session of some race kayakers.

When those waves hit the sea wall, we see nice reflections and a checkerboard pattern!

Btw, I find it super eye opening how the structures that, to me, absolutely dominate the picture above are not picked up as dominant in my comic app (see below). Of course the structures of the power plant or sea wall are more prominent, but the criss cross of the reflected waves is hardly visible. Maybe that’s how little it is visible to most people even in the photos? On my Instagram poll on whether comic-y-fying pictures helps to show what I am talking about, 60% of the responses were that yes, comics make it easier to see what I mean. Not quite sure yet what I will do with that information moving forward…

Something else really curious today, btw: The way the sea floor looked! Usually there are lots of ripples here, mostly parallel to the sea wall. And today these weird patterns of darker (finer?) sediment. I think I need to observe the sea floor more consciously to figure out what’s going on there!

And now the race kayak and accompanying boat are coming back. I love how you see their wake and then that bird’s wake in the picture below!

And then, finally, some larger waves, again the wake of some ship that I didn’t pay attention to.

And sometimes, the reflections are just suuuuuper difficult to see when you don’t know what you are looking for. Do you see them in the picture below?