Waves 101 (Part 1)

Learning about surface ocean waves is a little like learning the guitar: the basics are relatively easy and in some cases all you need to know. But sometimes you’ve got to probe a little deeper, and that’s where it gets tricky. So consider this post the “Smoke on the Water” of waves knowledge. In the case of surface waves, the basics are what is called “linear wave theory.” We rely on a few assumptions, namely that the wave heights are small compared to their wavelength and depth and that the water is incompressible (not easily squeezed), inviscid (not very viscous), and irrotational (no spinning of water blobs). It turns out that for most purposes, these conditions hold and linear theory is an accurate estimate of how the waves will move.

But I’m getting ahead of myself. If you’ve never thought much about surface waves you might be asking, “what causes the waves?” or “what’s actually happening in a wave?”. Surface waves are caused by gravity acting on the interface between the heavy water of the ocean and light air of the atmosphere. That’s why they are sometimes called “surface gravity waves.” It’s sort of a “what goes up must come down” situation: an upward disturbance in the water surface wants to return to its original state, but it overcompensates and so now it’s below the rest of the water, kind of like a person on a trampoline. Waves are described by their wavelength (the distance between two crests, it’s inverse is called wavenumber), period (the time it takes for two crests to pass a fixed point, it’s inverse is called frequency), or amplitude (the height of the waves from the still water level to the crest, the crest-to-trough height is called the wave height). Another important description is the phase speed (literally the wavelength divided by the period, or the speed of a crest). Here’s a handy schematic:

I need to return to linear wave theory to answer the question of what’s happening in a surface wave. Without going too far down the rabbit hole, it turns out that using the assumptions I described in the first paragraph you can simplify the complicated equations that govern water motion down to a form that can be solved by hand. There are two important results: wave orbital motion and the dispersion relation. Orbital motion means that in waves, water actually moves in closed loops (“orbits”). If you’ve done much wading in the ocean you’ve felt the horizontal “pull” of waves as a crest approaches. In deep water, where we’ll be making measurements, this orbit is nearly circular, meaning there’s as much back-and-forth motion as up-and-down. This has very important ramifications for one of our instruments that I’ll talk about sometime later.

The dispersion relation is a little trickier to describe without math, but the punchline is this: if you know one of those wave characteristics I mentioned above (period, wavelength, or phase speed), you know all of them. You might be aware of this without even really knowing it. If you drop a stone in a lake, you have a basic idea of what speed the waves are going to propagate at. That’s because it’s all controlled by gravity. So if you had a tank of water on the moon, for example, things would be look much different. It’s called the “dispersion” relation because different length waves move at different speeds. So a group of waves will “disperse” or spread out, with long waves moving faster than short ones. This is different than most waves you know about (light, sound, waves on a string), which move at the same speed no matter the wavelength.

Alright let’s wrap this up for now. There’s only a little bit more you’ll need to know before I can talk about more of the fun stuff we’ll be doing on the cruise, but I’ll save that for a later date. If you made it this far, I’m very proud of you.

– Mike

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4 thoughts on “Waves 101 (Part 1)

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