Some Water Waves

Curiosity Unbound.

Waves come in all shapes and sizes. Mister Fourier showed, via the Fourier transform, that by using a sum of enough sine waves you can create any arbitrary (periodic) function. That (plus linear systems theory) is about where everyone seems to leave it.

Linear systems theory is where a system is anything that turns inputs into outputs, and where being linear means the output of a sum is the sum of the outputs. So you can separate the inputs, get an output for each one, and the output of the sum of the inputs is just the some of their respective outputs. So you can analyse an input into simple parts that scale and shift and add up to make the complex input, and then consider the system as just what it does to any single simple input, and then figure out the complex output by scaling up and adding up the simple outputs one for each scaled-and-shifted simple input. Magic! It turns out the arbitrary complexity of any (linear) system whatsoever can be replaced by a simple Fourier transform (a sine-wave analysis) with a matrix multiply and you're done. Like that.

But the truth is that water waves are only locally periodic, if at all, and the linear systems theory of waves doesn't actually apply to water waves, because water isn't even a linear medium. (The long waves travel way faster than the short waves, so the other side of the ocean is NOT a simple sum of outputs over there for the different inputs initiated here. The inputs scale and shift differently, they spread out in frequency, and the different frequencies travel at different speeds: it's a mess.)

So what's actually going on with water waves?

Here is a place to start: a typology of water waves that I noticed at a trip to the beach, with some questions that are a mystery to me, if not to you. Science should have figured all this out already, but there is current research on wave packets and stuff mentioned here coming out even now in 2020, so maybe let's try to provide a little more insight than to just reference the Navier-Stokes equation and then withdraw in arrogant silence. So my challenge to you is, can you explain these facts, not just with physics equations but also with underlying intuitions that derive them?

Here:

1. Splash cylinder. Like a crown of milk with point droplets all around, as made famous in early high speed photography, the cylinder radius depends on the energy of the splash and the amount of material disturbed. What predicts the height? The wall thickness? The number and size of droplets?

2. Splash wave response. After the cylinder collapses and a regular outbound wave within the water body begins to spread, the outermost crest leads the way for a limited distance, then collapses, only to be succeeded by the one after which extends the effected frontier a bit further, before it too collapses.
   1. What causes the collapse?
   2. How come the effected frontier is NOT always defined by a crest?
   3. How come tidal waves seem to always be preceded by a trough?

3. Horizontal sheet flow.
   1. On a beach, apparently formed after the turbulent collapse of a breaking wave crest at the foot of the sand, a horizontally moving sheet passes inland above a return flow sheet layer or even just over the beach sand surface itself. The inbound front edge is approximately a cliff, bubbly and turbulent, while the body of the sheet is a settling/clarifying bubble zone that can be relatively quite flat and featureless even at high velocity. Outbound big ones could knock you down and drag you out to sea. Watch out!

   2. Horizontal sheet flow would also include the pyroclastic lava flows famous for forming much of the geology of Eastern Washington State, which may have had horizontal velocities of 200m/s (450mph), and depths of 10 meters.

   3. Tidal waves seem to be HSF's after a giant outflow event. Why always a preceding outflow?

   Maybe these are just planar instances of a squirt event, just like when you squirt caulk out of a caulking gun with a big squirt, force applied moves the material in that direction constrained by a channel, except that the squirt force is a momentary pile of water pushing down and the channel constraint is the ground below.

4. Surface-following wave shapes, where a sheet flows over the features of an underwater surface, and has shapes related to those underwater features.
   1. A beach rock may be inferred from a non-travelling bump in an otherwise flat horizontal sheet flow. No question here because you keep watching until it passes and there is your rock.

   2. River rock pillow wave. Watch for these while kayaking so you don't get stuck on one, especially upside down.

   3. The eddy line, with optional vertical edge on the air surface, variable entrainment direction and flow volume, and vertical vortices. This is a pillow wave with the rock entirely obstructing the flow. Kayakers need a bit of technique to cross the eddy line into or out of the eddy, leaning against the dragging side and sticking the blade across the line to have the other side pull you over. Sometimes there's a surfing opportunity the the top of an eddy line, or a nose-down squirt spot which is super fun and also rather safe. No mystery here.

5. Anti-causal edge ripples, in number maybe 7, maybe 3 to 15, in amplitude maybe 1/16", or less, in spatial period the crest-to-crest distances are maybe 3/8", according to a plumber's eye. I call them Anti-causal because the effect precedes the cause: these little ripples precede rather than follow the main energy-source crest or edge, and because the following surface after that primary crest is dead flat. Usually when some physical cause occurs, the biggest effect is the leading part and there are smaller effects which follow behind, but this is opposite to that. On the beach in 1/8" deep water, on the pool surface over 12' deep water, even on the wet bits on top of the pool edge drainage cover bars, when the pool water splashes over, an ACER travels along the top of the bar. They are everywhere once you start to see them. ACERs occur both in the shallowest water, even in wet sand, and in deep water. Explain these, young scientist!

   Does the top surface of the water form a skin which squeezes ahead of a discontinuity to form a wrinkling, and stretches behind to form a flatness? Is this the same thing as surface tension?

6. Boat wave train. Say a boat passes by a sleeping duck, trailing a wave train. The duck rides a wave packet, which is a small number of crests and troughs, and its body moves up and down and sideways in the direction of the wave in something like a circular or elliptical movement. Three, five, nine bounces, it's not infinite, it's not actually periodic, because periodic means infinitely periodic. Notice that the line along any given crest is not parallel to the line connecting the front edges of all the packets. Why? Why? Why!!?

7. Wind friction on flatwater. You know how the surface of a quiet lake may have some sections all glossy and sections matte? You can almost see the wind moving in the behavior of the matte areas. No mystery, air passing over water has friction.

8. Mid-ocean fixed-period field-flow. If it's fixed period then why do the big waves arrive first after a storm out to sea, and then they get shorter and choppier over time? How do the waves actually shorten?

9. Breaking waves transform deep water waves into shallow water turbulence, destroying enormous amounts of useable mechanical energy. But isn't there more to it? Why does it get pointy? What tips over at the top? How is it different from two horizontal sheet flows going in opposite directions? Can energy be extracted at the break, other than by surfers?

10. The mysterious Soliton. Does a single wave-crest, traveling along, without a trough or any other crest preceding or following it, exist as a type of water wave in nature? I'm pretty sure I saw some on the beach last week, travelling across, diagonal to the body flow of, horizontal sheet flow waves, but when I realized what I was looking at and wanted to be be sure, the only ones I saw came in twos or threes, hump-only, and far from sinusoidal, like a traveling triangular crease. Let me know if you see one.

11. Impulse response in a long skinny pool. Set a depth. Form one end of the pool with a hydraulically actuated vertical end-wall. Program a pulse whereby the end-wall moves linearly into the pool a unit distance in a unit time. Observe the resulting surface movements over time. That counts as a pretty good impulse response (best is if the unit time is infinitesimal, but we'll get close enough by making the movement quick.)

In linear systems (i.e., input-output) theory the impulse response is sufficient to characterize the system, since any input function or waveform can be constructed as a sum of shifted and scaled impulses, and for a linear system you can do the shifting and scaling either on the input before it goes in or on the output after it comes out and the end result is the same either way. That is, a sum of a bunch of copies of shifted and scaled outputs of the unit impulse is the same as the output coming from a shifted and scaled input to the system, by definition. Well, unfortunately water is non-linear. Like, you'll notice that the different wave frequencies travel at different speeds not at the same speed, so the further out you go the longer waves are leading and the short ones haven't gone as far yet. (All the frequencies travel at the same speed in a linear medium.) Well, so the impulse response maybe doesn't tell you everything about how the whole medium works, but still it'll isolate the causal chain so you can see what happens after what else and maybe you can figure out why.

I think these are super fun and pleasantly puzzling, and I hope you will advance our common knowledge by puzzling about them too. Let me know what you find out!

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