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 a weighted sum
of outputs is the output of a weighted sum of inputs. So you can
analyse an input into simple parts that scale up 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 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 you can't figure out what's
going on by adding the different waves when some of them haven't
even gotten to you yet.)
So what's going on with water waves?
Here is a place to start: a typology of water waves that I've 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?
- 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?
- 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.
- What causes the collapse?
- How come
the effected frontier is NOT always defined by a crest?
come tidal waves seem to always be preceded by a trough?
- Horizontal sheet flow.
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
contrained by a channel, except that the squirt force is a
momentary pile of water pushing down and the channel constraint
is the ground below.
- 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
- 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.
- Tidal waves seem to be HSF's after a giant outflow event.
Why always a preceding outflow?
- Surface-following wave shapes, where a sheet flows over the
features of an underwater surface, and has shapes related to those
- 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.
- id=4.2 River rock pillow wave. Watch for these while kayaking so you
don't get stuck on one, especially upside down.
- 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.
- 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,
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?
- 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!!?
- 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.
- 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
- 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?
- 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.
- 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.