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Waves Unit I: The Nature of Waves (Topic 5A-1) – page 1
Waves Unit II: Waves in the Ocean
Changes in Wave Height, Wavelength, and Wave Speed at the Shoreline
As a wave moves into shallow water, its orbitals “feel the bottom,” causing it to slow down. The
wave crests that are closer to the shore (“in front”) are in shallower water, so they are moving
slower than the wave crests farther out in the ocean (“behind”). This allows the wave crests out
in the ocean to get closer to the wave crests near the shore, reducing the wavelength (the distance
between the crests). This “squeezes” the water in-between the two wave crests horizontally; the
water cannot go down (the ocean is getting shallower as it approaches the shore!), so it goes in
the only direction it can: up! This is why waves grow larger at a beach.
I like to say that the reduction in wavelength at the shoreline is like a traffic jam. Imagine that
you are driving north on the 405 towards the junction with the 105 by LAX. You begin with lots
of distance between you and the car in front of you, but as you come around the bend near LAX,
you’ll see a sea of red tail lights in front of you: they’re all slowing down. Both you and the car
in front of you put on your brakes, but he saw the tail lights first, so he starts stopping first.
While you’re going faster, you get closer to the in front of you, just like the wave crest “behind”
gets closer to the wave crest “in front” of it. In other words, the “wavelength” (the distance
between you) gets smaller.
Here is another way to think about why
waves grow at a beach: the front part of
a wave crest is in slightly shallower
water than the back part of the wave
crest, so it is always going a little slower
than the back part of the crest. As the back part of the crest catches up to the front part of the
crest, more and more of the water that was spread out over a wide area gets concentrated in a
narrower area. (See the picture above: 1 foot × 120 feet becomes 30 feet × 4 feet) I always
Waves Unit I: The Nature of Waves (Topic 5A-1) – page 2
hesitate to use this explanation because many students think that a wave crest that is “behind”
another wave crest can actually catch up to the wave crest that is “in front” of it. This cannot
happen , because the wave crest that is “behind” slows down more and more as it enters
shallower water, so it can never actually catch up to the wave crest “in front” of it. Instead, it is
always “catching up,” but never can quite do so. The key thing to remember to avoid confusion
is that the back part of a wave crest is “catching up” to the front part of the same wave crest; two
separate wave crests are not merging.
In fact, one wave crest can catch up to another wave crest if two groups of waves from different
places are arriving at the same time. In shallow water the orbitals of the waves with a shorter
wavelength do not reach down as far as the waves with a longer wavelength and thus do not “feel
the bottom” quite as much. So, on a beach the waves with a shorter wavelength waves move a
little faster than the waves with a longer wavelength! This allows a wave crest of the waves with
a shorter wavelength to catch up with a wave crest of the waves with a longer wavelength. Since
their speeds are pretty close, the crests can stay together for quite some time, leading to a
significant increase in wave height due to wave interference for much longer than a “moment.”
This is why there are “sets” of waves with larger heights and why some surfers have rules of
thumb like “every seventh wave will be larger than the rest.”
Note: Waves do not grow at a beach because the bottom pushes them up (as if they are hitting
the bottom and bouncing upwards). Also, they are not gaining energy as they grow. Wave
growth as an example of the conservation of energy: the forward “motion” energy of the wave
(kinetic energy) is being converted into “gravitational potential energy” (it goes upward, fighting
gravity), like a ball being thrown upward loses speed (“motion”) as it goes upwards fighting
gravity. However, as you know, what goes up, must come down: the wave eventually becomes
too steep and it gets pulled down by gravity, causing it get all of its “motion” energy back.
1. As waves approach a beach, do the waves speed up (go faster) or slow down?
2. Why does the speed of waves change as they approach the beach?
3. As waves approach a beach, does their wavelength get longer (increase) or shorter
(decrease)?
4. Why does the wavelength of waves change as they approach a beach?
5. Why do waves’ height grow when they approach a beach?
Waves Unit I: The Nature of Waves (Topic 5A-1) – page 3
Wave Refraction
Most ocean waves are created by winds (often during storms). Winds can blow from any
direction, so at any given spot in the ocean, waves are typically coming from and going in a wide
variety of directions. However, if you think of waves at a beach, you know that they pretty much
go right towards the shoreline. In other words, they go towards you when you are standing on the
beach; they do not go up or down the coast. This means that as waves approach a shoreline, they
turn or “bend” towards the shoreline, a process we call wave refraction. (Not reflection:
reflection is when a wave bounces off something, like a wall.) The result is that wave crests tend
to parallel or “match” the shape of the shoreline as they break.
To understand why this happens, imagine
a line of soldiers marching towards some
mud at an angle. (Why march into the
mud? Because Sarge told them to.) A
soldier at one end of the line will reach
the mud first, slowing him down, while
the other soldiers continue moving
forward. Each soldier spends a little more
time on the grass than the soldier to his
left, so he moves farther than his
neighbor, getting ahead of him. The
result is that by the time the last soldier steps into the mud, the line of soldiers has been stretched
out and now makes a new angle: the line has “turned” or “bent.” The only difference between
this example and waves is that they actually change direction, unlike the line of soldiers. A better
example is to imagine pushing or driving a two-wheeled object like a hand truck (“dolly”) or
Segway into the mud. The wheel in the mud would get stuck, but the wheel on the grass would
keep moving and turn the object towards the mud.
As a wave crest approaches the shoreline, typically
one end of the line is closer to the shoreline than the
other. This end is in shallower water, so it slows down
while the other end, in deeper water, moves forward
faster. This swings the wave crest towards the shore,
since the end in deep water covers a larger distance
towards the shore than the slower-moving end in
shallow water.
If the seafloor slope around an island is not too steep,
wave refraction can actually cause waves to turn all the way around (“wrap around” an island)
and hit the opposite side. However, waves tends to be weaker on the side of the island facing
Waves Unit I: The Nature of Waves (Topic 5A-1) – page 4
away from the original waves, because the waves are getting “stretched out” over a larger area
(as in the soldier example where the line got longer). This means that the energy in a wave is also
spread over a larger area, which reduces the height of the waves. In other words, refraction often
makes waves smaller. However, if waves bend towards one another and begin to come together,
they interfere, creating a higher wave crest. Thus, wave refraction can also increase the height of
waves. This typically happens near the shallow water of a point or headland (a place where the
land sticks out into the ocean). The waves bend towards the headland, causing them to come
together, increase in height, and pound the headland even more fiercely than the rest of the coast.
If waves completely “wrap” around an island and come together on the other side, they will
become larger as well.
Typically, waves break before refracting completely (becoming parallel to the shore – that is,
perfectly matching the shape of the shoreline), which allows them to push sand down the coast,
our next topic.
6. What is wave refraction?
7. Where are waves typically moving faster, in the shallower water closer to the shoreline,
or deeper water farther from the shoreline?
8. Where does wave refraction make waves’ height smaller, where the wave crests bend
together (as at a headland) or when the wave crests get stretched out (as happens along a
ordinary straight shoreline)?
9. True or false? “Wave refraction can cause waves to go around an island and meet another
part of the same wave crest on the other side of the island.”
Waves Unit I: The Nature of Waves (Topic 5A-1) – page 5
Longshore Transport of Sand by Waves
Waves often do not refract completely, and come into the shoreline at a small angle, allowing
them to push sand along the shoreline. We say that the waves are transporting sand down along
the coast, and hence call it the longshore transport of sand.
A breaking wave pushes sand up the slope of the
beach at an angle. The water and sand then slide
back down the beach slope into the ocean,
pulled down by gravity, but they goes straight
downhill, taking the fastest route back into the
ocean. Thus, they are not back where they
started. (This motion is often called “zig-zag”
motion.) Each breaking wave pushes sand a little
bit down the coast. This may seem like a small
effect, but waves endlessly pound the shore, day
after day, year after year, slowly pushing sand
down the coast: inch after inch eventually becomes mile after mile.
As a non-breaking wave goes by, the water beneath the wave moves in a circle. In shallow water,
though, the bottom gets in the way, distorting the circle into an ellipse (oval). At the very bottom,
the ellipse is completely squished, so that the water hardly moves up and down at all: instead it
goes side-to-side or back-and-forth as the wave goes by. This water motion pushes the sand
beneath it, causing the sand to wiggle back-and-forth as well. Even though the sand moves, it
does not go anywhere: like a child on a swing, it goes back-and-forth but does not actually travel
from place to place. (Non-breaking waves moving sand back-and-forth does cause the sand
migrate or drift a little bit, but this is very small and slow compared to longshore transport.)
10. What is longshore transport?
11. What causes longshore transport?
12. Why do waves have to break for longshore transport to occur? In other words, why don’t
non-breaking waves cause sand to move along the coast?
13. True or false? “Waves that approach the coast at a steep angle push more sand down the
coast than waves that come into the coast nearly parallel to the coast (match the shape of
the shoreline).”
Waves Unit I: The Nature of Waves (Topic 5A-1) – page 6
Wave Period and Wave Frequency
Another way that scientists describe waves is to measure how quickly they cause the sea surface
to bounce up and down. One strategy for measuring this feature of waves is to watch a single
spot, and to count how long it takes for one wave crest to be replaced by the wave crest behind it.
(For example, watch a surfer or a bird bob down and then back up again.) This is called the wave
“period,” the period of time is takes for a wave crest to go by (travel the distance of 1
wavelength). Alternatively, you can measure the wave frequency, how often a wave comes by.
(For example, 6 waves pass by in a minute.) They are inversely proportional to one another:
frequency = 1/period, meaning that if one is high, the other is low; if waves pass often – high
frequency – then small amount of time between them – a short period of time. Note that if you
measure one, you can calculate the other.
Scientists measure wave period or frequency for several reasons. One reason is that wave period
and frequency never change as waves travel, unlike wave height or wavelength. Secondly, they
are much easier to estimate from a distance than wave height or wavelength. Finally, if you
measure the wave period or frequency and know the depth of the water, then you can calculate
the waves’ wavelength and speed. The relationship between wave period and wavelength is: the
longer the period, the longer the wavelength. Think about it for a moment: the larger the distance
between the crests, the longer the time it takes for one crest to replace the one in front of it. Since
long-wavelength waves move faster than short-wavelength waves, long-period waves move
faster than short-period waves.
For those you who are interested, this is the formula that relates wave characteristics to one
another: ω2 = gk tanh(kh), where ω is the angular frequency, g is gravitational acceleration, k is
the angular wavenumber (wavelength), and h is the depth of the water. In deep water, it is simply
ω2 = gk. In shallow water, it is ω2 = ghk2.
Wavelength changes as waves approach a beach, because the waves are slowing down. The
period does not change at all; the wavelength does all the adjusting that is necessary. The period
does change when the waves break, but by this point it no longer matters: once they break, they
are no longer waves. In short, the waves’ height, wavelength, and speed change as they approach
the beach – but one wave characteristic does not change: wave period (and frequency).
Waves Unit I: The Nature of Waves (Topic 5A-1) – page 7
14. What is the period of a wave?
15. What is the frequency of a wave?
16. If a wave has a long period, does that mean it will have a high frequency or a low
frequency?
17. Which wave characteristic is easiest to measure, height, period, speed, or wavelength?
18. True or false? “If you know the period of a wave and the depth of the water, then you
also can calculate the waves’ wavelength and speed.”
19. If a wave has a long period, does that mean it will have a long wavelength or a short
wavelength?
20. Which move faster, waves with a long period or waves with a short period?
21. Which waves created by a storm will reach the coast first, the waves with a long
wavelength or the waves with a short wavelength?
22. Which wave characteristic does not change as waves approach a beach, the waves’
height, period, speed, or wavelength?
Waves Unit I: The Nature of Waves (Topic 5A-1) – page 8
Making Waves
Most ocean waves are created by the wind blowing over the surface of the ocean. The largest
waves are created by the strong winds of storms. The wind enhances small differences in the
surface of the ocean by pushing the top of the waves forward and making the waves grow via the
Bernoulli Effect. Have you ever been standing by the side of the road when a fast-moving truck
goes by and felt “pulled” towards the truck? This is owing to the Bernoulli Effect. The truck
pushes air out of the way quickly, and you get sucked in with the air moving in to replace it. In
the same way, a fast wind “sucks” the surface of the ocean upward.
Bernolli Effect Experiment: Take a piece of paper (lighter is better), and hold one end with both
hands just beneath your mouth. Now, blow across the top of the paper. Notice how the paper
rises, pushed upwards by the air beneath. The wind causes waves to grow in the same way.
Waves must go through a cycle of growth and breaking many times to become large (tall height
and long wavelength). As a small wave grows higher, it becomes too steep and breaks, causing it
to lose the height it gained. However, the breaking causes its wavelength to stretch out (gets
longer), so when it grows again, it can get taller before becoming too steep and breaking again.
The wavelength gets stretched again and again, and the wave grows again and again. Remember:
the wavelength of a wave affects its speed: the longer the wavelength, the faster the speed.
Eventually, the wave moves as fast as the wind, so the wind can no longer push it, and the wave
stops growing.
The largest waves are created by strong, steady winds that blow over a large area called the
fetch. Since waves grow until they match the speed of the wind, strong (fast) winds make the
biggest waves. If the strong winds keep shifting – first making waves going one direction, then
another – then the wind will create small waves going in many directions, not large waves going
in one direction. It takes time for large waves to grow, so longer the winds blow in one direction,
the bigger the waves can become. Waves stop growing if there is no wind, so once they leave the
area where the wind is blowing (the fetch), they cannot grow any more. Winds with a large fetch
can help the waves grow for a long time before they leave the wind behind.
Winds over the ocean are strongest near the Poles, and waves are also largest where the winds
are strongest. Storms are common at these latitudes, especially in the winter, though storms are
more common in subtropics during the winter as well. During the summertime, tropical storms
are common near the Equator, and tropical storms that grow into hurricanes can create huge
waves. Tropical oceans are much calmer than polar oceans most of the time. Thus, most of the
large waves that strike the coast come from the Poles.
Waves Unit I: The Nature of Waves (Topic 5A-1) – page 9
23. What create most of the waves in the ocean?
24. Which produces waves with a larger height, a strong (fast) wind or a weak wind?
25. Which produces waves with a longer wavelength, a strong (fast) wind or a weak wind?
26. Which produces waves with a larger height, a steady wind or a wind that changes
direction over time?
27. Which produces waves with a larger height, a wind the blows over a large area or a wind
the blows over a small area?
28. True or false? “Waves grow until their speed matches the speed of the wind.”
29. Where are most of the big waves created, closer to the Poles or closer to the Equator?
30. During which season are waves especially large near the Poles, summer or winter?
Waves Unit I: The Nature of Waves (Topic 5A-1) – page 10
Waves across the Ocean
Once waves leave the fetch, waves lose very little energy as they travel across the ocean. Like a
row of dominos, water molecules bump into one another, passing the disturbance and its energy
from one to the next quite efficiently. Waves may lose energy if wave interference makes them
large enough to break, or winds encountered in their journey add or remove energy – and of
course they lose all their energy once they finally break along a shoreline.
As a wave moves away from the storm that created it, the wave is spreading out (like ripples on a
pond), so the wave height do decreases. The reduced wave height does not mean that the waves
are losing energy, though. They have the same total amount of energy, but it is spread over a
larger area. Thus waves from a nearby storm are larger (as you might have guessed) but affect a
smaller section of the coast than waves from a far-away storm.
The winds of a storm create waves with a variety of wavelengths and heights, but as they travel
across the ocean, they begin to sort themselves out by wavelength. The longer-wavelength waves
are faster, so they leave the shorter-wavelength waves behind. We call this wave dispersion
(disperse = too go apart, like the police telling a crowd to “disperse”). When all the waves of
different wavelength are all jumbled together, it can produce a very complex sea surface (see the
section on “wave interference”). Once waves separate (disperse), the sea surface becomes more
regular, resembling the nice smooth patterns in my side-view sketches of waves. We call such
nice, regular waves swell. When jumbled together, we say that they are sea waves.
31. True or false? “Wave typically lose little or no energy as they travel across the ocean.”
32. Does waves’ height increase, decrease, or stay about the same as they travel across the
ocean?
33. How are swell different from sea waves?
Waves Unit I: The Nature of Waves (Topic 5A-1) – page 11
Wave Groups
As waves move outward away from the storm that created them, a curious phenomenon can be
observed in the wave groups. A crest will emerge from the back of the group, move all the way
to the front, and disappear. This keeps happening again and again, with wave crests emerging at
the back of the group and disappearing at the front. Overall, this leads to a reduction in the
group’s speed (the group speed is ½ of the wave speed). The best – though confusing – way to
understand this is that a wave group is composed of several waves of slightly different
wavelengths all mutually interfering. New wave crests appear because the longer-wavelength
waves are moving faster: instead of crests overlapping with troughs (canceling both), the crests
begin to overlap with crests and troughs with troughs. The longer wavelength waves are moving
slightly faster, leading to a wave crest moving forward through the group. However, the
interference at the front of the group causes the waves to cancel out again as crests again line up
with troughs, so the group does not move forward as much an individual wave crest. Note that
the largest wave crest is in the center of the group. This is one reason surfers observe that every
3rd wave crest of a set (or every 7th wave crest, or other rules I’ve heard them speak of) tends to
be larger than the rest: two or more waves are coming into the beach at the same time and
interfering, growing, and breaking together.
The Importance of Waves in the Ocean
Waves have several important effects on the ocean, particularly breaking waves. Once a wave
breaks, the wave motion of the orbitals breaks down. Instead of moving in a circle, water surges
forward (just like water surges up the slope of a beach). Thus, a breaking wave becomes a
current.
In addition, as water falls down the front of a breaking wave, it captures air. (In other words,
bubbles form: the white foam that you see on a wave crest and is left behind as it moves onward
towards the shore.) If these bubbles break underneath the surface of the water, then the ocean
water captures gases from the atmosphere (they become dissolved gases). Similarly, breaking
waves disturb the surface of the ocean, sending a spray of water droplets into the air. The water
molecules, salts, and gases in the droplets now have a much easier time evaporating (there are
more directions in which to fly away), allowing them to enter the atmosphere. Salts, of course,
tend to settle back into the ocean or on land over time, but this is a major way in which the
oxygen made by ocean algae (like phytoplankton) enters the atmosphere for us to breathe!
(Thank you, waves!) Of course, air molecules can also strike the surface of the ocean, or break
free from the ocean surface, but this process is much less likely (and thus slower) than the
exchange of air molecules mediated by ocean waves.
Finally, breaking waves and wave orbitals stir up the surface of the ocean (appropriately called
the mixed layer), making it fairly uniform in temperature, salinity, and other characteristics.
Their mixing brings up both sinking phytoplankton and unused nutrients from down deep, and
Waves Unit I: The Nature of Waves (Topic 5A-1) – page 12
sends down abundant oxygen from the surface. Since waves make the water move in circles,
waves also push some phytoplankton floating at the surface down, but because phytoplankton
tend to sink, there are more phytoplankton who need to be brought up than there are floating near
the surface. Thus, overall waves tend to bring up more phytoplankton than they push down – and
bring up more nutrients than they push down as well.
It should be noted that other phenomena help mix the mixed layer: for example, when surface
water cools and sinks (higher density) and is replaced by the somewhat warmer water from
below, similar to the effect of wave orbitals.
34. True or false? “Breaking waves help create ocean currents.”
35. True or false? “Breaking waves help the ocean absorb carbon dioxide from the
atmosphere.”
36. True or false? “Breaking waves help the ocean release oxygen made by phytoplankton
into the atmosphere.”
37. True or false? “Waves tend to harm phytoplankton, making it more difficult for them to
get sunlight and nutrients needed for photosynthesis.”