oceanography ch06

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Formed by strong winds in ocean storms, long, graceful swells

can travel hundreds of miles across the open ocean—leaving

the storm that created them far behind. But, upon encountering

coastlines and shoaling depths, they come ashore and die as

surf, such as this spectacular plunging breaker.

Contents

The Basics of Ocean Waves 193

Tides 210

6CHAPTER





194 Chapter 6

Capillary waves

The tiniest waves on the surface of the ocean are capillary waves. These are thefirst waves to form when the wind blows, as energy is transferred from themoving air to the water. What happens is best visualized this way: if you wereto blow across the surface of a cup of coffee or tea in order to cool it down a

bit, you would see small ripples develop as the energy from your blowing istransferred to the liquid. This transfer of energy happens because even thoughthe air you are blowing is directed parallel to the surface of the liquid, air that

is moving has lower pressure than non-moving air (the air above the liquid just before you started to blow). Therefore, right above the air–water interface,

Direction of propagation

L

A H

Sea level at rest

Crest

Trough

(A) (B)

(C) (D)

FIGURE 6.2 Ocean waves can take many forms, asthese examples show. (A) Waves during a relatively calmperiod with only light winds. Smaller waves can be seenriding on top of larger waves, creating a confusion ofsurface wave patterns and making it difficult or impos-sible to determine the direction any are coming from.(B) Closer to shore, wave patterns are different; they

usually arrive from offshore, such as these surf wavesbreaking on a rocky coastline. (C) An almost perfectlycalm sea with little or no wind blowing except near thehorizon, where a slight disturbance can be seen. Only agentle swell is obvious on the sea surface. (D) A stormwave sending spray over the bow of a 176-foot-longresearch ship.

FIGURE 6.1 A rock tossed into a calmbody of water generates surface gravitywaves that propagate outward in all direc-tions. The parts of such waves includethe crest, or top, of the wave; the trough,the lowest point; the wavelength (L), thelength from crest to crest or trough totrough; the wave height (H ), the verti-cal distance from the wave crest to thetrough; the wave amplitude (A), one halfthe wave height.





Waves and Tides 195

slight differences in wind speed and the resulting differencesin lowered air pressure lifts some of the water upward intoripples. These ripples that form initially are capillary waves(also called cat’s-paws); their wavelength is less than 1.73 cm,which is just over ¾ inch. It is these tiny capillary waves thatmark the beginning of a process that, under the influence of

a wind, eventually builds larger waves such as those we aremore accustomed to seeing on the surface of the ocean. FIGURE

6.3 shows these little capillary waves beside our research shipone day when a gentle breeze was blowing.

For these very small capillary waves, the restoring forceis dominated by surface tension, which results from the pullof hydrogen bonds at the air–water interface (see Figure4.10). Gravity also pulls the wave downward, but at thesevery small wave sizes, surface tension pulls harder. Gravity

becomes important when the waves get bigger. When thewave’s initial upward bulge is formed, the water’s skin-likesurface is stretched, but returns to its original, flat positionas the hydrogen bonds pull the surface back. The alternat-

ing dominance of those two forces creates an oscillation asthe tiny wave is first formed. Such a forced displacement of a surface andits subsequent return to its original position is analogous to what happenswhen we beat a drum. The drumskin is depressed by the drumstick, andthen the drumskin’s tension restores the drum surface to its original flatshape, but then it overshoots the level of the original surface and oscillatesor vibrates, creating sound waves. The same thing happens when we plucka guitar string; the string’s tension pulls it back to where it started, and thenit overruns that position, oscillating back and forth in a vibration that createsa musical tone. Both of these examples using musical instruments representthe action of stationary or standing waves, but they illustrate nicely thesurface-tension-restoring force that creates capillary waves in water. Whilethere can also be standing waves in water (called seiches; see page 207), we

are more accustomed to seeing waves that move across the surface of theocean: progressive waves. So, back to our discussion of how wind waves areformed on the ocean.

Surface gravity waves

If the wind continues to blow across the surface of the ocean, the small capil-lary waves, which now have an inclined surface for the wind to blow against(FIGURE 6.4), will continue to acquire energy from the wind, which results inthe waves growing bigger. The wave will increase both in height and wave-length, and once the wavelength exceeds 1.73 cm, the restoring force is nolonger dominated by surface tension, but by gravity. Beyond 1.73 cm, theextremely weak surface tension of water is no longer capable of pulling thesurface back to a flat position. The wave transitions from a capillary wave toa surface gravity wave. Gravity causes the crest of a surface gravity wave to,quite simply, fall downward. As with the capillary wave, the gravity wavefalls, its momentum causes it to overshoot its original position, producing anoscillating wave. Under the influence of a wind blowing that wave horizontallyacross the surface of the ocean, we have a progressive wave which moves, orpropagates, away from its point of origin.

It is important to realize that progressive waves are moving energy, notwater. It is the wave form itself (the shape of the wave) and the energy thewave carries with it that are propagated. This phenomenon is illustrated inFigure 2.23, which shows how we can force a wave down the length of a rope

FIGURE 6.3 Capillary waves are very short-wavelengthwaves that can eventually transition to surface gravitywaves.

Wind

1.73 cm

FIGURE 6.4 A capillary wave on thesurface of the ocean provides a face forthe wind to blow against, making for amore efficient transfer of wind energy tothe ocean.





196 Chapter 6

attached to a wall. The wave form moves along the length of rope, but the ropeitself remains in your hand at one end and attached to the wall at the other.

If we were to examine closely the particles of water in a surface gravitywave both at the sea surface and immediately beneath, we would see thatwater particles don’t just move up and down as waves pass by; they actuallytrace out a nearly perfect circle called a wave orbit (FIGURE 6.5). To explainwhat happens, let’s start by considering just the surface of the ocean; we’llconsider subsurface waters later. Figure 6.5 shows that in the crest of the wave,

the water particles are moving forward in the direction of propagation withthe wave itself, but in the trough, they are moving backward (!), and theycomplete the trace of a circle as the wave passes. With each passing wave,the water particles return almost exactly to their starting position—the wavemoves, but the water does not. While waves in general behave according tothis principle, there is an exception in the case of these surface gravity wavesin water, whereby there is a very small but significant net movement of thenear-surface water itself in the direction the wave is moving. This phenomenonis known as Stokes Drift (see FIGURE 6.6).

Wave direction

1.

2.

3.

4.

5.

Wave

height

Sea level

One wavelength

FIGURE 6.5 The orbital path, equal to the wave height, traced by a particle of water onthe surface of the ocean as a wave passes from left to right. (1) A wave crest approachesfrom the left, carrying the water particle up and forward to the right. One-quarter wave-length later (2) the particle has continued to move forward to the right, but also down-ward. (3) As the next quarter-wavelength passes, the particle continues to fall downward,but now is being pulled backward into the wave trough. (4) As the next crest approaches,the particle is lifted upward, while continuing to be pulled backward. (5) As the wavecrest approaches, the particle is lifted upward and forward. When a full wavelength haspassed, from crest to crest, the particle has completed its circular wave orbit. The waveitself has moved horizontally from left to right, but the water, as shown relative to a sta-tionary dashed line in the background, has not. The water particle did not move any netdistance horizontally or vertically.

(A)

(B)

Strand line

Wave direction

Close up of the gradual

leftward movement

of a particle as four

waves pass by

Stokes drift after

10 wave crests

have passed

FIGURE 6.6 Have you ever wondered why floating debris, such as seaweed, driftwood,and trash accumulate on beaches rather than being washed out to sea? Two phenom-ena are responsible: the refraction of waves toward beaches and Stokes Drift. Eventhough we know that waves transport energy and not materials, that isn’t strictly truewhen it comes to matter in the near-surface waters. Because wave orbits decreasein size beneath the surface, an individual wave orbit will have a slightly smaller arc atits base, in the trough, than when it is at the crest. This produces a corkscrew actionand very slowly transports water and debris in near-surface waters in the direction ofwave propagation (A). These diagrams illustrate the principle, with the most recentorbit in red and the earlier orbits in gray. The inset shows a close up and movement ofa particle (the scale is exaggerated; actual movement is slight). In near shore waters,this direction is toward the shoreline, where various materials can collect, such as theseaweed in a strand line on this beach (B).





Waves and Tides 197

This phenomenon of wave orbits is not confined to the surface of the ocean;as you might logically expect, water particles beneath the surface also travel inorbits. But, unlike the surface-wave orbits, in which the diameter of the orbitis equal to the wave height, wave orbits beneath the surface have diametersthat are greatly diminished with increasing depth (FIGURE 6.7). At a depth ofonly one-half the wavelength, the orbital diameters are reduced to a mere 4%

of what they are at the surface.Scuba divers are well aware of this phenomenon of wave orbits. In arough sea, or with swells passing (more on swells below), divers at or near thesurface will be jostled about with these wave orbits, but upon diving deeper,they quickly find themselves in relatively calm waters. They do not have togo very deep to experience this diminished wave action.

Except for those of us who have experienced this phenomenon firsthand,the concept of wave orbits is somewhat difficult is to imagine. Figures 6.5 and6.7 help to visualize the phenomenon, but they aren’t nearly as effective asdirect observation. Apart from scuba diving, one of the best ways to observethese orbital motions is to stick your head over the edge of a dock with yourface just a few inches above the water and look at the waves passing by. As-suming that there are waves present, watch how objects on and just beneath the

surface move about. Look at the little air bubbles just below the surface of thewater as well as the tiny flecks of particulate material suspended in the water,and you’ll see the wave orbits. Better still: go to the beach and tread water just

beyond the breakers—if you can swim, that is. You can feel yourself travel-ing in these orbital motions as waves approach and pass by you. Or, stand inwaist-deep water. As the trough in front of a swell approaches you, and beforethe crest arrives and then breaks, you will feel yourself being pulled towardthe oncoming wave as water rushes by you toward the oncoming crest. Andafter the crest passes you will feel a momentary pull toward shore as waterrushes past you in the opposite direction. The main point here, and in Figures6.5 and 6.7 is that water particles return to very near their original positionsafter a wave passes by. Again, this is because, in general, the wave transportsenergy, not material (with Stokes drift being the exception).

Because wave orbits extend beneath the surface, it follows that a surfacegravity wave requires some minimum depth of water in order to exist. Contactwith the bottom generally interferes with the wave orbits, resulting in thewave losing energy due to friction, and slowing down. Waves in deep waterdo not encounter this problem. For this reason, we categorize surface gravitywaves according to the relationship of their wavelength to the water depth.

DEEP WATER GRAVITY WAVES As the name implies,deep water gravity waves are surface waves that occurin water that is deeper than one-half their wavelength;the bottom does not interfere with the wave orbits.The speed, or velocity, of propagation of a deep waterwave is 1.25 times the square root of the wavelength;that is, V = 1.25 √ L (BOX 6A). Notice that depth is notin the equation. When a deep water wave propagatesto where the water’s depth becomes less than one-half its wavelength, it will begin to be impeded byfriction, and its speed will diminish.

SHALLOW WATER GRAVITY WAVES In shallow wa-ter, the wave orbits of a deep water wave will bedeformed and flattened into ellipses (FIGURE 6.8).This frictional encounter with the bottom causes

Wave

direction

One wavelength

One half

wavelength

FIGURE 6.7 Wave orbits continue withdepth beneath a surface wave, but theirdiameters quickly diminish. At the surface,the diameter of a wave orbit is equal tothe wave height. But the diameter of waveorbits at a depth equal to one-half thewavelength is only about 4% the diameterof those at the surface.

Wave direction

FIGURE 6.8 In a shallow water wave, the bottom causes the waveorbits to flatten. This happens when the bottom depth is on the orderof 1/20 that of the wavelength. Just off the bottom, the water motionsflow back and forth. Note: This diagram is distorted vertically in orderto show more clearly the flattened orbits. In nature, the orbits would be

even flatter than shown here, where we have drawn the bottom depthto be nearer one-third the wavelength.





198 Chapter 6

the wave to lose energy and diminishes the wave speed. As the water gets

shallower than 1/2 the wavelength, the bottom becomes more and more im-portant in controlling the speed; the circular orbits become flattened and thewave loses more energy due to friction with the bottom. Eventually, when thewater shoals (becomes shallow) to a point where the depth is less than 1/20the wavelength, the waves become shallow water gravity waves. The velocityof these waves is controlled entirely by the water depth.

INTERMEDIATE GRAVITY WAVES Between the depths for deep water wavesand shallow water waves—between 1/2 and 1/20 of the wavelength—wehave intermediate gravity waves. Intermediate waves “feel” the bottom; theirproximity to the bottom distorts their wave orbits, and to varying extents, thisdistortion affects the speed of these waves. But wavelength also is important,and so the velocity of intermediate gravity waves is determined by both thewater depth and the wavelength (see Box 6A).

Seas and swells

Thus far in our discussion of wind-generated waves, we have illustratedthem as smooth sinusoidal curves, but in fact, locally generated wind wavesdon’t look like that. Those smooth sinusoidal waves are actually swells, whichare surface waves that have a relatively long wavelength, are symmetricallyshaped, and have a relatively long period, on the order of 10 seconds or so.Swells are generated remotely, some distance away from where they assumethese characteristics. Locally-generated waves created by the wind blowing on

BOX 6A The Speed of Ocean Waves

W

ave velocity (V , in meters per

second, m/s) can be described

as a function of wavelength (L,

in meters, m) and period (T , in sec-onds, s). These are related as:

We can rearrange this equation to solve

for T and L, as:

In terms of its wavelength (L), the full

wave equation for velocity (V ) is:

Here, g is the acceleration of gravity,

9.8 m/sec2; L is the wavelength; π is

3.14; h is the water depth; and tanh is

the hyperbolic tangent.

If the water is very deep, or greater

than one half the wavelength—that

is, if we are dealing with a deep water

wave—then the quantity tanh2πh/L is

approximately equal to 1.0, and so we

can drop that part from our equation.

Thus, the speed of a deep water wavewill depend only on its wavelength, and

not on the water depth, as:

In very shallow water, where the

depth is small compared with the wave-

length, that is, when the quantity 2πh/L

is small, then tanh2πh/L will be ap-

proximately equal to 2πh/L. Therefore,

the wave equation above reduces to:

This is the equation for the speed of

shallow water waves, where the wa-

ter depth is shallower than 1/20, or

0.05×, the wavelength (that is, where z

= 0.05L).

For intermediate water waves, where

the depth is between 1/2 and 1/20 the

wavelength, the full equation must be

used to solve for wave speed.

Velocity Wavelength

PeriodV

L

T ( )

( )

( )=

Period T L

V ( )=

Wavelength L VT ( ) =

22V gLtanh hLπ

π =

2

9.8

6.281.25V

gL LL

π

=

=

×

=

V gh=





Waves and Tides 199

the sea surface are known as seas, which are relatively short and steep waves,with an abrupt crest (FIGURE 6.9).

Swells begin as seas that propagate across the ocean to an area miles awayfrom the influence of the wind that created them, where they “flatten out” intoswells. So, capillary waves become seas, which can eventually become swells,

but there are a few more details we need to discuss with respect to the evolu-

tion of each of these developmental stages in the formation of wind waves.Formation and evolution of wind waves

The minimum wind speed required to generate a wave disturbance is on theorder of 1–2 knots. (One knot, or 1 nautical mile per hour, is about 51 cm/s,which is a slow walking speed for an average person.) You and I will only

begin to feel a breeze on our faces when the wind increases to a speed of about1 knot. Wind blowing at this speed across the surface of a body of water willgenerate capillary waves. Then, as the wind continues to blow, adding energyto the wave and growing it longer than 1.73 cm, the wave will transition froma capillary wave to a gravity wave (FIGURE 6.10A).

Once the early gravity wave has formed, its initial velocity is only aboutone-third the velocity of the wind that formed it. But as the wind continues to

blow at the same velocity, all the while continuing to add energy to the wave,the wave will grow in both wave height and wavelength. And as it grows, acurious phenomenon will occur: the surface gravity wave’s height ( H ) willincrease faster than its wavelength (L). This means that as the wave grows, itgets steeper. A wave’s height cannot continue to increase faster than its wave-length indefinitely; if it were to do so, we would eventually have sky-high

but pencil-thin slivers of waves on the surface of the ocean. Instead, the waveeventually collapses on itself—it breaks—when it reaches a critical steepness,which is determined by its height relative to its length (FIGURE 6.10B). Thatcritical steepness is reached when the ratio of wave height to wavelength( H :L) is 1:7; at this point the wave crest will exhibit a steep profile, with acharacteristic angle of about 120°. Once the critical steepness is reached, eventhough the wind may continue to blow at a constant velocity, the wave cannot

grow any higher; it has reached its maximum height, and the energy added

Swell

Sea

FIGURE 6.9 Shapes of swells and seas.Swells are generally longer-wavelength,gently sloping symmetrical waves, where-as seas are asymmetrical, with steepercrests than troughs.

(A)

(B)

7

1 120°

Wind direction

Wave direction

Capillary waves Gravity waves

Critical steepness

H / L = 1:7 Whitecap

1.73 cm

H

L

FIGURE 6.10 Stages in the develop-ment of a sea. (A) Capillary waves growto gravity waves once their wavelengthexceeds 1.73 cm. Wind continues to addenergy to the gravity waves and so theygrow, gaining wave height faster thanwavelength (vertical scale exaggerated).Eventually, the ratio of wave height towavelength exceeds a critical value of 1:7and the wave breaks, forming a white-cap. (B) The critical steepness diagramto scale.





200 Chapter 6

by the wind after that simply causes the wave to break,producing whitecaps, which dissipate the wave’s energy(FIGURE 6.11). After whitecaps form, we have an equilib-rium, where the energy imparted to the sea by the windmatches the energy dissipated by the breaking whitecaps.If the winds do not increase in velocity, the waves cannotget any bigger—wave height will not increase. This seastate is called a fully developed sea.

The attainment of a fully developed sea for a particularwind speed depends on two factors: the duration of time the

wind is blowing, and the length of ocean over which that wind blows, thefetch (FIGURE 6.12). In other words, it takes some time for the wind to build

the seas, and the span of ocean over which it blows has to be big enough. A brief high-velocity wind gust will not build seas that are nearly as big as asustained high-velocity wind will build; but even a sustained high-velocitywind will not make big seas in a swimming pool. For there to be a fullydeveloped sea when the wind is blowing 20 knots (kt), for instance, which isquite common out on the open ocean, we see in Figure 6.12A that the windspeed must be maintained for a minimum of 10 hours, and it must blow overa fetch of nearly 100 nautical miles. We can also see in Figure 6.12A that forwind speeds higher than 20 kt, the required fetch and duration become quitelarge—conditions that are only rarely met in nature.

But when those conditions are met—and sometimes they are—the seas can become very large. How large? Seas that can be produced in fully developedsea at different wind speeds are shown in Figure 6.12B. A fully developed

sea in a 40 kt wind, for example, will have waves that average 7 meters (m)in height—and one in ten of those waves will exceed 16 m, or about 50 feet.That is a big wave by any standard; fortunately, such waves are relativelyrare. Even though 40 kt winds are not uncommon in ocean storms, we don’tsee 16 m waves very often because both the fetch and duration criteria for afully developed sea are rarely met. Storms are usually on the order of 100 to200 miles wide, which is less than the required fetch, and they usually last lessthan 24 hours, which is less than the required duration for a fully developedsea at this wind speed. Nevertheless, ocean waves can still get to be quite largein a storm, even if conditions for a fully developed sea are never reached.

The sizes of waves that can be expected for a given wind speed—eventhough conditions for a fully developed sea may not be met—are given in theBeaufort Wind Force Scale (TABLE 6.1). The Beaufort Scale is an empirical scale

based on observed sea conditions and provides an estimate of wave sizes onemight expect to see for a particular wind speed. In general, the wave heightof the seas that result from the wind is related to: (1) the wind speed; (2) theduration of that wind; and, (3) the fetch. When extreme values for each of

FIGURE 6.11 Photograph taken from a ship at sea where thesea has become fully developed for that wind speed, which on thisparticular day was about 20 knots. The seas have reached theircritical steepness for that wind speed, and energy added to thesea as the wind continues to blow at 20 knots is dissipated aswhite caps.

1600

1400

1200

1000

800

600

400

200

F e t c h ( n m i )

0 10 20 30 40 50 60 70

Duration (hr)

(A)

Wind speed (kt)Wind speed (kt)

(B)

45

40

35

30

25

20

15

5

10 W a v e h e i g h t ( m )

0 10 20 30 40 50 60

Wind speed (kt)

Highest 10%

Average

Highest 10%

Highest1/3Highest1/3

Average

50

40

30

2010

FIGURE 6.12 (A) Plot of the required fetch (in nautical miles) and duration (in hours) forthere to be a fully developed sea at the indicated wind speeds (in knots). (B) Wave heights(m) for a fully developed sea at the indicated wind speeds, presented as the height of thehighest 10% of waves, the highest 1/3 of waves, and the average wave heights.





Waves and Tides 201

these three criteria occur, waves can get to be quite large. AsTable 6.1 reveals, hurricane-force winds (greater than 63 kt)can build seas with average wave heights greater than 46 feet!

In case you are wondering just how big a wave can get:the biggest wave on record was observed by the crew of theUSS Ramapo during a storm in the Pacific Ocean in 1933 that

was estimated at some 34 m (112 feet) in height. (That waveis now thought to have been a rogue wave; see page 204).The estimate was based on observations made by an alert(and obviously not seasick) watch officer on the bridge, wholined up by sight the ship’s crow’s nest with the horizon,while at the same time lining up the bow with the horizon(FIGURE 6.13); this allowed a calculation of the wave height.But, let’s face it: the biggest wave that actually has occurredout on the ocean, and which was also observed by peopleaboard a ship at the time, probably sank the ship and with itthe crew who saw it.

WAVE DISPERSION After wind-generated surface gravity

waves are formed, they propagate away across the sea at aspeed that is dictated by their wavelength, which grows lon-ger over time. And as those waves move away, a few otherinteresting phenomena occur.

Once waves propagate to an area where the wind is nolonger adding energy to the waves—as they leave the areaof the storm that generated them, for instance—the waves

begin to flatten and lengthen. Their wave heights decrease in

TABLE 6.1 The Beaufort Scale of average wave

heights for given wind speedsa

Wind speed(kts)

Miles per hour(mph)

Average waveheight (ft)

<1 <1 0

1–3 1–3 0

4–6 4–7 <0.3

7–10 8–12 0.3–1.6

11–16 13–18 1.6–4

17–21 19–24 4–8

22–27 25–31 8–13

28–33 32–38 13–20

34–40 39–46 13–20

41–47 47–54 13–20

48–55 55–63 20–30

56–63 64–72 30–46

>63 >73 >46

aThese data are based on observations, unlike those in Figure 6.12,

which are based on theoretical calculations.

(A)

(B)

HorizonBridge

Crow’s nest

34 m(112 ft)

FIGURE 6.13 (A) The USS Ramapo ,built and launched in 1919, and whichremained in service until 1946, observedwhat is still believed to be the largestwave ever recorded. (B) The 112-footwave occurred in 1933 in the PacificOcean, and was measured using thegeometry shown here and as explainedin the text.





202 Chapter 6

response to the pull of gravity, and, in order to conserve energy, their wave-lengths necessarily increase. Their period, however, does not change, and sothe waves now move faster than they did at the point of generation. If theperiod (T ) doesn’t change, then increasing the wavelength (L) will increasethe velocity (as explained in Box 6A, V = L/T ).

This decreasing wave height and increasing wavelength decreases thewaves’ steepness and the sea transitions to a swell. So areas of the ocean awayfrom the area of wave generation will experience long-wavelength, fast-movingswells; the smaller, slower waves will not have had time to get there, or theymay have dissipated (lost their energy and just plain died) before getting there.The waves are said to have dispersed, making groups or trains of swells thathave become separated out into groups of similar wavelengths and so theytravel at similar speeds.

Wave direction

Start

A few

seconds later

A few

seconds

after that

(A) (B)

123

1234

2345

New wave appears at back Wave 1 has disappeared

New wave appears at back Wave 2 has disappeared

Group propagation

Individual wave propagation

(twice that of group)

12

123

123

234

234

234

234

345

FIGURE 6.14 (A) Diagram of wave speed and group speed with time. Three wavesare present in the top panel, all moving to the right; a moment later, the first wavedisappears, while a fourth wave appears at the back. Notice how wave 3 moves twiceas far as the group as a whole. (B) Frames from a video clip, selected at approximate-ly one second intervals from top to bottom, after a rock is tossed into the water. (Thecamera was not stationary, but followed the waves as they propagated left to right.)Notice that wave 1 soon disappears, as wave 3 appears at the back; this trend canbe followed through time, revealing the same phenomenon in the diagram.





Waves and Tides 203

An important point here is that the longer-wavelength, faster-movingswells often arrive on coasts that are hundreds of miles away from the stormsthat created them, and they always precede the arrival of the storms that cre-ate them! This is why we see big surf on some beaches far removed from, orprior to the arrival of, a hurricane or other large ocean storm. These are the

big, well-formed waves that surfers dream of.

There is another curious phenomenon at work here as well: within awave train, the wave speed of individual waves is twice as fast as the group

speed. This means that the waves created by a storm far out at sea take twiceas long to arrive on the beach as we would expect them to based on theirindividual wave velocities (FIGURE 6.14). You can observe this phenomenonof the group speed being half the individual wave speed by simply tossinga rock into a calm body of water and watching closely what happens. Asthe waves spread outward, the front wave always disappears—and just

before it disappears, you will notice that it is moving twice as fast as thegroup. At about the same time that the leading wave disappears, a newwave appears at the back of the train to replace it. The new wave seemsto appear magically out of nowhere, but its appearance is simply a way ofconserving energy. You would also notice that as the waves move away

from the disturbance, they get longer and move faster. The important pointshere are: the group speed is slower, by one half, than the individual wavespeeds; and, the farther away a swell propagates, the longer its wavelengthand the faster its speed.

Surf

As waves approach the coast and encounter shallower water where thedepth decreases to less than one-half the wavelength, they begin to feel the

bottom and transition from deep water waves to intermediate water waves.These waves continue to slow down as they approach the shore, losingenergy through friction with the bottom. Thus, the waves following in thewave train will “catch up” with the front waves, and they will all begin topile up (FIGURE 6.15). As the waves slow down, they get steeper: their wave-

lengths shorten and their height increases. Then, when the water depth (Z)is approximately equal to 1.3 times the wave height ( H ) (or, when H /Z = ¾)the wave reaches its critical steepness, and like the white caps we just dis-cussed for conditions offshore, the wave breaks, creating surf . So, if you seea wave breaking on the beach, you can estimate the water depth where the

As a deep water gravity

wave approaches shore it

has a constant wavelength.

1

When water depth shoals to

one-half the wavelength (L /2),

the wave transitions to an

intermediate water wave.

Wave velocity decreases and

wavelength shortens.

2

Other waves behind begin to catch

up with the leading waves, causing

the wavelengths to shorten and

the wave heights to increase.

3 Wave steepness becomes

critical when the depth is

approximately 1.3 × H, the

wave will break as surf.

5

Wave transitions to a shallow

wave when depth is 1/20 the

wavelength.

4

Still water level

L /2

H

FIGURE 6.15 As a deep water waveapproaches shore and the water depthshoals to one-half the wavelength (L/2),the wave will begin to feel the bottomand transition to an intermediate waterwave and then to a shallow water wave.This causes the wave to slow down,which in turn compresses the energy andtherefore increases the wave height (H ) .

Stated another way: as the wave speeddrops, other waves behind begin to catchup with the leading waves, causing thewavelengths to shorten and the waveheights to increase. Eventually, when thebottom depth becomes critical (i.e., whenthe depth is approximately 1.3 × H ), thewave will break as surf.





204 Chapter 6

wave begins to break. For example, a 10-foot wavewill break when the depth is about 13 feet. This iswhy in the old days, ship captains would post alookout to spot breakers as the ship approachedan unknown or uncharted shore. The size of the

breakers (the surf) would give a measure of the

bottom depth and thus allow the lookout to spotshoal areas and reefs.

Wave interference

Waves on the surface of the ocean interact with oneanother and with objects such as shorelines and seawalls in interesting and important ways. This is

because waves are additive. That is, when the crestof one wave propagates past, or intersects, anotherwave, the two intersecting waves produce a newwave, the components of which are the sum of thetwo; the two crests are added to one another, andthe two troughs combine to produce a single trough

that is proportionally deeper.When two waves that are in phase with one an-

other—their crests and troughs coincide—intersect,the result is constructive interference, where a newwave is formed with a new wave height that is the sumof the two intersecting wave heights (FIGURE 6.16).If the waves are out of phase—with crests coincid-ing with troughs—there is destructive interference,which tends to cancel out the two component waves.This constructive and destructive wave interference

explains why you sometimes see waves at the beach comingin sets of several larger waves arriving for a minute or two,for example, followed by a period of relative calm, with only

smaller waves coming ashore. You are seeing the effects ofaddition and subtraction among random groups of waves,perhaps arriving from different points of origin far offshorewhere they were created, and as such they may alternate inand out of phase with one another.

ROGUE WAVES Wave interaction may also explain the exis-tence of rogue, or freak, waves. Rogue waves are huge wavesthat, although rare, are sometimes unexpectedly encountered

by ships at sea (FIGURE 6.17). The causes of these unusualwaves are complex, and we do not understand them wellenough to predict their occurrence, but many are thought to

be the result of constructive wave interference of multiplewaves. Recent research on these phenomena is revealingthat rogue waves occur in both standing wave forms, theresults of interference of multiple waves, and progressivewave forms, about which even less is known.

(B) Resulting wave

(A) Component waves

Constructiveinterference(addition)

Constructiveinterference(addition)

Destructiveinterference(subtraction)

1 2

Wave direction

FIGURE 6.16 (A) Diagram of how two sets of waves of equal waveheights but unequal wavelengths would interfere with one another toproduce a wave that is the sum of the two original waves (B). At point1, the two waves, each propagating to the right, are in phase with oneanother; the crests of each wave coincide, as do the two troughs. Thus,when they interfere, they produce a larger wave that is the sum of thetwo, with twice the wave height of either of the two original waves. Thisis constructive interference. Because the two wave types have differentwavelengths, they eventually propagate to point 2, where they are nowout of phase. Their crests coincide with the other’s troughs, cancel-ing out one another. This is destructive interference. Still later, the twowave sets propagate to a point where they are back in phase with oneanother, again producing waves that are the sum of the two componentwaves.

(A)

(B)

FIGURE 6.17 (A) Photograph taken in the Bay of Biscay, off thecoast of France, of a merchant ship in heavy seas as a rogue wavelooms astern. (B) Photograph taken from the SS Spray in 1986 in theGulf Stream off of Charleston, South Carolina.





Waves and Tides 205

Given the extremely large number of wave-generating phenomena (storms)out in the oceans, and the tendency for them to produce waves that can travelhundreds, even thousands of miles, there is a low, but very real statistical prob-ability of multiple large waves coinciding with one another at the same timeand place. When they do coincide, unusually large waves seem to appear out ofnowhere. The biggest rogue waves are so powerful they can break the hulls of

large ships, and in fact, the largest ships are especially vulnerable. Because of theunusually large wave heights and long wavelengths of rogue waves, large shipscan find themselves improperly supported as they encounter such phenomena,sometimes resulting in their hulls cracking.1 It has been reported that one totwo large ships a week (that’s 50–100 ships every year!) sink under mysteriouscircumstances, and of those, about 10% are supertankers.2

Some rogue waves are really quite common—the smaller ones, that is. Forexample, look out the window if you happen to be on airliner crossing theocean. The broadly-spaced patches of whitewater (e.g., foam from breakingwaves) you see on the ocean surface some six to seven miles beneath the planeare often the result of breaking rogue waves.

Most photographs taken of rogue waves at sea (see Figure 6.17) are of poorquality, for obvious reasons—there’s not much time to think about photography

when one approaches your ship! An eye-witness account of the wave hittingthe SS Spray in Figure 6.17 follows:

A substantial gale was moving across Long Island, sending a very long swelldown our way, meeting the Gulf Stream. We saw several rogue waves dur-ing the late morning on the horizon, but thought they were whales jumping.It was actually a nice day with light breezes and no significant sea. Only thevery long swell, of about 15 feet high and probably 600 to 1000 feet long.This one hit us at the change of the watch at about noon. The photographerwas an engineer (name forgotten), and this was the last photo on his rollof film. We were on the wing of the bridge, with a height of eye of 56 feet,and this wave broke over our heads. This shot was taken as we were div-ing down off the face of the second of a set of three waves, so the ship justkept falling into the trough, which just kept opening up under us. It bent the

foremast (shown) back about 20 degrees, tore the foreword firefighting sta-tion (also shown) off the deck (rails, monitor, platform and all) and threw itagainst the face of the house. It also bent all the catwalks back severely. Laterthat night, about 19:30, another wave hit the after house, hitting the stackand sending solid water down into the engine room through the forced draft

blower intakes.3

SEA WALLS Less monstrous, but still destructive examples of wave interferenceoccur when waves reflect off barriers such as sea walls. Sea walls are walls ofrocks intended to protect shore-side homes from damaging waves during astorm as well as to prevent shoreline erosion from storm waves. However, theysometimes promote wave interference by reflecting incoming waves back outto sea, where they meet and combine with their sister waves coming in fromoffshore. When the crests of incoming and outgoing waves combine, we getlarge standing waves that often break and create additional turbulence that

1 If a rogue wave elevates the midsection of a large ship, leaving the bow and stern nearlyout of the water, the ship’s hull may crack in the middle; the same result ensues if the shipencounters a deep trough such that only the bow and stern are properly supported, with themidsection nearly out of the water. Large ships are most vulnerable because they are morelikely to be similar in length to the wave length, and they are simply not designed to withstandsuch unusual stresses.2 Source: British Broadcasting System.3 Account by Captain G. A. Chase when aboard the SS Spray in 1986, personal communicationwith author. See also Rainey, R. C. T. 2002. 17th International Workshop on Water Waves andFloating Bodies (Peterhouse, Cambridge, England).





206 Chapter 6

resuspends beach sand, thus contributing to additional storm erosion (FIGURE

6.18). Sea walls will protect your house, but they may also help to wash awaythe beach in front of your house.

Wave refraction and diffraction, and longshore currents

Like the seismic waves we discussed in Chapter 3, ocean waves also bend, orrefract, toward an area where the wave speed is slower. As we just discussed,surface gravity waves slow down as they near the shoreline because of fric-tion with the shoaling bottom. Waves that initially approach the shoreline atan angle, as diagrammed in FIGURE 6.19, will be slowed down more on theshallower shoreward side, and thus will bend, or refract, toward that side.This is why waves always seem to arrive on beaches with their fronts nearlyparallel to the shore. An example of this kind of refraction can be seen in Fig-ure 6.19C. Waves will also diffract when they pass by an obstacle such as a

jetty (Figure 6.19B).Wave refraction does not always steer waves such that they arrive 90°, or

perpendicular, to the shoreline (with their wave fronts parallel to the shoreline);

FIGURE 6.18 Sea walls in front of homes onan eroding beach in Southern Maine. Stormwaves can reflect off the sea walls and, withconstructive interference with oncomingwaves, build even larger waves that erodethe beach even further, undercutting anddestroying the sea wall. The remains of twoearlier sea walls that have been destroyed are

indicated by the arrows. This photograph wastaken at about half-tide; at high tide the wateris up against the sea wall, leaving no beachsand exposed, and making the wall vulnerableto erosion.

(A) Wave refraction (B) Wave diffraction

(C)

Shallow

water

inshore

Deep

water

offshore

Waves

propagating

to right

Jetty

FIGURE 6.19 (A) Waves refract toward the shallower water depths wherethe shallow water gravity wave speed is slower; the result is that waves orientthemselves such that they arrive from offshore almost parallel to the shoreline.(B) Viewed from above, waves will diffract around an obstruction, such as a

jetty extending out into the water. As the wave fronts pass the object, they areeffectively cut off on one end, but they continue to propagate from the edge ofthe jetty as circular wave fronts emanating from a point, which results in thewaves effectively curling around the end of the obstruction. (C) Aerial photo-graph of diffraction and refraction.





Waves and Tides 207

often waves arrive at an angle. In such instances, theoblique arrival of waves on a beach can produce alongshore current. A breaking wave that arrives fromdirectly offshore, and not at an angle, washes straightup the beach face (in the swash zone), comes to a stop,and then washes back down the beach face. But wavesthat arrive at an angle wash up the beach face at anangle, and thus there will be an alongshore—along the

beach—component to the water motion, as well as anet transport of water along the beach (FIGURE 6.20).This produces a net current that flows along the shore,not just in the swash zone, but also in the strip of wateradjacent to the beach. You may notice this effect whileswimming at a beach, as you are gradually carriedalong the shoreline. These longshore currents alsotransport sediment that is resuspended by the wavesin the swash zone, often carrying significant loads of

beach sand along the beach.

Seiches

A seiche (pronounced “saysh”) is a form of standingwave that can be caused by a storm surge or simply

by a steady wind blowing toward one end of an en-closed body of water such as a lake (FIGURE 6.21) or

(A)

(B)

Net longshore currentPath of waves in swash zone

along with suspended sand particles

FIGURE 6.20 Waves that arrive atan oblique angle on a beach create alongshore current in the swash zone. (A)Waves arriving at an angle on a beach.(B) The same beach on the same day.The arrows indicate the path of water andsuspended particles washing first up thebeach with an alongshore component,

slowing down as the water curls to theright, before flowing back down the beachface with some complementary along-shore component.

Standing waveNode

One wavelength = 2× lake length

FIGURE 6.21 A seiche in a lake, where the water sloshes back andforth about a node (which is where the depth remains constant),where the water levels rise at one end of the lake while dropping atthe other, back and forth. Notice how the standing wave of this seiche(as all seiches) has a wavelength that is twice the length of the lake.





208 Chapter 6

a semi-enclosed body such an a coastal bay or inlet. The wind, if it is blow-ing at the right speed and direction relative to the size and shape of the bodyof water, can pile water up at the downwind end. When the wind stops, thewater sloshes back as a standing wave. The sloshing will occur at the resonant

frequency of the body of water in question (we discuss resonance in greaterdetail on page 220).

We’ve all experienced this effect of resonance when we slosh a cup of coffeeor a pan of water back and forth, setting in motion a seesaw-like wave. Buta seiche doesn’t have to occur an enclosed basin, like a lake. Seiches also areformed in semi-enclosed bodies of water (harbors, bays, and inlets) with theoutside sea level serving as the outer boundary within which the standingwave oscillates.

Tsunami (or seismic sea waves)

Incorrectly called tidal waves, tsunami (singular,tsunami) are extremely long-wavelength oceanwaves, with wavelengths that can be some 200km (125 miles), but very small wave heights, of-ten less than 1 m. They are usually caused by an

undersea earthquake or landslide that displacesa large volume of sea water, thus disturbing theocean surface and creating a huge wave. The wavepropagates across entire ocean basins at an ex-tremely fast speed. Because the wavelength is solong, it propagates as a shallow water wave: 1/20of a 200 km wavelength is 10,000 m, and the aver-age depth of the oceans is about 4000 m. Thus, thespeed of a tsunami is determined by the depth;as shown in Box 6A, the speed will be equal tothe square root of g (gravity) times h (the depth),or V = 3.1 √ h. For the Pacific Ocean, which has anaverage depth of 4600 m, the speed of a tsunami

is on the order of 210 m/s, or some 470 mph—thisis the speed of a jet airliner.

Tsunami are notoriously destructive (FIGURE

6.22). Because of their small wave height (<1 m)and long wavelength (200 km), they can pass be-neath ships at sea without being noticed; however,they wreak havoc when they come ashore. Likethe waves diagrammed in Figure 6.15, their wave-lengths will shorten and their wave heights willincrease as the bottom shoals, creating wave heightsthat reach several tens of meters high as they comeashore. But just before the first tsunami wave crestarrives on shore, there will be a trough—a verydramatic trough—that will appear to drain thecoastal ocean in a matter of minutes. Tragically,this phenomenon draws curious onlookers, whosoon find themselves facing monster wave creststhat follow each trough.

The great Alaska earthquake of 1964 (see Chap-ter 3) generated a tsunami that propagated sometwo thousand miles across the Pacific Ocean to theHawaiian Islands in only about five hours. Along

FIGURE 6.22 (A) Photograph taken as the first waves from 2004 IndianOcean tsunami came ashore. (B) The destruction left behind in BandaAceh, Indonesia, in early January 2005.

(A)

(B)





Waves and Tides 209

the way, the massive wave passed by ships at sea completely unnoticed, butwhen it arrived at Hawaii, it inundated Hilo on the Big Island with a seriesof huge waves that came ashore.

One of the most powerful and destructive tsunami in history occurredin the Indian Ocean on December 26, 2004. That day a massive earthquakecaused by an extensive subduction event beneath the sea floor off Sumatra

registered a magnitude between 9.1 and 9.3, making it one of the most power-ful earthquakes ever recorded. It created a series of tsunami throughout theIndian Ocean, and was responsible for the deaths of about 250,000 people.Indonesia was hit hardest by the waves, which were estimated at greater than30 m high; Sri Lanka, India, and Thailand also suffered extensive damage andloss of lives (FIGURE 6.23).

The 2004 Indian Ocean Tsunami was detected by both warning centersoperated by NOAA, the Alaska Tsunami Warning System, and the PacificTsunami Warning Center in Hawaii. They responded almost immediately,within minutes, with a limited warning that was revised several hours later,which called for a possible six-foot wave in Hawaii. Hawaii did experience aminor tsunami, but it was only a three-foot wave, which did not cause majorproblems there.

The Indian Ocean tsunami was a wake-up call for the U.S. National Oceanicand Atmospheric Administration (NOAA), and for the U.S. Congress, whichresponded by investing in much-needed improvements to our system ofmonitoring for tsunami. Congress appropriated millions for NOAA to makeupgrades to the warning system, which depends on seismograph stations,sea-level and tide gauges, and a system of 39 tsunami detection buoys in thePacific and Atlantic Oceans. The buoy system has bottom pressure sensorsthat can detect even small changes in bottom pressure that result from thepassage of tsunami.

Most recently, on March 11, 2011, a 9.0 magnitude earthquake struck theisland nation of Japan. With the epicenter just 70 km off Japan’s eastern coast,it is the most powerful earthquake ever recorded in Japan, and is one of thefive most powerful earthquakes ever recorded anywhere. The earthquake

caused extensive damage to buildings and homes, but the worst was yetto come. Minutes after the earthquake, tsunami waves estimated at some40 m in height came ashore, inundating coastal communities and causingextensive damage to a nuclear power plant. Japan has yet to fully recoverfrom this disaster.

1

2

3

4

5

67

89

10

8

910

FIGURE 6.23 Map of the first ten hours of propagation ofthe 2004 tsunami across the Indian Ocean, as simulated bya computer model; the wave continued to propagate beyondthe Indian Ocean and was detected around the world.





210 Chapter 6

Internal waves

An interesting category of ocean waves are internal waves, which are ubiqui-

tous throughout the world ocean, although most often they are not obviousto an observer on the surface. Usually, these are interfacial waves, similarto surface gravity waves, but instead of propagating along the air–waterinterface, these waves propagate along a pycnocline (such as created by athermocline; FIGURE 6.24). Recall from Chapter 4 that a pycnocline is a zoneof rapidly changing water density below the surface. But because the den-sity differences of water on either side of the pycnocline are extremely smallcompared with the difference between water and air, these waves tend tomove much more slowly than surface waves, and appear to move almost inslow motion. You can witness this for yourself by making internal waves ina bottle of salad dressing. For example, if you allow a bottle of oil and vin-egar (vinegar is basically acidic water) to separate into layers, with the oilsitting on top of the vinegar, and then gently jostle the bottle, you can pro-

duce slow-motion waves that are clearly visible at the interface between thelayers. Also, scuba divers may have experienced internal wave phenomenawithout understanding what happened. If a diver is at a depth at or near thethermocline as an internal wave passes, he or she will experience a periodic,abrupt change in temperature of the water as the thermocline oscillates upand down with the passing wave.

As internal waves propagate, they sometimes leave evidence of theirsubsurface presence in peculiar patterns on the surface of the ocean. Thesepatterns result from alternating vertical water motions as the waves’ crests andtroughs pass beneath, pushing water upward and then downward. Deeperwaters rising upward to the surface tend not to support surface waves, andthese features appear as long streaks of calm water, or surface slicks. Whenthose same waters spread horizontally at the surface and eventually pass

beneath adjacent water that is warmer and lighter, foam lines may be formed.Too buoyant to sink, the foam, as well as other floating materials, collects andaccumulates at the surface along the area of convergence.

Tides

Tides are the largest of all ocean waves in terms of their wavelength, andmost of the world ocean experiences them. They are large-scale water mo-tions that result from the gravitational attractions between the Earth and its

Surface waves

Internal waves

Pycnocline

Low-density

water

High-density

water

FIGURE 6.24 Diagram of an internalwave propagating along a pycnocline,separating lower-density water at the sur-face from higher-density water at depth.





Waves and Tides 211

nearest celestial bodies, especially the Sun and the Moon, all of which pullon one another and influence each other’s orbits in space. The gravitationalpull on the oceans produces tidal waves and their associated tidal currents,which result in a massive sloshing about of the oceans as the Earth rotates.As the Earth, Moon, and Sun change positions relative to one another overthe course of a day, a month, a year, and even longer periods of time, the tidal

waves and currents vary in their size and velocity. These variations are theastronomical high and low tides you often hear about, especially when a tropi-cal storm or hurricane looms in the forecast; storm surges are either added toor subtracted from the tides, and coastal flooding can be exacerbated duringastronomical high tides.

The water motions that result from gravitational forces are also influenced bythe shapes and sizes of oceans and seas and their coastlines, which act togetherto produce tides that don’t just go up and down, as most people tend to thinkof them—they create strong horizontal tidal currents. It is the flow of thosetidal currents in response to tidal waves that cause the tide to flood and ebb, orto come in and go out, thus changing local sea level over the course of hours.And those same tidal currents are very important to a lot of oceanographicprocesses, which in turn influence ocean biology, as we’ll see later.

Around the world’s ocean, tides vary greatly in their tidal range—the dif-ference in local sea level between high and low tides—with some parts of theworld experiencing tides that are barely noticeable, while other areas have hugetides (FIGURE 6.25). Tides also vary throughout the world in their periodicity:the tide may come and go once a day, twice a day, or exhibit different propor-tions of each pattern (FIGURE 6.26A). Parts of the world where tides come andgo twice a day have a lunar, or semidiurnal, tide. Such regions are “in tune”4 with the gravitational attraction of the Moon. In other places, there may beone high tide and one low tide each day—this is a solar, or diurnal, tide. Theseareas are more in tune with the gravitational attraction of the Sun. In parts ofthe world where both the Sun and Moon exert significant influence, we havemix of solar and lunar tides; these are called mixed tides.

Much of the west coast of North America experiences mixed tides (FIGURE

6.26B). Notice that Los Angeles, California, experiences approximately two

(A) (B)

FIGURE 6.25 High tide (A) and low tide(B) in the Bay of Fundy, Canada, whichhas the greatest tidal range in the world,exceeding 15 m (50 feet).

4 By being in tune, we are referring to the resonant frequency of a region. This concept ofresonance is explained further below in this chapter.





212 Chapter 6

high tides and two low tides each day (each lunarday, that is),5 but that one of the high tides andone of the low tides is dramatically bigger thanthe other. Diurnal tides dominate in the Gulf ofMexico, as shown for Mobile, Alabama, where thetide comes and goes once a day. The Gulf of Maine

(in Eastport, Maine;FIGURE 6.26C

), on the other hand,has semidiurnal tides, with two high tides and twolow tides each day. But notice that one of the twohigh tides is higher, and one of the two low tideslower, in an alternating pattern. Because of theseinequalities, the Gulf of Maine actually has a mixed,semidiurnal tide, similar to that of Los Angeles, butnowhere near as extreme. Notice also in Figure 6.26that the tidal range in Eastport is on the order of 7m (23 feet), whereas the tidal ranges in the Gulf ofMexico (Alabama; FIGURE 6.26D) are only about 65cm (just over 2 feet). This variation gives us a hint ofthe unequal distribution of tidal energy in different

parts of the world ocean. The tides in the Gulf ofMaine are not only large; they increase toward itsupper reaches in the Bay of Fundy, where we findthe largest tidal ranges in the world (we’ll come

back to the tides in Gulf of Maine and the Bay ofFundy below).

Understanding the forces at work

In order to understand and appreciate the forcesthat create tides, we’re going to rely on yet anotherthought experiment. Our approach is based on whatis generally known as the equilibrium theory of thetides, which was first developed by Isaac Newton

in 1687 as part of his universal theory of gravity.But unlike Newton, we aren’t going to derive themathematical formulation of the tide-generatingforces; instead we’re going to use diagrams andlogical arguments in order to gain a conceptualunderstanding (a more quantitative description isgiven in Box 6B.) In short, Newton showed that thegravitational attraction between two celestial bod-ies is proportional to the product of their massesdivided by the square of the distance between them.That means that more massive objects, like the Sun,have a much stronger gravitational attraction thansmaller bodies, such as the Moon. However, theirgravitational attractions on other objects diminishwith distance. Because the Sun is so much fartheraway from Earth than the Moon, its gravitational

(A)

(B) Los Angeles, California

(C) Eastport, Maine

(D) Mobile, Alabama

0.4

0.2

–0.2

0

T i d a l h e i g h t ( m )

r e l a t i v e t o M S L

8/04 8/05 8/06 8/07 8/08 8/09 8/10 8/11

Date

4.0

2.0

–2.0

0

T i d a l h

e i g h t ( m )


1.5

1.0

0.5

–0.5

–1.0

0

T i d a l h e i g h t ( m

)


60ºN

30º

60ºS

30º

0º

Los Angeles,California

Mobile, Alabama

Eastport,Maine

Semidiurnal tides

Diurnal tidesMixed tides

FIGURE 6.26 (A) Distributions of the types of tides around the world,with examples of each: (B) Los Angeles, with a mixed tide; (C) Eastport,Maine, with a semidiurnal tide; and (D) Mobile, Alabama, with a diurnaltide. Observed tidal heights for the week of August 4–12, 2010, areplotted in meters relative to mean sea level (MSL). 5 Because the Earth is rotating as the Moon orbits it, we need

to introduce the lunar day, also called a tidal day—the time ittakes for the Earth to complete one full rotation on its axis withrespect to the Moon, about 24 hours, 50 minutes, 28 seconds.





Waves and Tides 213

effect on the tides is only about half (46%) that ofthe Moon.

Our thought experiment will examine whathappens to the tides on Earth in response to thegravitational attraction of the Moon, ignoring forthe moment the influence of the Sun. To do so, we

are going to imagine a hypothetical, simplifiedEarth without any continents, and only a thinskin of water covering the entire surface of theplanet (FIGURE 6.27A). The Moon, shown off in thedistance, exerts a gravitational attraction, which,as Newton explained, is inversely proportional toits distance from Earth. The different lengths ofthe vectors in the sketch are intended to show therelative proportions of the strength of the Moon’sgravitational attraction at various positions aroundthe world. It shows that the Earth experiences—everywhere around the world—a gravitationalpull by the Moon in the directions indicated, being

greatest at points on Earth that are closest to theMoon. Notice that the pull of the Moon’s gravityis felt all the way through to the opposite side ofthe world—the presence of the Earth is no shieldagainst gravity. The key point here is that locationson the Earth closest to the moon experience thegreatest gravitational pull.

Given the pull of the Moon’s gravity as dia-grammed in Figure 6.27A, it would logically followthat anything not tied down to the surface of theEarth, such as an ocean, would tend to slide alongthe Earth’s surface in the directions indicated bythe arrows in FIGURE 6.27B. Therefore, that gravi-

tational attraction would tend to pull water such that it flows to one side ofthe spherical Earth; that is, it would flow toward the spot directly under theMoon and pile up, producing a bulge. Thus we would expect a high tide onthe side of the Earth facing the Moon; everywhere else around the Earth, from90° away and beyond, we would have low tides, as shown in the sketch.

We can demonstrate the phenomenon illustrated in Figure 6.27B withanother brief thought experiment. Imagine a bowling ball (call it “Planet 1”)that we have attached to a length of wire. Imagine also that we dip that ballinto a large bucket of latex paint, and then pull it out, leaving it hanging fromthe wire; the thin layer of wet paint coating the ball would constitute Planet1’s ocean. We would see what was initially a uniform layer of wet paint beginto slide across the surface of the bowling ball where it would collect on the

bottom of the ball in response to the gravitational attraction of Earth. That is,the paint would pile up and create a bulge—the paint would be thickest, ordeepest, on the side of the ball facing the Earth; this phenomenon is analogousto our hypothetical high tide in Figure 6.27B.

Based on our brief painted bowling ball thought experiment, it wouldseem that we have confirmed what we expected, based on the argumentsin Figure 6.27: that there must be a single tidal bulge on one side of ourhypothetical Earth that results from the Moon’s gravitational pull of watertoward the point on the Earth closest to the Moon. So far so good, exceptthat this is wrong. Here on the real Earth, there are actually two tidal bulges

High tide?

Low tide? To Moon

(A)

(B)

Gravitational attractionof the Moon

Earth

Moon

FIGURE 6.27 (A) Sketch of an imaginary Earth with all the continentsremoved and a single, uniform ocean covering its surface. Vectors rep-resent the relative pull of the Moon’s gravity, which is greatest at pointsclosest to the Moon. (B) Sketch of the tidal currents on the surface of ahypothetical Earth that might be expected to result from just the gravita-tional attraction of the Moon as in (A). We would expect a flow of watersaround the surface of the Earth to a point in line with the Moon, wherethey pile up, creating a high tide. But this is not what actually happens,as explained in the text.





214 Chapter 6

attributable to the pull of the Moon. In reality, thereis another high tide on the opposite side of the globe(FIGURE 6.28), due to a second set of forces on thesurface of the Earth, which we explain next.

To understand why there is this second set offorces, we need to appreciate the fact that not only

does the Moon orbit the Earth, but the Moon andEarth orbit each other, around a point that is thecenter of mass of the Earth–Moon pair. Continuingour original thought experiment: If the Earth andMoon were the same size and mass, they would orbitaround one another the way a dumbbell would spinon its handle, as shown in FIGURE 6.29A. Becausethe mass of each of the two balls on the ends of thehandle is the same, the balance point (the fulcrum)is exactly halfway between them. Imagine how thedumbbell might rotate around that center of gravity,thus mimicking two planets of identical mass that

orbit around one another. As the two planets of equal mass orbit one another,

they are held in place by their mutual gravitational attraction, which exactly balances the centrifugal force pulling them outward, indicated by vectorsof equal length in Figure 6.29B (see Appendix A). But if we make one of theplanets smaller (now we will call it a “moon”), the balance point—the centerof gravity—moves closer to the larger ball (Figure 6.29C). Eventually, if thesmaller ball (the moon) gets small enough, the balance point, the center ofmass, will be located beneath the surface of the larger ball. This is actuallythe case for the Earth–Moon pair. This is illustrated in the sketch in Figure6.29D, where the fulcrum point is set at the center of mass of the Earth–Moonpair, 2903 miles away from the exact center of the Earth. In other words,the balance point is about 1000 miles, or one-eighth of the Earth’s diameter,

beneath the Earth’s surface.This means that as the Moon orbits the Earth, the Earth orbits the Moon as

well, but it isn’t as noticeable because, rather than tracing out a large orbitalcircle as the Moon does, the Earth orbits a much tighter circle around the center

High tide

High tide

Low tide

Low tide

To Moon

FIGURE 6.28 The directions of water motions on the surface of thereal Earth under the influence of the Moon’s gravitational attraction.Unlike our hypothetical Earth in Figure 6.27, there are actually twoidentical but opposite forces pulling water to opposite sides of theEarth, the result of which produces two high tides and two low tides.

(A)

(B)

(C)

(D)

Earth

Earth

Moon Earth

Moon

Earth

Moon

CF g

To Moon

FIGURE 6.29 (A) Sketch of a dumbbellwith two weights of equal mass. Theywould balance at a point in the middle ofthe handle, such that a fulcrum placedthere would allow the dumbbell to spinaround that point. The spinning dumbbellis analogous to an “Earth–Moon pair” inwhich both bodies are of the same mass.The dashed arrows show the orbital pathof the two. (B) As the Earth and Moonorbit one another around their centerof mass, they would each experiencea gravitation attraction (g ) to the othermass and a centrifugal force (CF) equal

but opposite to that force, as shownhere for just the Earth. (C) Reducing themass of the Moon causes the balancepoint to move closer to the Earth. (D) Theactual Earth–Moon pair has a center ofmass that is beneath the surface of theEarth. Even though the center of mass ofthe Earth–Moon pair is no longer out inspace between them, they each experi-ence a centrifugal force that is equal tothe gravitational attraction by the othermass, as in (B).





Waves and Tides 215

of mass deep inside itself, and wobbles (FIGURE 6.30).But the Earth still has a centrifugal force as a result ofthat wobble, and that centrifugal force is still exactlyequal to the Moon’s average gravitational attraction.6

Therefore, the entire Earth and everything on it experi-ences a centrifugal force as the Earth wobbles aroundthe center of mass of the Earth–Moon system, whichis equal in magnitude, but opposite in direction, to theaverage force of the Moon’s gravitational attraction.

These two forces, the Moon’s gravitational attrac-tion and centrifugal force, are equal to one anotheronly along a plane that runs through the center of the

Earth and as an average for the whole Earth (FIGURE6.31). Everywhere else on Earth the two forces arenot equal, and it is this inequality that creates thetwo tidal bulges. This is because while the centrifugalforce on Earth is the same everywhere—directly underthe Moon, on the opposite side of the Earth from theMoon, anywhere—the Moon’s gravitational attrac-tion is not (BOX 6B). It varies with the distance to theMoon, which, obviously, isn’t the same everywhereon the surface of the Earth. On the side of the Earthnearest the Moon, the Moon’s gravitational attrac-tion is greater than the Earth’s centrifugal force; onthe side of the Earth opposite the Moon, the Moon’s

gravitational attraction is less than centrifugal force.But, again, on average for the whole planet Earth,they both are equal, which must be the case in orderfor them to orbit one another.

This inequality in forces on opposite sides of Earthcreates the tidal flows in Figure 6.28 that form thetwo identical bulges—two high tides—on oppositesides of the Earth. As the Earth rotates once eachday, the two bulges (the two high tides) will always

be on the side of the Earth toward the Moon and onthe opposite side away from the Moon. Therefore,the rotating Earth essentially slides beneath each ofthe two bulges once every day (every lunar day), as

(A)

(C)

(B)

Earth Moon

CF

g

Fulcrum

CF

CF

g

FIGURE 6.30 (A,B) As the Moon orbits around the center ofmass of the Earth–Moon pair, the Earth orbits around the samepoint, which is beneath the surface of the Earth. Earth’s orbitaround this point, then, will appear as a wobble (C) that will cre-ate a centrifugal force (CF) that is equal to but opposite to theaverage gravitational attraction (g ) of the Moon.

To MoonCF g

Net force Net force

CF g CF g

6 The point inside the Earth that is the center of mass of the Earth–Moonsystem is not a stationary, or fixed, point inside the Earth; it essentiallymoves through the Earth as the Moon orbits such that it is always

beneath the surface of the Earth that is closest to the Moon. This isan important detail because it influences the way the planet wobblesaround the point: it is as if you were to hold firmly a basketball with

both hands, and then move it in circles, like the “Earth” in Figure6.29C. This is not the same as drilling a hole off-center in the basketball,inserting a stick, and then spinning the basketball around the stick.

FIGURE 6.31 Diagram of the relative importance of the Moon’sgravitational attraction (g , red arrows) and the Earth’s centrifugalforce (CF, blue arrows). CF is the same everywhere on the surfaceof the Earth, as indicated by the blue arrows of equal length, andin the center of the Earth, CF and g are equal. But, because loca-tions on Earth on the side of the Moon are closer to the Moon, theyexperience a greater g , and so g exceeds CF there. On the oppositeside from the Moon, the greater distance from the Moon results ina smaller g , and so CF is greater than g there. The net forces thatresult are indicated by the black arrows. These forces are explainedfurther in Appendix A.





216 Chapter 6

BOX 6B The Tide-Generating Forces

I

saac Newton showed that the

gravitational attraction between two

bodies is directly proportional to

their masses, M , (more mass meansgreater gravitational attraction), and

inversely proportional to the square of

the distance between them, d , (greater

distance means less attraction). As-

signing Earth’s mass a value of 1,

then the gravitational attraction of the

Moon, g , at the center of mass is pro-

portional to

This means that the net force is

In that equation, G is the gravitational

constant. Because the Earth and

Moon orbit one another around their

common center of mass, the Earth

and the Moon experience a centrifu-

gal force (CF), which at the center

of the Earth, is equal to the Moon’s

gravitational attraction (g ), as shown in

FIGURE A. The variable r is the radius

of the Earth.

CF is the same everywhere on the

Earth, as in FIGURE B, but g is not

everywhere the same. It varies accord-

ing the distance from the center of theMoon. At Position 1, which is the clos-

est point to the Moon, g is greater than

CF. The Moon’s gravitational attraction,

g , is therefore proportional to:

The Earth’s CF is proportional to

The sum of the two forces, CF and g ,

gives the net force at Position 1 as

proportional to:

Using the same logic, the resulting

force at Position 2, on the opposite

side of the Earth, is proportional to:

At positions along the vertical dashed

line, which runs through the Earth’s

center of mass, CF and g are equal and

are proportional to:

However, g is directed along the line

indicated toward the center of mass

of the Moon, and so it has a small

component directed toward the centerof the Earth.

Forces at other points can be deter-

mined trigonometrically. The result-

ing relative forces are illustrated in

FIGURE C, and are responsible for the

tidal currents illustrated in Figure 6.28.

Figure A

Earth (mass=1)

Moon (mass=m)

CF g

r

d

d–r

Figure B

Moon

CF g CF g

On side awayfrom moon, gis less than CF

On side towardmoon, g isgreater than CF

At center of mass, CF and g are equal

Position 1

Position 2CF

g

CF g

Figure C

Moon

Resulting tractiveforces

2

1 2F G

M M

d

=

2

2 2 3( )−

− =

m

d r

m

d

mr

d

2

2 2 3( )+

− =−m

d r

m

d

mr

d

2

m

d

2 2

1 2 M M

d

m

d=

2

m

d r( )−

2

m

d−





Waves and Tides 217

illustrated in the sketch in FIGURE 6.32. (We should note here that the actualtidal heights that result will be related to the angle between the Earth’s orbitalaxis and the Moon’s orbital plane; this is an additional complication, but itis a relatively minor one that we don’t need to consider in this discussionof the basics of tides.)

Combined influences of the Sun and Moon

While the strongest force producing the Earth’s tides is the Moon’s gravity,which is about twice as strong as that of the Sun, the Sun nonetheless exertsa significant gravitational force. The Earth’s tides are therefore the result ofa combination of gravitational attraction by the Sun and Moon, such thatwater flows toward each of them as they and the Earth fly through spacerelative to one another. Once we add the gravitational pull of the Sun to oursimple conceptual model of the lunar tides, and incorporate a considerationof the orbital dynamics of both the Moon and the Sun, the tides get muchmore complicated.

The Sun and Moon don’t “go around” the Earth at same rate. The Sun“goes around” once a day, while the Moon “goes around” about once ev-ery month. (Of course, the Moon orbits the Earth and they both orbit the

Sun.) This means that there are important tidal phenomena that occur atfortnightly (bi-weekly) intervals in relation to the phases of the Moon. Theorbital alignments of the Sun–Earth–Moon system are lined up in a straightline twice a month—when we have a full moon and when we have a newmoon (FIGURE 6.33A). Each results in the gravitational attractions of the Sunand Moon effectively working together. During a full Moon, for example,the Moon pulls in a direction that is 180° opposite from the pull of the Sun,thus reinforcing the two bulges on opposite sides of the Earth. Likewise, theircentrifugal forces are also directed 180° from one another, also reinforcingthe double bulges. During a new Moon, the effect is asif the Moon and Sun were welded together into a single,larger body. In each case, the gravitational attractionsand centrifugal forces are additive, which gives us tides

that are significantly larger than average during a fullor a new Moon. That is, the bulges of water—the hightides—are higher, which means that the low tides must

be lower. These are called spring tides.

(A) Spring tides (B) Neap tides

Extremelow tide

Extremelow tide

NewMoon

FullMoon

Sun

E x t r e m e h i g h

t i d e

E x

t r e m e h i gh t i d e

W a x i n

g

W a

n i n g

W a x i n g

W a

n i n

g

Modestlow tide

Modestlow tide

First quarter

Third quarter

Sun

M o d e s t h i g h t i d

eM

o d e s t h i gh t i d e

W a x i

n g

W a x i n

g

W a

n i n

g W

a n i n g

FIGURE 6.33 (A) Weekly orientations of the Sun, Earth, and Moonorbital system with phases of the Moon indicated. Twice a month,the Sun, Earth, and Moon line up, giving us a full moon and a newmoon. In those instances, the gravitational attractions reinforce

each other such that we see spring tides—high tides are higherand low tides are lower than average. (B) When the Sun, Earth,and Moon are oriented at right angles, their gravitation forces areperpendicular and there are no additive effects; these are whenwe have the first quarter and third quarter phases of the Moon,which correspond with the neap tides—the lowest high tides of themonth and the highest low tides of the month.

FIGURE 6.32 An idealized Earth rotat-ing beneath an ocean without continentswould have two high tides and two lowtides. As the Earth rotates an imaginaryperson standing on it would experience

two high tides and two low tides eachlunar day. That is, the tides aren’t movingin this conceptual model; the Earth ismoving under them.





218 Chapter 6

Using the same logic, we will also see twice a month the Sun, Moon, andEarth oriented at right angles to one another, which is when we have “halfmoons” (also known as the first and third quarter phases of the Moon; FIGURE

6.33B). When this happens, the gravitational forces are not additive; becausethey are at right angles, they have no effect on one another. But that orientationdoes make the tides a bit more complicated. Because the gravitational forcesare working at right angles, we have the Sun (or Moon) pulling the tide toone side of the Earth, while the Moon (or Sun) pulls the tides to a point 90°away. Twice a month, then, we have minimal tidal ranges, when high tidesare not very high and low tides are not very low. These are called neap tides.

An example of this spring–neap cycle over the course of a month can be seenin the sampling of tide records for Eastport, Maine, and Boston, Massachusetts,in 2010 (FIGURE 6.34). Notice that the fortnightly pattern of spring and neap tidesis apparent for both ports, with spring tides on about July 14 and 28, and neaptides on July 5 and 19. In addition, notice how the tides are mixed semidiurnal,with slightly different levels for successive high tides and low tides. Finally, noticethat the tidal ranges are markedly different from one another; the tidal rangesin Eastport are twice those in Boston. The reasons for differences in tidal rangessuch as these are related to the shapes of ocean basins and their coastal areas.

Tides in ocean basins

Our thought experiment on the forces controlling the tides has produced aconceptual model of what we might expect on our planet if there were nocontinents and only one big ocean. Once we add the continents, the Earth’soceans are no longer just a wet surface on a smooth ball; there are now obstaclesthat prevent the easy flow of tidal currents. The addition of the continents

Spring tides

Neap tides

(A) Eastport, Maine

(B) Boston, Massachusetts

4.0

3.0

2.0

1.0

–1.0

–2.0

–3.0

–4.0

0



4.0

3.0

2.0

1.0

–1.0

–2.0

–3.0

–4.0

0


r e l a t i v e t o

M S L

07/02 07/07 07/12 7/287/237/17

Date

FIGURE 6.34 Observed tides recordedat (A) Eastport, Maine, and (B) Boston,Massachusetts, for the month of July2010. The tidal height relative to meansea level (MSL) is plotted; high tides aregiven as meters above MSL, and low tideas meters below MSL.





Waves and Tides 219

and consideration of bottom friction as it impedes tidal currents produces aninteresting variation in the nature of tides.

Because of the Earth’s rotation, the tides respond to the Coriolis effect and donot flow in straight lines; instead they flow as waves that tend to slosh aroundthe insides of embayments (bays), both large and small. As a flooding tide (orrotary tidal wave) enters a coastal embayment, the tide enters as a wave that

tends to hug the right side of the bay, keeping the shoreline to its right as itrotates around the basin (FIGURE 6.35). (In the Southern Hemisphere, rotarytidal waves rotate in the opposite direction). Along the coastal edges of the

bay, the changes in depth between the crest and troughs of the tidal wave aregreatest, and toward the open end of the bay there is a point where there is nochange in water depths—this is known as an amphidromic point.

This phenomenon of a rotary tidal wave is easy to visualize or to demonstrateto yourself in your kitchen (for an enclosed body of water, not a semi-enclosed

bay). For example, if you hold a pan of water—an ordinary dishpan, preferablya round one for this experiment—and then swirl it around just right, you cancreate a very dramatic and well-formed wave that rotates around the edgesof the pan (Figure 6.35B). As that wave rotates, you will see the water depthsat the edges of the pan increase and decrease as the crests and troughs pass

(B)

(A)

(C)

1. Tidal wave

floods bay

Open endof bay

4.

AP

AP

3.

2. Depth at amphidromicpoint (AP) remains

unchanged

AP

Amphidromic point

High tide

Half tide

Low tide

0 hour11 hours

10 hours

9 hours

8 hours

7 hours

1 hour

2 hours

3 hours

4 hours

5 hours6 hours

3 m

2 . 5

m

2 m

1 m

1 m

FIGURE 6.35 (A) Diagram of a tidalwave entering a bay that is open to theocean at one end (1). In the NorthernHemisphere the incoming wave will hugthe edges of the bay (2, 3), keeping thebay to the right of the wave’s progres-sion as it rotates around an amphidromicpoint at the opening of the bay (4). (B)A person swirling a dishpan of water

just right can make a wave that rotatesaround the edges of the pan. The waterdepth in the very center of the pan willremain unchanged, while water depth atthe edges of the pan will increase anddecrease as the wave’s crest and troughpass by. This is analogous to a tidal wave

that rotates around an ocean basin. (C)A hypothetical ocean basin diagrammedlooking down from above as well as incross section. The solid lines are co-tidallines that mark the position of the crestof the rotary tidal wave each hour intothe lunar tide. The dashed concentriccircles give the tidal heights in meters(tidal height is the depth dif ferences be-tween high and low tide) across the basinfrom the center to the outer edge. Thereis no change in water depth in the center,which is the amphidromic point.





220 Chapter 6

by; these are analogous to high and low tides. And inthe center of the pan you will notice an amphidromicpoint, where the water depth doesn’t change. In anocean basin (versus a dishpan) we would see thetidal wave rotate similarly as in Figure 6.35A, butin this case, for the lunar tide. Figure 6.35C shows

a cross section of how the water depths change asthe tidal wave completes one cycle, and also a planview (Figure 6.35C), with locations of the crest of thetidal wave for each hour into the lunar tidal cycle;also shown in that diagram are concentric circles thatrepresent approximate tidal wave heights, from zeroat the amphidromic point to a meter or so aroundthe edges. These rotary tidal currents and their am-phidromic points are found throughout the worldocean (FIGURE 6.36).

It is this complication resulting from the pres-ence of shorelines, and the relative size scales ofthose coastal features, that help to alter the relative

importance of the gravitational forces from the Sunand Moon, thus giving us the various tide types(semi-diurnal, diurnal, and mixed tides) that arefound around the world (see Figure 6.26).

Tides in the Gulf of Maine and the Bay of Fundy

The rotary tidal wave in the North Atlantic Ocean rotates counterclockwise,thus forming a tidal wave that propagates down the east coast of NorthAmerica, from north to south (FIGURE 6.37A). When this North Atlantictidal wave crest is sitting outside the Gulf of Maine (and other coastal bays,harbors, and estuaries on the east coast of North America) the sea level will

be higher outside, farther offshore in the Atlantic Ocean, than inshore andinside the Gulf of Maine, thus creating a flow of water from high sea level

12

4 5

8

12

4

5

3

87

9

9

10

11

8

7

1011

6

2

45

3

768

9

10

11

1 24 5

3 8

76

9

1011

9

1011

1

2

4 5

3

8

76

1011

9

1

24

5

3

6

9

8

7 6

9 7

6

12

45

310

111011

1 29

10111

2 3

12

4 5

8

1

2

4

5

3

87

9

9

10

11

8

7

1011

6

2

45

3

768

9

10

11

1 24 5

3 8

76

9

1011

9

1011

1

2

4 5

3

8

76

1011

9

1

24

5

3

6

9

8

7 6

9 7

6

12

45

310

111011

1 29

10111

2 3

FIGURE 6.36 Amphidromic points (red points) in the world ocean,along with co-tidal phase lines, which approximate the location of thecrest of the tidal wave for each hour into the 12 hour lunar tidal cycle.

Height of wave (m)

Tidal wave crest location

(A) (B)

New York

Boston

Eastport

Bay of Fundy

N o r t h A

t l a n t i c

t i d a l w a v e

New York

Tangier

Lisbon

Brest

London

1. 0 m

0 . 6

m

0 . 2 m

0.2 m

0.6 m

1.0 m

0 h

2 h1 h

3 h

4 h5 h

6 h7 h8 h9 h

10 h

11 h12 h

12 h10 h

9 h

9 h

8 h

FIGURE 6.37 (A) The co-tidal phaselines (red), approximate locations of thecrest of the tidal wave each hour intothe lunar tidal cycle around the NorthAmerican amphidromic point. The co-range lines (blue) give the tidal height inmeters, and the black arrows indicate thecounterclockwise movement of the tidalwave crest. (B) Currents in the Gulf ofMaine and Bay of Fundy.





Waves and Tides 221

toward low sea level (downhill); this creates a tidal current that floods theGulf (FIGURE 6.37B). Those tidal currents will continue to flow into the Gulfof Maine until the trough of the tidal wave moves into position outsidethe Gulf, at which time the Gulf drains. It turns out that this time period

between successive Atlantic tidal wave crests and troughs in the NorthAtlantic Ocean is about the same as the resonant frequency of the Gulf of

Maine—about 12.4 hours, or half a lunar day, the time between successivelunar high tides. And this is one of the factors—a major one—that producesthe biggest tides in the world in the upper reaches of the Gulf of Maine, inthe Bay of Fundy.

As we saw in Figure 6.34, the tides in the Gulf of Maine are quite large, withspring tides ranging from about 4 m in Boston, Massachusetts (the westernGulf), to more than 7 m in Eastport, Maine (the eastern Gulf). But the biggesttides in the world are still farther to the east in the upper reaches of the Bayof Fundy at the extreme eastern end of the Gulf of Maine (FIGURE 6.38A). Thereasons for these exceptional tides are two-fold: first, as just mentioned, theresonant frequency of the Gulf of Maine is very close to that of the lunar tide.Second, the Bay of Fundy’s location within the Gulf of Maine and its narrow-ing shape focus the tides, as we explain next.

We are all familiar with the principle of resonant frequencies, but mostof us don’t think about it very often. Examples include the tone that can

be achieved when we blow air across the top of an empty bottle at just theright speed—we get a standing wave pattern in the bottle that produces atone unique to that bottle. It resonates at the bottle’s resonant frequency.We can also often get a standing wave resonating in a coffee cup—usuallyunintentionally, spilling the coffee (FIGURE 6.38B). That frequency is some-thing close to 0.2 seconds. And, of course, who hasn’t made dramatic waveswhen sitting in a bathtub, forcing a wave that sloshes back and forth, at afrequency of about 1.5 seconds? OK, maybe you haven’t; if that’s the case,then try this little experiment (but do it standing up in the tub; you’ll get

better results). First, bend over and move your hands forward and backwardthrough the water, at a fairly fast rate. You’ll find that all you have done is

make a mess by splashing water everywhere. Then do the same but moveyour hands very slowly; if you do this right, all you will see is your handsdragging through the water, with no waves or splashing. But if you move

(B)(A)

(C)

The Gulf

of Maine

Georges

Bank

Browns

Bank

Minas BasinChignecto Bay

FIGURE 6.38 (A) The Gulf of Maine.The undersea features, Georges Bankand Browns Bank, make the Gulf semi-enclosed. (B) The resonant frequencyof a coffee cup is about 0.2 second; it

is short because the cup is so small.(C) The larger bathtub has a frequencyof about 1.5 seconds; this is still quiteshort. The much larger Gulf of Maine (A),while not a well enclosed container likethe coffee cup or bath tub, has bottomfeatures and offshore banks that makeit semi-enclosed (as indicated by thedashed line). The resonant frequencyof the Gulf is estimated at about 12.4hours—which is very close to the fre-quency of the North Atlantic tidal wave.





222 Chapter 6

your hands back and forth at just the right frequency—about 1.5 seconds,the resonant frequency for the tub—you will get a very nice standing wavesloshing back and forth in the tub, as in FIGURE 6.38C, with the water risingup on one end and then the other (small children perform this experimentmasterfully, by the way).

A similar phenomenon happens in the Gulf of Maine. The North Atlantic

tidal wave is just about the right frequency such that no sooner has the highersea level offshore from a passing tidal wave crest filled the Gulf of Maine,which takes some time, than the trough of the tidal wave passes by, with justenough time to drain the Gulf. These successive crests and troughs are at theright frequency that they reinforce the successive filling and draining of theGulf, analogous to our hand motions in the bath tub experiment, thus creatingfairly large tides in the Gulf of Maine.

But why are the tides in the Bay of Fundy so much bigger than tides inother places? The answer is in the bay’s shape and its position inside the Gulfof Maine. For example, looking back at the sketch of the bathtub in Figure6.38, imagine that the tub is made of soft modeling clay, such that we couldpinch one end of the tub into a narrow point. This would be analogous to theshape of the upper reaches of the Bay of Fundy, with its two pointed bays,

Chignecto Bay and Minas Basin. Then, when the next wave crest arrives atthat end of our clay bathtub, you can imagine what will happen: the wavewill focus water up the narrowing wedge, reaching a very high water levelat the far end, and perhaps even shooting water out that end of the tub. This

basically is what happens in the Bay of Fundy.

The importance of tides

Tidal motions on Earth represent huge amounts of energy. In fact, the en-ergy dissipation associated with those tidal flows rubbing against the ocean

bottom is slowing down the rotation of the Earth. A few hundred millionyears ago (380 mya), there were about 400 days in a year—the Earth rotated400 times for each complete orbit of the Sun—and a day was only about 22hours long. Today the Earth’s rotation has slowed such that it takes 24 hours

to rotate once, and it completes only 365 rotations for each orbit aroundthe Sun. The effect the tides have on the Earth’s rotation is a lot like whathappens to a baseball in flight when its cover is partially torn off; the ad-ditional friction will slow the ball’s rotation. Similarly, the constant frictionof tides rubbing against the ocean floor is progressively slowing down theEarth’s rotation.

TIDAL POWER DEVELOPMENT As oil reserves continue to deplete and asworld oil markets keep raising the price of a barrel of oil, a number ofcompanies are looking very closely at harnessing tidal power. The kineticenergy associated with the tides, which is simply lost in slowing the Earth’srotation, is the target of their efforts. The idea is simple: by placing a turbinein the flow of tidal currents, it is possible to extract energy for use in gen-erating electricity. There are a number of technical variations of this basicprinciple. For example, one major tidal power station in operation today isthe 240 megawatt (MW) facility in St. Malo, France, on the Rance Estuary,which was built in 1966. There are also several experimental facilities, suchas the 20 MW unit at Annapolis Royal in the Bay of Fundy, Canada, builtin 1984. These two examples were built with dams that capture the tidalflow on one or both tides (flood and ebb). But dams are not the only ideas:tidal turbines are also being proposed in various configurations throughoutregions where there are sufficient tidal current speeds. Areas of the worldocean where tidal power development is considered feasible require a





Waves and Tides 223

minimum 4 m tidal range which, it turns out, isquite common (FIGURE 6.39).

TIDAL MIXING Tidal currents are important to the bi-ology of the ocean in a number of ways. The currentsare important, of course, in transporting plankton andsmaller organisms, especially in their early develop-mental stages. But tidal currents are also one of themain reasons why some coastal areas are very pro-ductive biologically: tidal currents create tidal mixing.

Tidal currents run from the surface to the bot-tom—they encompass the entire water column.

Because tidal currents extend to the bottom, theyrub along the bottom, and in the process, they createa turbulent mixing action. Rivers are analogous:they also flow from top to bottom, and where theriver bottom is shallow enough, the river currentspeed swift enough, and the bottom rough enough,we often see rapids, or whitewater. Vertical mixingthere extends from the bottom all the way to thesurface, giving us the visible turbulence.

Likewise, where coastal waters are shallow enough, we sometimes seetidal currents that completely mix the waters from the bottom to the surface,homogenizing both the temperature and salinity (and hence, the density). Thetidal mixing process is illustrated in FIGURE 6.40. The arrows represent turbulent

mixing currents that extend upward from the bottom to a fixed distance from the bottom. That distance is determined by the tidal current speeds and the bottomroughness. For example, inshore waters that are shallow enough for mixing toextend to the surface, as in the left side of Figure 6.40, will not exhibit a thermo-cline; despite the Sun’s rays warming of the surface waters, the waters stay wellmixed. Instead, heated surface waters are simply mixed with the colder watersat depth, keeping the surface waters from becoming very warm. But fartheroffshore, the waters become too deep for this to happen; the tidal mixing doesnot extend far enough up off the bottom, and surface heating is unaffected. Herewe routinely see warm surface waters separated fromcolder deeper waters by a thermocline. The overallpattern is one of cool surface water temperatures ininshore areas, warm water at the surface offshore (atthe surface above the thermocline), and cold waterson the bottom offshore. It is often in these tidallymixed areas that deep nutrient-rich waters are mixedupward, stimulating biological productivity—thesubject of our next chapter.

Target sites

1. Siberia2. Inchon, Korea3. Hangchow, China4. Hall Point, Australia5. New Zealand6. Anchorage, Alaska7. Panama8. Chile9. Punta Loyola, Argentina

11. Bay of Fundy12. Frobisher Bay, Canada13. Wales, UK 14. Antwerp, Belgium15. Le Havre, France16. Guinea17. Gujarat, India18. Burma19. Semzha River, Russia20. Colorado River, Mexico21. Madagascar

1

23

4

5

6

7

8

9

10

11

12 1314

15

16

17

18

19

20

21

10. Brazil

Cool near-shore water

Warm surface water

FIGURE 6.40 Schematic diagram of tidal mixing in coastalwaters, creating a mixed zone against the coast of coolerwaters. In deeper water, the surface waters offshore arewarm, too far removed from the bottom to be mixed.

FIGURE 6.39 Areas of the world ocean where tidal energyis considered sufficient to make tidal power developmentfeasible. Notice that all are in coastal and inshore areas,where tidal current speeds are greatest, as modified byshoaling depths and a concentration of those flows, incontrast to the open ocean.





224 Chapter 6

Chapter Summary

Waves have a crest and a trough. The horizontal

distance between one crest and the next, or be-

tween one trough and the next, is a wavelength.

The period is the time it takes for one completewave cycle, usually measured as the time be-

tween two successive crests or troughs.

The displacement of the sea surface created by a

wave can be measured as the wave height, which

is the vertical distance from the top of a crest to

the bottom of a trough, or the wave amplitude, the

vertical distance that the sea surface is displaced

from a position at rest. The wave’s amplitude is

equal to one-half the wave height.

Waves are the result of a disturbing force and a

restoring force. One disturbing force is the wind,

which gives us wind waves; one restoring force is

gravity.

As wind blows and imparts energy to waves,

waves grow in length and height, but they grow

faster in height than in length. Eventually a criti-

cal steepness value of the ratio of wave height to

wavelength is reached (when H :L = 1:7) and the

wave breaks, forming whitecaps.

It is generally true that surface waves move en-

ergy, not mass, but there is an exception: Stokes

Drift, which is what brings flotsam to shore.

Wind waves begin as small capillary waves, with

a wavelength less than 1.73 cm, for which the

restoring force is the water’s surface tension.Once formed, and as the wind continues to blow

and their wavelength increases beyond 1.73 cm,

the waves transition from capillary waves to sur-

face gravity waves, for which the restoring force is

gravity.

There are two categories of surface gravity waves:

shallow water waves and deep water waves. These

waves differ in the relative importance of water

depth and wavelength in controlling the speed

they move across the sea surface.

Wave speed, or velocity, is determined by the

wavelength for deep water waves, for which the

depth is greater than 1/2 their wavelength; by thewater depth for shallow water waves, for which

the water depth is less than about 1/20 of their

wavelength; and by both factors for intermediate

water waves.

Surf results when surface gravity waves come into

shoaling water depths; the waves lose speed due

to bottom friction in the shallower water, whichshortens the wavelengths while increasing the

wave heights. Eventually, the waves reach their

critical steepness, become unstable, and break

as surf, usually when the water depth is about

1.3 times the wave height.

Seas are locally formed by the wind and have a

steep profile, whereas swells are more graceful,

symmetrically shaped waves. Swells develop from

seas that have propagated out of the area where

they were formed and have lost their steepness.

The size of wind waves is determined by three

major factors: the wind speed, the fetch (distance

over which the wind blows), and the duration of

that particular wind speed. Depth is also impor-

tant, in that deeper waters can support larger

waves than shallower waters can.

Waves are additive. When two sets of waves are in

phase, they produce a wave with a wave height that

is the sum of the two. When they are out of phase,

they produce a wave with a wave height that is the

difference between the two. It is this additive ef-

fect that sometimes produces rogue waves.

Tsunami are usually produced by earthquakes.

They have very long-wavelengths, on the order of

200 km or so, but very small wave heights, onthe order of 1 m, and propagate as shallow water

waves at high speeds. When tsunami encounter

shallow coastal waters, their wavelengths shorten

and wave heights increase greatly, sometimes to

more than 20 m.

Tides, the largest ocean waves, are the result

of gravitational attractions of the Sun and Moon

(and other celestial bodies as well) on the Earth’s

oceans. The Sun and Moon account for most of

the tide-generating forces, with the gravitational

attraction of the Moon accounting for more than

twice that of the Sun. There is a fortnightly cycle

of spring tides and neap tides.

Because of the natural resonant frequencies of

different areas of the world ocean, some areas

are more in tune with the daily cycle of the Sun’s



Waves and Tides 225

Discussion Questions

1. What are some examples of disturbing forces that can create waves

in the ocean?

2. What controls the speed of propagation of a surface water wave?

Why do surface water waves slow down as they move onto shallower

waters?

3. Make a sketch of a deep water gravity wave that is moving as a swell

and explain the changes that it undergoes as it moves into shallow

water on its way to a beach.

4. Explain what happens to the sea sur face as the wind begins to blow,

and as the wind continues to blow at a constant wind speed.

5. What factors are important in determining the maximum size a wind

wave can achieve?

6. What are wave orbits? How do they change with depth below the

surface?

7. What is the difference between a sea and a swell? Why do waves

from distant storms outrun the storm itself, producing large surf on

coastlines far removed from the storm?

8. What are whitecaps? How and why are they formed? What is meantby a fully developed sea?

9. Explain what is meant when we say that the group velocity of waves is

equal to one-half the speed of individual waves in that group.

10. Why do some regions of the world ocean experience semidiurnal

tides, others diurnal tides, and still others mixed tides? What is

meant by these terms?

11. What might be the environmental implications of harnessing a

significant fraction of the tidal energy in an area with large tides, such

as the Bay of Fundy?

Further Reading

Bascom, Willard. 1964. Waves and Beaches: The Dynamics of the Ocean Surface. Garden City,NY: Anchor Books, Doubleday.

Gonzalez, F. 1999. Tsunami! Scientific American 280(5): 56–65.

Junger, Sebastian. 1997. The Perfect Storm. New York: W. W. Norton. This is the real-lifethriller on which the movie (2000) was later based.

Knaus, John A. 2005. Introduction to Physical Oceanography. Englewood Cliffs, NJ: Prentice-Hall.

Korgen, S. 1995. Seiches. American Scientist 83(4): 330–341.

Sverdrup, Harald U., Martin W. Johnson, and Richard H. Fleming. 1942. The Oceans: TheirPhysics Chemistry and General Biology Englewood Cliffs NJ: Prentice-Hall

gravitation attraction than other areas. In these

areas, we see diurnal tides, with just one high

tide and one low tide per day. In areas that are

more in tune with the Moon’s gravitational forces,

we see semidiurnal tides, with two high tides and

two low tides per day. Still other areas exhibit a

combination of both influences, producing mixed

tides. Extreme tidal ranges, such as in the Gulf of

Maine and Bay of Fundy, result when the resonant

frequency of a body of water closely matches the

lunar tidal frequency, and is modified by the basin

shape.

Tides are important for many reasons. In areas

of the ocean where tides are large, they may

be used to generate electricity. Also, tidal cur-rents mix coastal waters and deliver deep-water

nutrients to the near-surface waters, where they

stimulate biological production.

oceanography ch06

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