Tides -The Last Wave

Information by the University of California (Riverside)

updated in 2014

Tides are rhythmic variations of sea levelwith periods of approximately 6 or 12 hours, depending on where you are on Earth.

6 hours later

Tides are caused by an imbalance*between the gravitational attractionof the Sun and Moon and theinertia of water on a rotating Earth.

* forces must be unequal or no movement -- Newton’s 1st law (F=ma)

Some common ideas:

Does the Moon orbit around the Earth?

Tides are caused by the Moon’s orbit

The Earth and Moon form a systemthat rotates around a common center

of mass!

Tides result from the rotation ofEarth on its axis every 24 hours.

Let’s begin with the gravitationalpull of the Moon on the Earth:

gravity again!

F = Gmmoonmearth

r2

This explainsthe bulge of watertowards the Moon,but ….

… how do we explain the bulge away from the Moon?

This other bulge is the result of both the Earth andthe Moon orbiting around a common center of

mass:

The center of mass of the Earth-Moon systemis located 1650 km deep in the Earth. Both theEarth and Moon rotate around this same axis

(If they did not, then the unbalanced force ofthe Moon would cause the Earth to leave its orbit.)

Just as you seem to feel a force pulling you outwardon a merry-go-round, the water on the side away from the Moon experiences the same ‘force’:

It is not reallya force, but justthe tendency of anobject to continuein a straight lineunless acted on aby a force (Newton’s

2nd law). This is called inertia.

Moon’s gravity pull

Inertia

One bulge is due to gravity and the otheris due to inertia.

Because the Earth-Moon system completeone revolution every 27.3 days, the bulgegoes completely around the Earth in 27.3days. Right? NO!

The 12 hour period seen in tides is dueto the spin of the Earth on its axis, onerevolution every 24 hours. They are NOTdue to the orbit of the Moon.

Write this down:

Complications:

The lunar tidal period is not exactly 12 hours

This takes the Earth 24 hours * 360

2.373= 24h 53m (50)

The tidal period is thus 12h 26.5m (25 min)

In the time it has taken the Earth tocomplete one revolution, the Moonhas moved 360°/(27.3 days)= 13.2°farther along, so the Earth has to rotate360+13.2=373.2° for a point to movedirectly beneath the Moon again.

MORE complications:

The Earth’s spin axis is tilted, sothe Moon moves from 28.5° belowthe equator to 28.5° above the equatorevery 27.3 days.

At onepoint, dayand night tides differ

But the BIGGEST complication issomething you see every day -- the Sun!

The Sun also exerts a tidal force (gravity ALWAYS sucks), but to understand this we must first examine the equation for the combined gravity-inertia force.Both of these forces depend on the inverse square distance and on the mass of the Moon, so the difference can be shown to be proportional to the inverse cube of distance:

Net tidal force mobject

r3

Taking the ratio of the tidal force of the Sun to that of the Moon (msun = 27,000,000mmoon but rsun = 387rmoon):

3moonmoon

3sunsun

moon

sun

r/m

r/m

Tidalforce

Tidalforce

3

6

3moonmoon

3moonmoon

6

387

27X10

r/m

)r387/()m*(27X10

0.1

46.

The Sun exerts a tidal force approximately HALFof that of Moon.

Oh God, tell me he doesn’t want

me to do this math!

No, but I expect students toanswer questions like:

What is the ratio betweenthe tidal forces of the Moon and Sun?How can the Sun exert a smaller tidal force than theMoon when it is SO much bigger.

Only once every 27.3 days are the tidalforces of the Moon and Sun aligned:

Depending on the time of month, wecan have large or small tides:

Spring (‘jump’) tide large ht. new,full moon

Neap (‘hardly disturbed’)tide

short ht. waxing,waning moon

There is a seasonal dependence.Earth is 3,700,000 km farther from the Sunin summer than in Winter:

Winter tides in the northern hemispheretend to be larger than summer tides.

Another complication:

Now, let’s consider the wave propertiesof tides*:

*Remember, tides ARE waves!

Period? 12 h 25 m (=43,518 s)

Wavelength?1/2 circumference ofEarth = 20,000 km

Speed? Governed by rotationrate of Earth - 1600 km/hrat Equator (444 m/s)

Does C=L/T? L/T = 460 m/s

Not quite!

Are tidal waves deep water or shallowwater waves?

Consider wavelength = 20000 km

For deep water, ocean depth > 20000/2 km

Max ocean depth = 11 km and average ~3.8 km

For shallow water, depth < 20000/20 = 1000 km

Tides are shallow water waves!

Tides have lots of energy!

But small wave height: solar lunar

0.24 m 0.55 m(theoretical max)

Why are tides so much larger than theorypredicts?(Compare 2 m in open ocean versus theoretical maxima of 0.55 m and 0.24 m predicted for lunar and solar tides)

This led to Laplace proposing the“Dynamic Theory of Tides”

Dynamic Theory of Tides

Tides are shallow water waves andare forced waves because the driving forces

are always there

Tidal waves get reflected and refractedby land masses, and these waves constructively and destructively interfere with each other. This leads to…

.. a complicated pattern of tides.

Semidiurnal tides - 2 high and 2 low per day (12 hr period)Diurnal tides - 1 high and 1 low per day (24 hr period)Mixed tides - unequal high and low (12 hr periods)

More reality:

Tides get amplified in confined basinsif the resonant period in the basin (remember seiches?) is close to thetidal period:

tidal waves in open ocean = 2 m

coastal tides as large as 8 m but usually only 1 m

tidal bore = true tidal wave moving up river inlet

Tidal bore = true tidal wave (now youcan say you’ve seen one!)

1-8 m high (this one is 2-3 m)

Bay of Fundy, New Brunswick, Canada:

resonant period = tidal period

BIGGEST tides in world (15 m)

Low tideHigh tide

Energy flow in tides

Tides acquire gravitational potential energyfrom pull of Sun and Moon (water higher)

Tidal waves transmit energy to coasts

Energy dissipated as heat andturbulence at shores

+slowing Earth’s rate of rotation!!

(22 hours/day 450 My ago)