San José State University
Thayer Watkins
Silicon Valley
& Tornado Alley

Tides and Their Explanation

In some primordial time humans living near seas noticed that the water rose and fell twice a day. For those living near a steep coast the water merely rose and fell but for those living near coastal flats the water came in and went out. It was a long time before more precise time measuremnt told them that the time between tides was not exactly one half of a day (12 hours) but instead twelve hours and twenty five minutes. A morning tide would come fifty minutes later the next day. After millenia humans came to understand what causes the tides, although perhaps at an earlier stage they perceived that the cause had something to do with the moon. The moon rises fifty minutes later each day. Even with modern education in science the general population is not aware of this fact, but astute ancient observers might have known.

I became aware of this fact only at an age of nearly forty because of a unique episode. One day about dusk I was hiking on the fire trails in the Berkeley hills above the Botanical Gardens and saw an exceptionally beautiful moonrise. I wanted my children, Sam (age 9) and Storm (age 6), to see it so I brought them to the same area the next day at dusk. I was expecting the moon to rise at the same time again. We waited and waited but no moon. Finally after about an hour it appeared and they got to see it and agreed it was truly a beautiful moonrise. The only problem was that now it was almost night and we had to navigate the trail back in near darkness. There was really no danger but Sam and Storm perceived this journey as one fraught with danger and I let them think that. They were very brave in the face of what seemed to them as real danger and I was extremely proud of them. Ever afterwards we remembered the beauty of the moonrise and the tension of our journey out of danger. When we made it back to the car I started calculating on the basis that the moon circles the Earth about once every 30 days so each day it should lag about 1/30 of a day each day. One thirtieth of a day is 0.8 of an hour which is 48 minutes. So I had my explanation for the moon rising nearly an hour later each day.

The phase of the moon at a particular location affects the extent of the tide. If the moon is directly overhead at the time of high tide the tide is higher than it is at other times. But tides are also exceptional high when the moon is at the opposite point in its orbit from the point of being directly overhead. This unintuitive fact has a cogent explanation. The fact that there are two tide cycles a day instead of just one also can be cogently explained.

Explanation of Tides

Although we commonly think of the moon as revolving around the Earth they both revolve around their common center of gravity. The gravitational attraction between them is counterbalanced by the centrifugal force they experience as a result of their revolution about their common center of gravity. This balance of gravitation and centrifugal force is exact at the centers of the Earth and moon. But at points closer or farther away than the center there is a slight imbalance that produces the tides. Consider a point on the Earth's surface on the side facing the moon. At this point, being closer to the moon, the gravitational attraction is greater whereas the centrifugal force is about the same as it is at Earth's center. Therefore at this point there is a net greater attraction toward the moon.

At the opposite point on the Earth's surface, on the side facing away from the moon the gravitational attraction iess than at Earth's center whereas the centrifugal force is about same. This means there is a net force away from the surface of the Earth. This net force raises water to a higher level just as the net force toward the moon raises water to a higher level there.

The combination of gravitational and centrifugal forces around the Earth results in water on the Earth assuming an ellispoidal shape. A cross section of the water and the Earth is shown below. The deviation of the water from the Earth is of course greatly exaggerated. This shows why there would be two high tides each day. The black lines represent tidal gauges.

On the moving surface of the Earth we perceive the water rising and falling but the reality is the water ellipse stays stationary (more or less, it rotates about once every thiry days) and the Earth turns within the ellipse producing the perception of the water level moving. The following animation depicts the situation with respect to the tides. It shows the Earth with its black tidal markers turning into the hydrosphere at the equator producing two high tides each day. The magnitude of the tide is enormously exaggerated.

From the viewpoint of Earthings it seems that we are motionless and the hydrosphere is turning, as shown below.

It is little appreciated how fast we are moving with the Earth. At the equator this speed is close to a thousand miles per hour. Therefore that is the apparent speed of the tidal crest. The height of the tide is determined by the vertical and horizontal profiles of the boundaries of the ocean. For a coast with a sloping sea floor and/or a narrowing bay the height of the tidal crest is vastly amplified. It is similar to what happens with a tsunami. The height of the tsunami wave in the open ocean may be only a few inches, but where it comes into a bay the water may rise to tens of feet as the water piles up on the shore.

To get more precise we have to take into account the tidal effect produced by the Earth revolving around the sun, or rather around the common center of gravity of the sun and the Earth. The sun produces a tidal effect but it is only half as strong as that produced by the moon. The sun tends to create an ellipsoid of water which the Earth turns within. This is why the tidal effect of the sun is felt on a daily basis rather than over the period of time of Earth's revolution about the sun-Earth center of gravity.

Component Symbol Period
solar hours
Main lunar semi-diurnal M2 12.42 1.00
Main solar semi-diurnal S2 12.00 0.466
lunar distance N2 12.66 0.191
Solar-lunar variations K2 11.97 0.127
Solar-lunar K1 23.93 0.584
Lunar O1 25.82 0.415
Solar P1 24.07 0.193
Lunar fortnighly Mf 327.86 0.172

The Shape of the Earth's Hydrosphere

Having demonstrated that the points on the Earth facing the moon and the points facing away from the moon experience a net upward force the next question is to determine how high the water should rise. It has already been presumed that the surface of the hydrosphere is an ellipsoid. The analysis below provides the basis for this assumption and shows that while the shape is approximately an ellipsoid it is not precisely an ellipsoid.

First consider a cross section for a plane that passes through the polar axis of Earth and the line connecting the centers of the Earth and the moon. If the curved boundary of the hydrosphere is an ellipse then the hydrosphere would be an ellipsoid of revolution.

The form of the cross section curve may be found by determining the curve whose tangent is perpendicular to the force vector at each point. Two lines are perpendicular if the product of their slopes is equal to negative one. Let y(x) be the equation of the cross section curve. Its slope is dy/dx. The force vector through a point has components Fx and Fy. The slope of the force vector is Fy/Fx.

For perpendicularity

(dy/dx)(Fy/Fx) = -1
and thus
dy/dx = -Fx/Fy.

The force at a point is the combination of the gravitational force and the centrifugal force associated with the rotation of the Earth and moon about their common center of gravity. The gravitational attraction of the moon on a unit of mass is inversely proportional to the square of the distance.

For further analysis of the shape of Earth's hydrosphere click here: Hydrosphere

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