- Table of Contents
- Chapter 1 - Introduction
- Chapter 2 - The Astronomical Tide-Producing Forces: General Considerations
- Chapter 3 - Detailed Explanation of the Differential Tide-Producing Forces
- Chapter 4 - Variations in the Ranges of the Tides: Tidal Inequalities
- Chapter 5 - Factors Influencing the Local Heights and Times of Arrival of the Tides
- Chapter 6 - Prediction of Tides

### Chapter 3 - Detailed Explanation of the Differential Tide Producing Forces

The tide-raising forces at the earth's surface thus result from a combination of basic forces: (1) the force of gravitation exerted by the moon (and sun) upon the earth; and (2) centrifugal forces produced by the revolutions of the earth and moon (and earth and sun) around their common center-of-gravity (mass) or barycenter. The effects of those forces acting in the earth-moon system will here be discussed, with the recognition that a similar force complex exists in the earth-sun system.

With respect to the center of mass of the earth or the center of mass of the moon, the above two forces always remain in balance (i.e., equal and opposite). In consequence, the moon revolves in a closed orbit around the earth, without either escaping from, or falling into the earth - and the earth likewise does not collide with the moon. However, at local points on, above, or within the earth, these two forces are not in equilibrium, and oceanic, atmospheric, and earth tides are the result.

The center of revolution of this motion of the earth and moon around their common center-of-mass lies at a point approximately 1,068 miles beneath the earth's surface, on the side toward the moon, and along a line connecting the individual centers-of-mass of the earth and moon. (see G, Fig. 1) The center-of-mass of the earth describes an orbit (E1, E2, E3..) around the center-of-mass of the earth-moon system (G) just as the center-of-mass of the moon describes its own monthly orbit (M1, M2, M3..) around this same point.

*1. The Effect of Centrifugal Force.* It is this little known aspect
of the moon's orbital motion which is responsible for one of the two force
components creating the tides. As the earth and moon whirl around this
common center-of-mass, the centrifugal force produced is always directed
away from the center of revolution. All points in or on the surface of
the earth acting as a coherent body acquire this component of centrifugal
force. And, since the center-of-mass of the earth is always on the opposite
side of this common center of revolution from the position of the moon,
the centrifugal force produced at any point in or on the earth will always
be directed away from the moon. This fact is indicated by the common direction
of the arrows (representing the centrifugal force Fc) at points A, C, and
B in Fig. 1, and the thin arrows at these same points in Fig. 2.

It is important to note that the centrifugal force produced by the daily rotation of the earth on it axis must be completely disregarded in tidal theory. This element plays no part in the establishment of the differential tide-producing forces.

While space does not permit here, it may be graphically demonstrated that, for such a case of revolution without rotation as above enumerated, any point on the earth will describe a circle which will have the same radius as the radius of revolution of the center-of-mass of the earth around the barycenter. Thus, in Fig. 1, the magnitude of the centrifugal force produced by the revolution of the earth and moon around their common center of mass (G) is the same at point A or B or any other point on or beneath the earth's surface. Any of these values is also equal to the centrifugal force produced at the center-of-mass (C) by its revolution around the barycenter. This fact is indicated in Fig. 2 by the equal lengths of the thin arrows (representing the centrifugal force Fc) at points A, C, and B, respectively.

*2. The Effect of Gravitational Force.* While the effect of this
centrifugal force is constant for all positions on the earth, the effect
of the external gravitational force produced by another astronomical body
may be different at different positions on the earth because the magnitude
of the gravitational force exerted varies with the distance of the attracting
body. According to Newton's Universal Law of Gravity, gravitational force
varies inversely as the second power of the distance from the attracting
body. Thus, in the theory of the tides, a variable influence is introduced
based upon the different distances of various positions on the earth's
surface from the moon's center-of-mass. The relative gravitational attraction
(Fg) exerted by the moon at various positions on the earth is indicated
in Fig. 2 by arrows heavier than those representing the centrifugal force
components.

*3. The Net or Differential Tide-Raising Forces: Direct and Opposite
Tides.* It has been emphasized above that the centrifugal force
under consideration results from the revolution of the center-of-mass of
the earth around the center-of-mass of the earth-moon system, and that
this centrifugal force is the same anywhere on the earth. Since the individual
centers-of-mass of the earth and moon remain in equilibrium at constant
distances from the barycenter, the centrifugal force acting upon the center
of the earth (C) as the result of their common revolutions must be equal
and opposite to the gravitational force exerted by the moon on the center
of the earth. This fact is indicated at point C in Fig. 2 by the thin and
heavy arrows of equal length, pointing in opposite directions. The net
result of this circumstance is that the tide-producing force (Ft) at the
earth's center is zero.

At point A in Fig. 2, approximately 4,000 miles nearer to the moon than is point C, the force produced by the moon's gravitational pull is considerably larger than the gravitational force at C due to the moon. The smaller lunar gravitational force at C just balances the centrifugal force at C. Since the centrifugal force at A is equal to that at C, the greater gravitational force at A must also be larger than the centrifugal force there. The net tide-producing force at A obtained by taking the difference between the gravitational and centrifugal forces is in favor of the gravitational component - or outward toward the moon. The tide-raising force at point A is indicated in Fig. 2 by the double arrow extending vertically from the earth's surface toward the moon. The resulting tide produced on the side of the earth toward the moon is know as the direct tide.

At point B, on the opposite side of the earth from the moon and about 4,000 miles farther away from the moon than is point C, the moon's gravitational force is considerably less than at point C. At point C, the centrifugal force is in balance with a gravitational force which is greater than at B. The centrifugal force at B is the same as that at C. Since gravitational force is less at B than at C, it follows that the centrifugal force exerted at B must be greater than the gravitational force exerted by the moon at B. The resultant tide-producing force at this point is, therefore, directed away from the earth's center and opposite to the position of the moon. This force is indicated by the double-shafted arrow at point B. The tide produced in this location halfway around the earth from the sublunar point, coincidentally with the direct tide, is know as the opposite tide.

*4. The Tractive Force.* It is significant that the influence of
the moon's gravitational attraction superimposes its effect upon, but does
not overcome, the effect's of the earth's own gravity. Earth-gravity, although
always present, plays no direct part in the tide-producing action. The
tide-raising force exerted at a point on the earth's surface by the moon
at its average distance from the earth (238,855 miles) is only about one
9-millionth part of the force of earth-gravity exerted toward its center
(3,963 miles from the surface). The tide raising force of the moon, is,
therefore, entirely insufficient to "lift" the waters of the earth physically
against this far greater pull of earth's gravity. Instead, the tides are
produced by that component of the tide-raising force of the moon which
acts to draw the waters of the earth horizontally over its surface toward
the sublunar and antipodal points. Since the horizontal component is not
opposes in any way to gravity and can, therefore, act to draw particles
of water freely over the earth's surface, it becomes the effective force
in generating tides.

At any point on the earth's surface, the tidal force produced by the moon's gravitational attraction may be separated or "resolved" into two components of force - one in the vertical, or perpendicular to the earth's surface - the other horizontal or tangent to the earth's surface. This second component, know as the tractive ("drawing") component of force is the actual mechanism for producing the tides. The force is zero at the points on the earth's surface directly beneath and on the opposite side of the earth from the moon (since in these positions, the lunar gravitational force is exerted in the vertical - i.e., opposed to, and in the direction of the earth-gravity, respectively). Any water accumulated in these locations by tractive flow from other points on the earth's surface tends to remain in a stable configuration, or tidal "bulge".

Thus there exists an active tendency for water to be drawn from other
points on the earth's surface toward the sublunar point (A, in Fig. 2)
and its antipodal point (B, in Fig. 2) and to be heaped at these points
in two tidal bulges. Within a band around the earth at all points 90^{o}
from the sublunar point, the horizontal or tractive force of the moon's
gravitation is also zero, since the entire tide-producing force is directed
vertically inward. There is, therefore, a tendency for the formation of
a stable depression here. The words "tend to" and "tendency for" employed
in several usages above in connection with tide-producing forces are deliberately
chosen since, as will be seen below, the actual representation of the tidal
forces is that of an idealized "force envelope" with which the rise and
fall of the tides are influenced by many factors.

*5. The Tidal Force Envelope.* If the ocean waters were completely
to respond to the directions and magnitudes of these tractive forces at
various points on the surface of the earth, a mathematical figure would
be formed having the shape of a prolate spheroid. The longest (major) axis
of the spheroid extended towards and directly away from the moon, and the
shortest (minor) axis is center along, at right angle to, the major axis.
The two tidal humps and two tidal depressions are represented in this force
envelope by the directions of the major axis and rotated minor axis of
the spheroid, respectively. From a purely theoretical point of view, the
daily rotation of the solid earth with respect to these two tidal humps
and two depressions may be conceived to be the cause of the tides.

As the earth rotates once in each 24 hours, one would ideally expect to find a high tide followed by a low tide at the same place 6 hours later; then a second high tide after 12 hours, a second low tide 18 hours later, and finally a return to high water at the expiration of 24 hours. Such would nearly be the case if a smooth, continent-free earth were covered to a uniform depth with water, if the tidal envelope of the moon alone were being considered, if the positions of the moon and sun were fixed and invariable in distance and relative orientation with respect to the earth, and if there were no other accelerating or retarding influences affecting the motions of the waters of the earth. Such, in actuality, is far from the situation which exists.

First, the tidal force envelope produced by the moon's gravitational attraction is accompanied by a tidal force envelope of considerably smaller amplitude produced by the sun. The tidal force exerted by the sun is a composite of the sun's gravitational attraction and a centrifugal force component created by the revolution of the earth's center-of-mass around the center-of-mass of the earth-sun system, in an exactly analogous manner to the earth-moon relationship. The position of this force envelope shifts with the relative orbital position of the earth in respect to the sun. Because of the great differences between the average distances of the moon (238,855 miles) and sun (92,900,000 miles) from the earth, the tide producing force of the moon is approximately 2.5 times that of the sun.

Second, there exists a wide range of astronomical variables in the production of the tides caused by the changing distances of the moon from the earth, the earth from the sun, the angle which the moon in its orbit makes with the earth's equator, the superposition of the sun's tidal envelope of forces upon that caused by the moon, the variable phase relationships of the moon, etc. Some of the principle types of tides resulting from these purely astronomical influences are describe below.

Chapter 4 - Variations in the Ranges of the Tides: Tidal Inequalities