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 90o 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 |
