tides3

$e-MATH$
Fourier analysis of ocean tides

Some physics and some trigonometry

The tides are caused by gravitation. But in fact the "tidal force" depends on the gradient of the gravitational field. Because what creates the bulge in the Earth's oceans on the side facing the Moon is the fact that the surface of the Earth is closer to the Moon than the center, and is therefore attracted more strongly. This also explains the bulge on the opposite side: there the center is closer than the surface. In general a variable gravitational field will stretch bodies along its gradient; our ocean tides are a special case of this effect.

The gravitational force exerted by a body of mass m on one gram of matter at a distance L centimeters is

                       G m
                      -----   dynes,
                       L²

where G= 6.67x10^-8dyne-cm²/g² is Newton's gravitational constant. Suppose another gram of matter is one centimeter farther in the same direction. Then it feels a force

                      G m
                    -------   dynes.
                    (L+1)²

The difference between these two forces can be thought of as a force pulling the two grams apart. To first order approximation this difference is

                      -2 G m
                      -------   dynes,
                        L³

the derivative of the gravitational force with respect to L. This is the "tidal force" separating two one-gram objects at distances L and L+1 centimeters respectively.

This calculation explains why the Sun (m = 2x10³³, L = 1.5x10¹³, m/L³= 6x10^-7) and the Moon (m = 7.3x10²⁵, L = 3.8x10¹⁰, m/L³=13.3x10^-7) exert comparable tidal forces at the surface of the Earth.

This force is very tine, but if one of the grams is at the center of the earth and one on the surface, so the L values differ by 6x10⁸cm., In more detail. At any moment, at any point of the surface of the earth, the tidal force causing the ocean to bulge up at that point is the upward-pointing component of the sum of the Moon's and the Sun's contributions. As a function of time this will depend on

the time of day,
the direction and distance of the Moon,
the direction and distance of the Sun.

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