THE ORIGIN OF THE TIDES
                                                                                                                                           
                                Summary

The ocean tides occur because the gravitational forces of the Moon and
the Sun acting on the Earth vary from place to place over the surface of
the Earth. The result of lunar gravity is small horizontal forces (that
is, parallel to the Earth's surface) that accelerate parts of the oceans
toward the sub-lunar point and other parts toward the anti-lunar
position. This produces a build up of water at the places closest to and
furthest from the Moon, a double tidal bulge in the oceans. Solar
gravity also produces horizontal forces that accelerate the oceans
toward the sub-solar and anti-solar points, and yield a double tidal
bulge about half as high as the lunar bulge. The solid Earth rotates
under this complex ocean pattern, and the result is a semi-diurnal
pattern of approximately two high tides and two low tides each day at
most locations on the Earth.

                        Origin of the Lunar Tides 

At first we will ignore the rotation of the Earth about its axis, and
the motion of the Earth about the Sun, and concentrate on the motion of
the Earth and Moon under their mutual gravitation. The Earth and Moon
orbit their common center of mass ("COM," labeled "Z" below), each
taking a lunar month to make a full cycle. The COM lies ~1/82 of the way
from the center of the Earth toward the center of the Moon, a position
that is about 1600 km below the surface of the Earth. 

The first key to the tides is understanding the motion of the Earth as
it revolves about the COM of the Earth-Moon system. Figures 1-4 below
show the location of four places labeled "A"-"D" on the surface of the
Earth, as the Earth "E" and the Moon "M" orbit their mutual COM. In each
case we are looking down on the north pole of the Earth, and the Moon
and Earth each orbit the COM in a counter-clockwise direction. Since we
ignore the rotation of the Earth, the relative orientations of points A
through D do not change as the Earth revolves about the COM.

....................................
(1)
            A
   
         D  E ZB           M
   
            C
   
...................................
Figure 1: The Earth-Moon system viewed from above the Earth's north pole.
The points "E" and "M" are the centers of the Earth and Moon,
respectively. Points A-D lie on the equator of the Earth, separated by
90 degrees in longitude. The center of mass of the Earth-Moon system is
the point Z, which lies about 1600 km under the Earth's surface. The
relative size of the Earth (radius R) and the Earth-Moon distance d are
not drawn to scale; in reality d = 60 R.
  
  
...................................   
 (2)   
              M
              
             
            
                           
              A
              Z
           D  E  B
          
              C  
              
....................................
Figure 2: Same as Figure 1, but 1/4 lunar month later. Here we are
ignoring the rotation of the Earth about its axis, so the points A-D are
located in the same places as in Figure 1.


....................................
(3)
                A

 M           DZ E  B              
              
                C
                  
....................................               
Figure 3: The Earth-Moon system 1/4 lunar month after Figure 2.


....................................
(4)
              A
              
           D  E  B
              Z	 
              C
               
               
                              
               
              M 
               
 ...................................
Figure 4:  The Earth-Moon system 1/4 lunar month after Figure 3.
 
 
In this motion the COM remains fixed in space. A few moments reflection
should convince you that when the Earth moves in this way, each and
every part of the Earth moves in a circle whose radius is the distance
from the center of the Earth to the COM of the Earth-Moon system. (The
location of the COM inside the Earth is NOT a fixed point in the Earth,
but rather moves around in a circle as seen in the frame of the Earth.)

The motion of the Earth around in such a circle is produced by a
centripetal acceleration of each part of the Earth. Since the whole
Earth moves together, these centripetal accelerations are the same for
each point on and in the Earth (see Figure 5). These accelerations are
parallel to the line joining the centers of the Earth and Moon; they are
horizontal and to the right in Figure 1, vertical and upward in Figure
2, horizontal and to the left in Figure 3, and vertical and downward in
Figure 4. In Figure 5 we show the Earth-Moon system as seen from a point
in the plane of their orbits. With the Moon situated to the right of the
Earth, the centripetal acceleration of each part of the Earth is
horizontal and to the right.


....................................
(5)
                +
                +
             N-->
                +
                +
        D--> E  Z B-->           M
                +
                +
             S-->
                +
                +
....................................   
Figure 5: The Earth-Moon system viewed from a point in the plane of their
orbit at a moment when the Earth is approaching the viewer and the Moon
is receding. In this picture the north pole of the Earth is "N" and the
south pole "S" (this view is different from that of Figures 1-4!). The
axis about which the Earth-Moon system rotates is indicated by the
vertical line "+++". The centripetal accelerations "-->" experienced at
points N, B, S, and D are all the same since each point in the Earth
moves in the same circle about the COM of the Earth-Moon system.

 
These centripetal accelerations are, of course, caused by the
gravitational force of the Moon. According to Newton's second law, these
accelerations are equal to the average gravitational force/mass of the
Moon acting on the Earth. The average force/mass exerted by the Moon on
the Earth is that required to accelerate the center of the Earth in this
way, i.e.,

                      Fm(avg) = G M(moon)/d^2,
                        
where d is the distance between the centers of the Earth and Moon. This
force is parallel to the line between the center of the Earth and the
center of the Moon. However, the local forces/mass Fm exerted by the
Moon on different parts of the Earth are in general different from
Fm(avg) because in general (1) each place is at a different distance
from the center of the Moon, and (2) the lines between each place and
the center of the Moon point in slightly different directions. THE
DIFFERENCES BETWEEN THE LOCAL FORCES ACTING AT VARIOUS PLACES ON THE
EARTH AND THE AVERAGE FORCE DRIVE THE TIDES. These difference forces are
called "tractive forces." 

The local forces/mass Fm try to produce accelerations that are slightly
different from the mean acceleration of the Earth, but as the Earth is
solid it moves as a whole with a single average acceleration. The
additional forces needed to satisfy Newton's second law locally at each
place in the Earth (and to produce the average acceleration) are
provided by solid body (elastic) forces. On the other hand, THE OCEANS
ARE FREE TO RESPOND TO THE LOCAL FORCE/MASS EXERTED BY THE MOON, AND ARE
THEREBY ACCELERATED DIFFERENTLY THAN THE SOLID EARTH AROUND AND UNDER
THEM. 
                    
                  The Tractive Forces Due to the Moon

The vector nature of the tractive forces may be discerned by considering
separately the side of the Earth closest to the Moon (the "near side of
the Earth"), and the side of the Earth farthest from the Moon (the "far
side of the Earth"):  

(1) Everywhere on the Earth, the local force/mass of the Moon acting on
the Earth is a vector directed toward the center of the Moon.

(2) On the near side of the Earth the force/mass of the Moon is a vector
coming upwards out of the surface of the Earth. On the far side of the
Earth, the force/mass of the Moon is a vector pointing downwards into
the interior of the Earth. 

(3) On (almost all of) the near side of the Earth, the local force/mass
Fm is greater than the average force/mass Fm(avg), and points in a
different direction than Fm(avg). 

(4) On the far side of the Earth, the local force/mass Fm is smaller than
the average force/mass Fm(avg), and points in a different direction than
Fm(avg).

On each side of the Earth, the difference between Fm and Fm(avg) is
defined to be the local value of the tractive force/mass Ft due to the
Moon. At any location on the surface of the Earth, the tractive
force/mass may be resolved into a component directed towards the center
of the Earth (the vertical component), and a component perpendicular to
the vertical component, one that points along the surface of the Earth
(the horizontal component):

(5) Everywhere on the Earth, the vertical component of the tractive
force/mass due to the Moon is tiny compared to the gravitational
force/mass exerted by the Earth (which is the mean acceleration of
gravity g), and may be neglected.

(6) The horizontal component of the tractive force/mass is zero along
the boundary between near and far sides, and zero at the sub-lunar and
anti-lunar points. At all other points on the surface of the Earth the
horizontal component of the tractive force/mass due to the Moon has an
amplitude that is about 1 part in ten million of the force/mass due to
the Earth itself (i.e., it has a value of about one-millionth of a meter
per second-squared).

(7) Despite their relative size compared to g, the horizontal components
of the tractive force/mass due to the Moon have a significant effect on
the oceans, and may not be neglected. This is true for two reasons.
First, Ft is the only horizontal force since the Earth's gravity is
vertical, and second, Ft is a "body force," that is, a force that acts
on every part of the ocean, rather than simply acting at one of its
boundaries.

The direction of the horizontal component of Ft is different on the near
and far sides of the Earth from the Moon:

(8) On the near side of the Earth, the horizontal component of Ft at a
point P is directed along a great circle that flows from P towards the
sub-lunar point.

(9) On the far side of the Earth, the horizontal component of Ft at a
point P is directed along a great circle that flows from P towards the
anti-lunar point.

(10) Thus, on the near side of the Earth to the Moon, compared to the
solid Earth, the oceans are accelerated along the surface of the Earth
toward the sub-lunar point, and on the far side of the Earth from the
Moon, the oceans are accelerated along the surface of the Earth towards
the anti-lunar point.

(11) The result of these tidal tractive forces is that the oceans reach
a quasi-equilibrium in which there is a build-up of water around the
sub-lunar and anti-lunar positions, and a relative deficit of water
around the boundary between the near and far sides of the Earth. These
are the "lunar tidal bulges."


                          The Solar Tides

Exactly the same considerations apply to the gravitational force exerted
by the Sun at various places on the Earth. In other words, everywhere
above where we have written "Moon" and "lunar," you may substitute "Sun"
and "solar." Even though the Sun is far more massive than the Moon,
because it is also much more distant Earth, the resulting solar tidal
bulges are only about 50% as high as the lunar bulges. 
                        
                          The Net Tides 

The net tidal bulges are the sum of the bulges produced separately by
the Moon and Sun. Each of these bulges produces an approximately
semi-diurnal pattern of high and low tides. The rotation of the Earth
causes the solid portion of the Earth to move with respect to the
oceans, carrying a given location on Earth from low to high tide and
back about twice each day. The sum of the lunar and solar tides results
in a complex pattern since they have similar but slightly different
periods. Then the lunar and tidal bulges overlap the tides are
especially large, and when they are at right angles to each other the
tides are minimal.

A complete theory of the tides must include the effects of local
geographical features on the freedom of the ocean to be accelerated by
the tractive forces. It also includes the tilt of the Earth's axis, the
tilt of the orbit of the Moon, and the varying distance of Moon and Sun
from the Earth.


                         Solid Earth Tides

The same tractive forces that act on the oceans also act on the solid
Earth. The Earth does not deform as easily under tractive forces as do
the oceans. However, the deformation is measurable (even though it is
only a few centimeters in height) by lunar laser ranging or very high
precision radio astronomy.


                            Conclusions
                                
The lunar and solar tides on the Earth are the result of the different
gravitational force/mass that is exerted on different parts of the Earth
and its oceans by the Moon and by the Sun. The differences between the
actual forces/mass and the mean force/mass accelerate the mobile oceans
with respect to the immobile solid Earth. The direction of these
tractive forces is such that they drive the oceans on the half of the
Earth near the Moon toward the sub-lunar point, and the oceans on the
far side of the Earth toward the anti-lunar point, giving rise to a
double tidal bulge in the oceans. A similar but smaller double tidal
bulge is created by the gravitation of the Sun. The rotation of the
Earth carries the land under these tidal bulges, and produces
approximately two high tides and two low tides per day at typical
mid-latitude locations.                           


                          Acknowledgement

The first satisfactory explanation of the tides was (not surprisingly)
due to Isaac Newton. This treatment is based on S. Pond and G. L.
Pickard,"Introductory Dynamical Oceanography," 2nd ed., Pergamon (1983),
which in turn credits G. H. Darwin, "The Tides and Kindred Phenomena in
the Solar System," Houghton Mifflin (1911), reprinted by Freeman (1962).