Moon's gravity is only part of the story for the production of tides, the realistic situation is considerably more complicated. Actually, we can conclude that the tide is formed by 3 various kinds of forces with some assumption. Now, we are going to investigate the "secret" part of the tides.
In the Earth-Moon System, the Earth orbits the center of mass of the Earth-Moon System and it held its orbits by the lunar gravity. The center of the gravity for the Earth orbits the Moon so that the gravity of Moon on the Earth gives the exact acceleration needed to keep them in orbit.
The centrifugal force is produced as the Earth and the Moon spin around the center of mass of the system and this force is pointed away from the common center of rotation. Moreover, since the center of mass of the Earth is opposite to the center of mass of the system from the position of the Moon, the centrifugal force created on the Earth is pointed away from the Moon. Note that this centrifugal force is the same anywhere on the Earth.
Ignoring the the gravitational force of the Moon and the Sun, we can conclude that a bulge of water is developed around the equator due to the centrifugal force.
Figure2. Centrifugal Force is the same anywhere on the Earth
According to the Newton's Universal Law of Gravity, gravitational force decreases as the inverse square the distance from the attracting body:
Fg = Gm1m2 / R2
We then realize that the amount of gravitational force acts on the Earth may be different at different positions on the Earth's body because this force is affected by the distance of the attracting body. Generally, we would calculate the "gradient" -- variation in force over distance -- instead of the Fg . The following formula gives the strength of the gradient:
DF / DR = 2Gm1m2 / R3
The above equation shows that the tide-raising force varies inversely as the third power of the distance of the center of mass to of the attraction
In order to be more specific, we will divide the gravitational forces into three parts:
Gravity from the Moon:
The points nearer the Moon are affected by a stronger gravitational force, the net effects is that these points are pulled toward the Moon raising a tide.
Gravity from the Sun:
Even the Sun also rises the tides on Earth, tides are more influenced by the gravitational effect of the Moon than they are by the Sun because the distance between the Earth and the Moon outweighs by far the Sun's size. Compare to the lunar gravity, the solar gravity is less effective, with only 1/3 times of the Moon's gravitational effect!!
Anyway, both gravity have influence to the tides and they result different types tides at some specific moment. We will discuss types of tides in next section.
The following is a simple illustration of how the Moon and the Sun act together to raise the tides.
Figure3. Raise of Tides under the Influence of the Lunar Gravity and the Solar Gravity
Gravity from the Earth:
Actually, the Earth's gravity has no direct effect to raise the tides, but it does affect the tidal force of the Moon's gravity over the Earth. With the existence of Earth's gravity, the tidal force of the Moon is not strong enough to pull the waters out of the Earth. So, the tides are created by the tide raising force of the Moon which only pulls water of the Earth horizontally over the Earth's surface toward itself.
Net Tide-Raising Force
Now, we comes up with the following facts:
1. The centrifugal force from the revolution of the center of mass of the Earth around the center of mass of the Earth-Moon System is the same at any points on the Earth.
2. The individual centers of mass for the Moon and the Earth are separated at a constant distances from the center of mass of the system.
3. The centrifugal force acting on the center of the Earth as the result of the Earth and the Moon revolutions must equal and opposite to the gravitational force applied by the Moon on the Earth's center.
Therefore, we can come up with the conclusion that the tide-raising force is zero at the Earth's center.
Figure4. Combination of Forces which Form the Tides
From this graph, we have the following equations:
Centrifugal Force at A = Centrifugal Force at B = Centrifugal Force at O
Gravitational Force at A > Gravitational Force at B > Gravitational Force at O
Gravitational Force at A > Centrifugal Force at A > Tide-raising force at A
Gravitational Force at B < Centrifugal Force at A > Tide-raising force at A
Gravitational Force at O = Centrifugal Force at O > zero