# Why Are There Two High Tides Each Day?

This is a very frequently asked question! And a frequently answered question, too. However, many of the answers are different! Newton's equilibrium theory from 1687 used the differential of the gravitational force, but some people (especially oceanographers) also consider a centrifugal force caused by the rotation of the Earth around the Earth-Moon center of mass. However, non of the sources that explain it using centrifugal forces do any computations. The reason is simple. The computations would be wrong! I will describe some of the attempted explanations I found.

1. The correct explanation was given by Newton in 1687. The Moon's gravity pulls on the Earth and the water on it, but the force of the Moon's gravity varies across of the Earth. The pull is greater on the side facing the Moon, pulling the water there closer to the Moon, while the pull is weaker on the side away from the Moon, making the water there lag behind. This stretches out the Earth and the water on it, creating two bulges. Remember that both the Earth and the Moon are falling towards each other. The reason why they don't collide, is that they already have a motion perpendicular to the direction in which they are falling, so the falling only results in a change in that direction.

2. Some argue in the same way as above, but in order to get an outward pointing force at the far side, they consider a centrifugal force caused by the Earth's rotation around the Earth-Moon center of mass. This is not correct.

They argue that the centrifugal force is the same at every point of the Earth, and that it must be equal to the gravitational pull at the center of the Earth. I find it too much of a “coincidence” that the centrifugal force happens to equal the gravitational pull at the center (or the center of mass). In fact, a simple computation shows that the centrifugal force caused by the Earth's rotation around the Earth-Moon center of mass is tiny compared to the gravitational differential. Notice also that the angular momentum due to the revolution around the center of mass is about five times as big as the angular momentum due to the rotation of the Earth. (See Kibble's Classical Mechanics.)

This explanation seems to be popular among oceanographers. All the sites below use a centrifugal force. Except for that minor problem, they are excellent sources of further information.

3. Some people, including Aslamazov and Varlamov in the book “The Wonders of Physics” use a centrifugal force, but they let the force depend on the distance from the center of rotation, which is clearly wrong. If we ignore the rotation of the Earth, then all points on the Earth describes circles with the same radius, but different centers, in the course of the month.

4. Some argue that the gravitational pull from the Moon would only create one bulge by itself. They use the centrifugal force, not to cancel the gravitational pull at the center, but to create the bulge on the far side. This is wrong. It would only be true if the Earth and the Moon were stationary and held in place so that they couldn't fall towards each other.

• Motion of the Earth in Space from Introduction to Earth Sciences I by John C. Mutter of the Department of Earth & Environmental Sciences, Columbia University.
• Beyond the Moon by James Greig McCully is a very interesting book, but he unfortunately makes this error.

There are many other factors we need to look at, which I hope to explain in more detail later on.

• Spring and neap tides.

• The diurnal inequality.

• Tropical and equatorial tides.

The equilibrium theory is of course only a theoretical approximation. In order to actually predict tides, we need to consider a number of other factors.

Back to my page on Heavenly Mathematics & Cultural Astronomy.

Helmer Aslaksen
Department of Mathematics
National University of Singapore
helmer.aslaksen@gmail.com

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