The tale of the two tides

Last week, I was sitting on the beach looking at the "Oosterscheldekering", one of the largest engineering constructions on Earth. It was designed to protect Zeeland from flooding during large storms and extreme high tides. As a boy I was always remembered of the power of the water (my complete family is from the island Schouwen-Duiveland, where the storm of 1953 hit hard). It gave me a sense of aw and pride, that engineers designed and build these large constructions. Maybe even, it gave me motivation to go into engineering.

As I was admiring the view, I was thinking about tides and their cause. One of the most common questions about tides is: Why are there two high (and low) tides a day? If tides are caused by the Moon, due to its gravitational attraction, there should only be one tide a day, because the Earth rotates only ones a day, right? The answer to this problem is: Correct Frame of Reference!

Since we, humans, know the Earth is round, we tend to place our point of origin in the center of the Earth. We can stand on the surface of the Earth, because gravity is pulling us (and sea water) towards this center of mass. This is why you do not fall off at the other side of the globe (yes even in Australia, everything falls down, I tested this myself!). This same force, together with the angular momentum of the Moon, keeps the Moon rotating around the Earth.

The Moon also has its own gravitational attraction, pulling other mass particles towards it, even our own Earth. If we look at a mass particle at an arbitrary location around the Earth, it experiences several forces. The gravitational attraction of the Earth is counteracted by the normal forces of the surface (me) or centrifugal force due to rotating motion (our mass particle). Another force is the gravitational attraction of the Moon (red arrow). However, both the Earth and the particle feel this force. This is where the correct frame of reference is coming into play.

The Earth is pulled a little bit, which means that the center of mass shifts a little bit towards the direction of the Moon. But as I told you, we like to think that our point of origin is at the center of the Earth. We live on the Earth, that is our frame of reference, that is how we experience tides.  Therefore, we (mathematically) have to correct for this by adding a resulting force (green arrow) to the Earth and any other mass particle (you, sea water, or our mass particle in an arbitrary location), same in magnitude as the gravitational attraction of the Moon exerting on the Earth, but in the opposite direction. This correction sets us back in the reference frame that we like, Earth centered.

This correction introduces an additional force on our mass particle (and on you and sea water, you get the point). With elementary vector addition (black striped lines), we obtain the true tide force (black arrow) that the particle experienced, as seen in the reference frame we live in, Earth centered. Theoretically we can do this for any particle, so also for the sea water on the surface of the Earth. The following sea level curve (which should be an ellipsoid, but my paint skills are not excellent) is observed due to the attraction of the Moon:

As you can see there are two bulges, instead of one. Lets take a better look at both locations. At location 1 (closest to the Moon) the gravitational attraction of the Moon is slightly larger than that at the center of the Earth. Gravitational attraction looses strength quite fast with distance (this is a good thing, otherwise black holes would be devastating and walking on the Moon, even more difficult). This means that the resulting tidal force is towards the Moon, as we all expected. Location 2 is experiencing the opposite, the gravitational attraction at the center of the Earth is larger than the force exerted on the sea water. This results in a tidal force, similar in magnitude as location 1, but in opposite direction, away from the Moon. However, Living on the surface of the Earth (as we all do), both tidal forces look similar, so therefore we experience two tides a day. 

Of course, the complete problem is much more elaborated and complex, but than you have to follow courses like Planetary Sciences, Astrodynamics or even Satellite Orbit Determination. Or post a comment on this blog and I will try to see if I can explain some more.