Origin of the Tides, Part 1

While the previous post dealt with the issue of currents, we need to know how they arise. This set of posts deals with tides.   Here is a listing of topics: 

Tides and tidal currents

          Origin of the Tides, part 1

          Static theory of the tides (advanced, can be skipped) 

          Origin of the Tides, part 2

          Figuring out tides and tidal currents

Tides, part 1

In ancient times, tides were understood to be associated with the passage of the moon across the sky and also related to its phases.    With his theory of gravity, Isaac Newton described the simplest explanation for tides: the spreading out of lines of gravity from the moon to the earth creates the tides, and to a lesser extent, the sun’s lines of gravity. 

Figure 1  Gravitational field lines from the moon diverging over the surface (and volume) of the earth create a residual tidal force after accounting for the average force of attraction.

The figure above illustrates the static theory of tides as described by Newton.  The lines of the gravitational field of the moon spread out as you move away from the moon.  We can think of the density of field lines as representing the strength of gravity.   The closer together the lines, the stronger the gravitational force.  And, the more spaced out the lines are, the weaker the gravitational force.  The force is stronger on the side facing the moon and weaker on the away side. 

The total force on the earth comes from each little bit of mass that feels the gravitational pull from the moon.   In order to figure this out, you have to add each contribution by evaluating each field line at each bit of mass and finding that force.   You then add all this up to get the total force acting on the earth.

Now, the earth moves as a single mass, as if all the mass were concentrated at the center.   It turns out that there’s a way to figure out the sum of all those bits of force.   You take the value of the gravitational attraction of the earth at its center as if all the mass were concentrated there.   In this case, you’re effectively saying that the net gravitational attraction is as if the earth was in a uniform field and this causes it’s motion.   The earth-moon system has center called the barycenter, and although we’re accustomed to only think of the moon orbiting the earth, in reality the earth and the moon both orbit the same barycenter, which is actually somewhere in the middle of the earth, but not right at the center. 

So, the average field gets the earth orbiting this barycenter.   Once you take the average, there is, however, a residual field that acts over the surface of the earth and averages out to zero.   This is shown on the lower right hand side of Figure 1.   The residual force is just what’s left over when we subtract off the average field (shown on the lower left hand side of Fig. 1) from the diverging lines of force.

The residual field distorts the shape of the earth  - the lines pull it apart parallel to the earth-moon axis and push it together perpendicular to the earth-moon axis.   In between these there is some component of the tidal force that’s parallel to the surface.   

Although it might be surprising, the forces are completely symmetric front-to-back, so the amount of force pulling away from the surface is the same on the force on the side facing the moon as away from the moon.

These forces produce a distortion in the shape of the earth, giving a near-side bulge pointing toward the moon and an away-side bulge that are the same.   As the earth rotates under the near and far bulges, it experiences a distortion.   Normally the distortions of solid earth are quite small of order a few millimeters, but since the water in the oceans can flow, tidal forces have a more pronounced effect on the oceans – hence the tides.

Figure 2  The residual tidal forces are symmetric and cause a near-side and far-side bulge that the earth rotates under daily.

Note that in all of the above, not once did I mention the words “centrifugal force” or the phrase “centripetal acceleration”.    There are some derivations of tides that don't use the above formulation, but rather rely on evaluating forces in a rotating frame of reference.   In that derivation, people will often say that the near bulge is created by the stronger force of the moon's attraction and the away force is centrifugal.    Physicists typically try to avoid derivations in rotating or accelerating frames of reference.   It's not obvious with the wording about about the far side bulge in this description that the near and far bulge are symmetric. 


The next post can be skipped, but has the nitty-gritty math details of the origins of tides.  If you want to skip that, I can't blame you, and you can go on to the next installment of tides found here.