# Why Two High Tides a Day?

Long before Newton, it was obvious that the moon exerted a force on the oceans. After all, day in and day out, year after year, ever since the oceans formed, there have been two high tides a day, once with the moon at its zenith and once again nearly 12 hours later. When the concept of universal gravitation was introduced, it quantified the forces involved and neatly accounted for the ocean tides. Today, anyone can tell you that the tides are caused by the "gravity" of the moon.

Or are they?

The oceans form a thin, incompressible, but easily deformable layer on most of the earth's surface. One might expect that the moon's gravitational force would pull the entire ocean layer towards it, distending the layer (producing a high tide) in the direction of the moon on the near side of the earth, while simultaneously compressing it (producing a low tide) on the opposite side of the earth (see Figure 1, not to scale). This model is half right and half wrong: half of all high tides are generated in exactly this way, but instead of there being a low tide on the far side of the earth, there is also a high tide. There must be some other explanation.

Indeed there is. The motion of the earth-moon system, taken in isolation, is a classical two-body problem where each body exerts an attractive force (gravity) on the other. Solving the equations of motion, one finds that the two bodies rotate about their common centre of mass, like a rigid asymmetric dumbbell spinning around an axis perpendicular to the bar. As shown in Figure 2, the centre of mass of the earth-moon system turns out to be inside the earth, about three-quarters of an earth radius from its center, along the line joining the earth and moon.

The earth, therefore, orbits the center of mass in a tight circle (almost) while the moon orbits in a large one. This orbital motion causes the earth to experience a centrifugal (pseudo-) force, which distends the ocean layer in the direction away from the moon, not unlike what happens to the water in a pail when you swing it in a circle. Combining this orbital effect with the direct gravitational pull of the moon explains the simultaneous high tides on opposite sides of the earth: on the near side the direct pull dominates and causes the oceans to bulge in the direction of the moon; on the far side the centrifugal effect dominates and causes the oceans to bulge in the direction away from the moon. As the earth spins on its axis, a given seaside location will experience a high tide when the moon is at its closest, and then another one about 12 hours later when it is at its furthest.

Reality, of course, is never quite so simple. Although the sun is much farther away from the earth than the moon, it is also much more massive, so its gravitational pull on the earth is relatively large (almost half of what the moon exerts). It therefore plays a significant role in determining both the timing and strengths of the tides.

So, are the tides caused by the "gravity" of the moon? In large part, yes. But the gravity of the earth is as much responsible for the two-body rotation as that of the moon. The tides are caused by the combined gravitational forces of the moon, earth, and sun.

REFERENCES:

John L. Synge and Byron A. Griffith, "Principles of Mechanics" (McGraw-Hill 1949), section 6.5.