Tides

Why is it tides are semi-diurnal? That is, why do they occur every 12 hours when they are caused by the gravitational pull of a moon that we turn to face once a day?  The straightforward answer is that water bulges out on both the side nearest and the side furthest from the moon.  (We'll ignore the Sun for simplicity, but adding it in doesn't change anything.)

The Earth rotates while the bulges remain in place, causing tides 12 hours apart.  However, this doesn't explain why there should be two bulges.  The reason becomes clear when you change the frame of reference.  Instead of thinking about a frame in which both the moon and the Earth rotate, set the origin to be the centre of mass of the two objects, and choose a rotating frame in which the moon is stationary.  In this frame there is a "fictional" centrifugal force of $\omega^2 r$ which combines with the gravitational force from the moon, $GM_{\text{moon}}/(r-r_{\text{moon}})^2$

The two forces match at the centre of Earth, explaining why Earth doesn't fall towards the moon in this frame of reference:

This explains quite nicely why there are two bulges: the two forces are more or less linear across the width of Earth and cancel in the middle.  So the force pulling the water towards the moon on the near side must be more or less the same as the force pushing the water away from it on the far side.

But confusingly there is also a smaller diurnal signal:

 


Courtney, Elya & Courtney, Michael. (2015). A More Accurate Fourier Transform.

This is weird. If you drive a system with a forcing that repeats with period $1/f$, you'd expect the response to contain frequencies $f$, $2f$, $3f$, and so on, but not $f/2$.  It turns out that there is a daily forcing, and it comes from the tilt of the Earth

Consider the time of the month in which the moon is closest to the north pole.  The water on the Arctic circle will experience the greatest pull towards the moon when the moon is highest in the sky.  But, 12 hours later, instead of being pushed away it will experience no forcing at all.

There are other frequencies that appear in the response because the solar day and the lunar day are ever so slightly different lengths, and as a result of orbital eccentricities.  But the response is mostly a combination of diurnal and semi-diurnal frequencies, with the largest of the two depending on the resonant frequency which is in turn a feature of the local geography.


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