Tides — why two a day

Problem

Tides. Explain why there are two tides a day, one while the Moon is overhead and the other on the opposite side of the Earth.

The Earth deforms into two bulges: one towards the Moon, one on the opposite side.

Solution

The key is that the Moon’s gravitational force is not uniform across the Earth: it depends on distance as $1/r^2$, so it is stronger on the side near the Moon and weaker on the far side relative to the Earth’s centre.

Letting $d$ be the Earth-Moon distance and $R_T$ the Earth’s radius, the Moon’s acceleration at the three points (near side, centre, far side) is: $a_\text{vicino} = \frac{GM_L}{(d-R_T)^2} > a_\text{centro} = \frac{GM_L}{d^2} > a_\text{lontano} = \frac{GM_L}{(d+R_T)^2}$

It is best to reason in the reference frame of the Earth’s centre, which falls towards the Moon with acceleration $a_\text{centro}$. In this frame only the differences relative to the centre matter (tidal accelerations):

• On the near side $a_\text{vicino} > a_\text{centro}$: the water is pulled towards the Moon (relative to the centre) → tidal bulge.
• On the far side $a_\text{lontano} < a_\text{centro}$: the water is pulled less than the centre, so it “lags behind” and piles up on the opposite side → second bulge, symmetric.

Two water bulges thus form, aligned with the Moon, on opposite sides of the Earth. The Earth rotates on its axis in $24$ h: every point on the surface passes through both bulges in one day, thus experiencing two high tides (and two low tides) a day.

$\ev{\text{the } \sim 1/r^2 \text{ tidal force creates two symmetric bulges; Earth's rotation} \Rightarrow \text{2 tides/day}}$

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Topics: Gravitation Concepts: Law of universal gravitation Skills: Analysis of limiting cases and speculative physics