# 8.3.1: A Bit More of Physics

$$\newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$$

$$\newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$$

$$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$

( \newcommand{\kernel}{\mathrm{null}\,}\) $$\newcommand{\range}{\mathrm{range}\,}$$

$$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$

$$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}[1]{\| #1 \|}$$

$$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$$

$$\newcommand{\Span}{\mathrm{span}}$$

$$\newcommand{\id}{\mathrm{id}}$$

$$\newcommand{\Span}{\mathrm{span}}$$

$$\newcommand{\kernel}{\mathrm{null}\,}$$

$$\newcommand{\range}{\mathrm{range}\,}$$

$$\newcommand{\RealPart}{\mathrm{Re}}$$

$$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$

$$\newcommand{\Argument}{\mathrm{Arg}}$$

$$\newcommand{\norm}[1]{\| #1 \|}$$

$$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$$

$$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\AA}{\unicode[.8,0]{x212B}}$$

$$\newcommand{\vectorA}[1]{\vec{#1}} % arrow$$

$$\newcommand{\vectorAt}[1]{\vec{\text{#1}}} % arrow$$

$$\newcommand{\vectorB}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$$

$$\newcommand{\vectorC}[1]{\textbf{#1}}$$

$$\newcommand{\vectorD}[1]{\overrightarrow{#1}}$$

$$\newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}}$$

$$\newcommand{\vectE}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{\mathbf {#1}}}}$$

$$\newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$$

$$\newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$$

$$\newcommand{\avec}{\mathbf a}$$ $$\newcommand{\bvec}{\mathbf b}$$ $$\newcommand{\cvec}{\mathbf c}$$ $$\newcommand{\dvec}{\mathbf d}$$ $$\newcommand{\dtil}{\widetilde{\mathbf d}}$$ $$\newcommand{\evec}{\mathbf e}$$ $$\newcommand{\fvec}{\mathbf f}$$ $$\newcommand{\nvec}{\mathbf n}$$ $$\newcommand{\pvec}{\mathbf p}$$ $$\newcommand{\qvec}{\mathbf q}$$ $$\newcommand{\svec}{\mathbf s}$$ $$\newcommand{\tvec}{\mathbf t}$$ $$\newcommand{\uvec}{\mathbf u}$$ $$\newcommand{\vvec}{\mathbf v}$$ $$\newcommand{\wvec}{\mathbf w}$$ $$\newcommand{\xvec}{\mathbf x}$$ $$\newcommand{\yvec}{\mathbf y}$$ $$\newcommand{\zvec}{\mathbf z}$$ $$\newcommand{\rvec}{\mathbf r}$$ $$\newcommand{\mvec}{\mathbf m}$$ $$\newcommand{\zerovec}{\mathbf 0}$$ $$\newcommand{\onevec}{\mathbf 1}$$ $$\newcommand{\real}{\mathbb R}$$ $$\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}$$ $$\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}$$ $$\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}$$ $$\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}$$ $$\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}$$ $$\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}$$ $$\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}$$ $$\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}$$ $$\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}$$ $$\newcommand{\laspan}[1]{\text{Span}\{#1\}}$$ $$\newcommand{\bcal}{\cal B}$$ $$\newcommand{\ccal}{\cal C}$$ $$\newcommand{\scal}{\cal S}$$ $$\newcommand{\wcal}{\cal W}$$ $$\newcommand{\ecal}{\cal E}$$ $$\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}$$ $$\newcommand{\gray}[1]{\color{gray}{#1}}$$ $$\newcommand{\lgray}[1]{\color{lightgray}{#1}}$$ $$\newcommand{\rank}{\operatorname{rank}}$$ $$\newcommand{\row}{\text{Row}}$$ $$\newcommand{\col}{\text{Col}}$$ $$\renewcommand{\row}{\text{Row}}$$ $$\newcommand{\nul}{\text{Nul}}$$ $$\newcommand{\var}{\text{Var}}$$ $$\newcommand{\corr}{\text{corr}}$$ $$\newcommand{\len}[1]{\left|#1\right|}$$ $$\newcommand{\bbar}{\overline{\bvec}}$$ $$\newcommand{\bhat}{\widehat{\bvec}}$$ $$\newcommand{\bperp}{\bvec^\perp}$$ $$\newcommand{\xhat}{\widehat{\xvec}}$$ $$\newcommand{\vhat}{\widehat{\vvec}}$$ $$\newcommand{\uhat}{\widehat{\uvec}}$$ $$\newcommand{\what}{\widehat{\wvec}}$$ $$\newcommand{\Sighat}{\widehat{\Sigma}}$$ $$\newcommand{\lt}{<}$$ $$\newcommand{\gt}{>}$$ $$\newcommand{\amp}{&}$$ $$\definecolor{fillinmathshade}{gray}{0.9}$$

The physical mechanism of ocean tides has been described in detail in Chap- ter Two. Let’s briefly recall: two tidal bulges are created on opposite sides of the Earth, one on the side facing the Moon, the other on the “far side” with respect to Moon. A common misconception is to attribute each bulge to different mechanisms. The truth is that both are manifestations of the same force, referred to by physicists and astronomers as the tidal force. Any celestial body experiencing gravitational pull from another celestial body does experience the tidal force: it tends to stretch the body (actually, both bodies) along the line connecting the centers of mass of both bodies. For instance, if one of the bodies were a giant water balloon, perfectly spherical if no force were acting on it – then, subjected to a tidal force, this water balloon would change its spherical as if it tended to assume a football shape.

Well, one could say, if so, then the whole Earth should be stretched, and not only the waters. The answer is that not only the waters are experiencing the tidal force, the rest of the Earth is experiencing them as well. So how comes that only the water bulges up? This is not true, the rocky core of Earth also bulges up – yet, because its greater stiffness it bulges up less than the water does. Therefore, what we observe is the actual bulging up of water minus the bulging up Earth.

Because of the apparent motion of Moon across the sky, the tidal bulges also move, following the Moon’s position. Accordingly, for an observer at a fixed location on Earth the two bulges with the two low tides in between look like a “tidal wave” with two “crests” (one at the Earth’s near side and one at its far side, with respect to Moon) and two troughs. It takes the crests (and the troughs as well) 24 hours and 50 minutes to travel all the way around the globe (the same time period that separates successive Moon rises – note that if you observe a Moon rise one evening, next evening it will occur 50 minutes later). Or, the tide bulges arrive every 12 hours and 25 minutes.

There is one more interesting effect, known as the tidal bulge offset. Namely, the highest tide occurs not at the moment the moon is directly overhead. There is friction between the tidal waters and the Earth’s surface. The Earth rotation is much faster than the orbital rotation of Moon – Earth makes one revolution in one day = 24 hours, while it takes Moon 29 days to complete a single orbit. Due to the faster Earth’s rotation, the friction “captures” tidal bulge and moves it about 3 ahead of the line connecting the centers of the two bodies, as shown if Fig. 8.12. It means that for an ob- server on Earth the maximum bulging occurs not when the Moon is directly overhead, but when Moon is 3 behind its overhead position. To turn by an angle of 3 takes Earth about 12 minutes. Accordingly, for the observer the maximum tide “lags behind” the Moon’s overhead position by about 12 minutes. Well, one could say, if so, then the whole Earth should be stretched, and not only the waters. The answer is that not only the waters are experi- encing the tidal force, the rest of the Earth is experiencing them as well. So

how comes that only the water bulges up? This is not true, the rocky core of Earth also bulges up – yet, because its greater stiffness it bulges up less than the water does. Therefore, what we observe is the actual bulging up of water minus the bulging up Earth.

The existence of the tidal bulge offset has important consequences. Be- cause of the asymmetric position of the bulges relative to Moon, the Earth’s gravitational field at the Moon’s position is not perfectly symmetric – there appears in it an additional weak component tangent to the Moon’s orbit, oriented in the same direction as the Moon’s orbital velocity vector. This extra force continuously “pushing” Moon tends to increase its kinetic energy, which results in slow increase of the Earth-Moon distance – about 11 inch per year. A reaction to this force acting on Moon is a force of the same magni- tude, and opposite orientation (due to Newton’s 3rd Law of Dynamics), thus tending to slow down the Earth’s rotation. Such a process indeed occurs, 100 years from now the day will be longer than the present one by about 2 milliseconds (or will be longer than today by one second about 50,000 years from now). So, the process is very slow. However, it also provides an answer to the question of what is the source of the energy that drives all the tidal effects on Earth. The answer is:the resource from which energy is taken to move tidal waters is the kinetic energy of Earth’s rotation. Because energy is taken from this reservoir, the kinetic energy of Earth’s rotation decreases, resulting in slowing the rotation down.

We have not discussed the role of Sun in tidal phenomena on Earth. The contribution of Sun is 44 percent that of the Moon, weaker than that of the Moon. The Sun’s impact on the tides i.e., slightly less than half. The effects can add up, which produces the strongest tides – or partially subtract, producing the weakest one. Much information about the Sun’s role is given in this Wikipedia article.

8.3.1: A Bit More of Physics is shared under a CC BY 1.3 license and was authored, remixed, and/or curated by Tom Giebultowicz.