The theory of tides is the application of continuum mechanics to interpret and predict the tidal deformations of planetary and satellite bodies and their atmospheres and oceans, under the gravitational loading of another astronomical body or bodies. It commonly refers to the fluid dynamic motions for the Earth's oceans.

## Origin of theory

In 1616, Galileo Galilei wrote Discourse on the Tides (in Italian: Discorso del flusso e reflusso del mare),[1] a paper in which he tried to explain the occurrence of the tides as the result of the Earth's rotation around the Sun. However, Galileo's theory was, in the later Newtonian terms, an error.[1] Later analysis over the centuries had led to the current tidal physics.

## Tidal physics

### Tidal forcing

 A. Lunar gravitational potential: this depicts the Moon directly over 30° N (or 30° S) viewed from above the Northern Hemisphere. B. This view shows same potential from 180° from view A. Viewed from above the Northern Hemisphere. Red up, blue down.

The forces discussed here apply to body (Earth tides), oceanic and atmospheric tides. Atmospheric tides on Earth, however, tend to be dominated by forcing due to solar heating. Oceanic tides act according to the Dynamic theory of tides.

On the planet (or satellite) experiencing tidal motion consider a point at latitude $$\varphi$$ and longitude $$\lambda$$ at distance $$a$$ from the center of mass, then this point can be written in cartesian coordinates as $$\mathbf{p} = a\mathbf{x}$$ where

$\mathbf{x} = (\cos \lambda \cos \varphi, \sin \lambda \cos \varphi, \sin \varphi).$

Let $$\delta$$ be the declination and $$\alpha$$ be the right ascension of the deforming body, the Moon for example, then the vector direction is

$\mathbf{x}_m = (\cos \alpha \cos \delta, \sin \alpha \cos \delta, \sin \delta),$

and $$r_m$$ be the orbital distance between the center of masses and $$M_m$$ the mass of the body. Then the force on the point is

$\mathbf{F}_{a}= \frac{G M_m (r_m\mathbf{x}_{m}-a\mathbf{x})}{R^3}.$

where $$R = \|r_m\mathbf{x}_{m}-a\mathbf{x}\|$$

For a circular orbit the angular momentum $$\omega$$ centripetal acceleration balances gravity at the planetary center of mass

$Mr_{cm}\omega^2= \frac{G M M_m }{r_m^2}.$

where $$r_{cm}$$ is the distance between the center of mass for the orbit and planet and $$M$$ is the planetary mass. Consider the point in the reference fixed without rotation, but translating at a fixed translation with respect to the center of mass of the planet. The body's centripetal force acts on the point so that the total force is

$\mathbf{F}_p= \frac{G M_m (r_m\mathbf{x}_{m}-a\mathbf{x})}{R^3} -r_{cm}\omega^2\mathbf{x}_m.$

Substituting for center of mass acceleration, and reordering

$\mathbf{F}_p = G M_m r_m \left( \frac {1}{R^3} - \frac {1}{r_m^3} \right) \mathbf{x}_m -\frac{ ( G M_m a\mathbf{x})}{R^3}.$ In ocean tidal forcing, the radial force is not significant, the next step is to rewrite the $$\mathbf{x}_m$$ coefficient. Let $$\varepsilon= \frac {a} {r_m}$$ then

$R = r_m \sqrt{ 1+ \varepsilon ^2-2 \varepsilon ( \mathbf{x}_m,\mathbf{x} ) }$

where $$( \mathbf{x}_m,\mathbf{x} )= \cos z$$ is the inner product determining the angle z of the deforming body or Moon from the zenith. This means that

$\left( \frac {1}{R^3} - \frac {1}{r_m^3} \right) \approx \frac{3\varepsilon \cos z }{r_m^3},$

if ε is small. If particle is on the surface of the planet then the local gravity is $$g=\frac{ G M}{a^2}$$ and set $$\mu= \frac{M_m} {M}$$.

$\mathbf{F}_p = 3 g \mu \varepsilon^3 \cos z \mathbf{x}_m - \frac{ ( g \mu a^3\mathbf{x})}{R^3} + O(\varepsilon^4),$

which is a small fraction of $$g$$. Note also that force is attractive toward the Moon when the $$z<\pi/2$$ and repulsive when $$z > \pi/2$$.

This can also be used to derive a tidal potential.

### Laplace's tidal equations

in 1776, Pierre-Simon Laplace formulated a single set of linear partial differential equations, for tidal flow described as a barotropic two-dimensional sheet flow. Coriolis effects are introduced as well as lateral forcing by gravity. Laplace obtained these equations by simplifying the fluid dynamic equations. But they can also be derived from energy integrals via Lagrange's equation.

For a fluid sheet of average thickness D, the vertical tidal elevation ς, as well as the horizontal velocity components u and v (in the latitude φ and longitude λ directions, respectively) satisfy Laplace's tidal equations[2][3]: \begin{align} \frac{\partial \zeta}{\partial t} &+ \frac{1}{a \cos( \varphi )} \left[ \frac{\partial}{\partial \lambda} (uD) + \frac{\partial}{\partial \varphi} \left(vD \cos( \varphi )\right) \right] = 0, \\[2ex] \frac{\partial u}{\partial t} &- v \left( 2 \Omega \sin( \varphi ) \right) + \frac{1}{a \cos( \varphi )} \frac{\partial}{\partial \lambda} \left( g \zeta + U \right) =0 \qquad \text{and} \\[2ex] \frac{\partial v}{\partial t} &+ u \left( 2 \Omega \sin( \varphi ) \right) + \frac{1}{a} \frac{\partial}{\partial \varphi} \left( g \zeta + U \right) =0, \end{align} where Ω is the angular frequency of the planet's rotation, g is the planet's gravitational acceleration at the mean ocean surface, and U is the external gravitational tidal-forcing potential.

William Thomson (Lord Kelvin) rewrote Laplace's momentum terms using the curl to find an equation for vorticity. Under certain conditions this can be further rewritten as a conservation of vorticity.

## Tidal analysis and prediction

### Harmonic analysis

Laplace's improvements in theory were substantial, but they still left prediction in an approximate state. This position changed in the 1860s when the local circumstances of tidal phenomena were more fully brought into account by William Thomson's application of Fourier analysis to the tidal motions. Thomson's work in this field was then further developed and extended by George Darwin: Darwin's work was based on the lunar theory current in his time. His symbols for the tidal harmonic constituents are still used. Darwin's harmonic developments of the tide-generating forces were later brought up to date with modern developments by A T Doodson whose development of the tide generating potential (TGP) in harmonic form was carried out and published in 1921:[4] Doodson distinguished 388 tidal frequencies.[5] Doodson's analysis of 1921 was based on the then-latest lunar theory of E W Brown.[6]

Doodson devised a practical system for specifying the different harmonic components of the tide-generating potential, the Doodson Numbers, a system still in use.[7]

Since the mid-twentieth century further analysis has generated many more terms than Doodson's 388. About 62 constituents are of sufficient size to be considered for possible use in marine tide prediction, but sometimes many less even than that can predict tides to useful accuracy. The calculations of tide predictions using the harmonic constituents are laborious, and from the 1870s to about the 1960s they were carried out using a mechanical tide-predicting machine, a special-purpose form of analog computer now superseded in this work by digital electronic computers that can be programmed to carry out the same computations.

### Tidal constituents

Tidal constituents combine to give an endlessly-varying aggregate because of their different and incommensurable frequencies: the effect is visualized in an animation of the American Mathematical Society illustrating the way in which the components used to be mechanically combined in the tide-predicting machine. Amplitudes of tidal constituents are given below for the following example locations:

ME Eastport,
MS Biloxi,
PR San Juan,
AK Kodiak,
CA San Francisco, and
HI Hilo.

#### Higher harmonics

Darwin

Period

Phase

Doodson coefs

Doodson

Amplitude at example location (cm)

NOAA

Species

Symbol

(hr)

rate(°/hr)

n1 (L) n2 (m) n3 (y) n4 (mp)

number

ME

MS

PR

AK

CA

HI

order

Shallow water overtides of principal lunar M4 6.210300601 57.9682084 4 455.555 6.0 0.6 0.9 2.3 5
Shallow water overtides of principal lunar M6 4.140200401 86.9523127 6 655.555 5.1 0.1 1.0 7
Shallow water terdiurnal MK3 8.177140247 44.0251729 3 1 365.555 0.5 1.9 8
Shallow water overtides of principal solar S4 6 60 4 4 -4 491.555 0.1 9
Shallow water quarter diurnal MN4 6.269173724 57.4238337 4 -1 1 445.655 2.3 0.3 0.9 10
Shallow water overtides of principal solar S6 4 90 6 6 -6 * 0.1 12
Lunar terdiurnal M3 8.280400802 43.4761563 3 355.555 0.5 32
Shallow water terdiurnal 2"MK3 8.38630265 42.9271398 3 -1 345.555 0.5 0.5 1.4 34
Shallow water eighth diurnal M8 3.105150301 115.9364166 8 855.555 0.5 0.1 36
Shallow water quarter diurnal MS4 6.103339275 58.9841042 4 2 -2 473.555 1.8 0.6 1.0 37

#### Semi-diurnal

Darwin

Period

Phase

Doodson coefs

Doodson

Amplitude at example location (cm)

NOAA

Species

Symbol

(hr)

(°/hr)

n1 (L) n2 (m) n3 (y) n4 (mp)

number

ME

MS

PR

AK

CA

HI

order

Principal lunar semidiurnal M2 12.4206012 28.9841042 2 255.555 268.7 3.9 15.9 97.3 58.0 23.0 1
Principal solar semidiurnal S2 12 30 2 2 -2 273.555 42.0 3.3 2.1 32.5 13.7 9.2 2
Larger lunar elliptic semidiurnal N2 12.65834751 28.4397295 2 -1 1 245.655 54.3 1.1 3.7 20.1 12.3 4.4 3
Larger lunar evectional ν2 12.62600509 28.5125831 2 -1 2 -1 247.455 12.6 0.2 0.8 3.9 2.6 0.9 11
Variational MU2 12.8717576 27.9682084 2 -2 2 237.555 2.0 0.1 0.5 2.2 0.7 0.8 13
Lunar elliptical semidiurnal second-order 2"N2 12.90537297 27.8953548 2 -2 2 235.755 6.5 0.1 0.5 2.4 1.4 0.6 14
Smaller lunar evectional λ2 12.22177348 29.4556253 2 1 -2 1 263.655 5.3 0.1 0.7 0.6 0.2 16
Larger solar elliptic T2 12.01644934 29.9589333 2 2 -3 272.555 3.7 0.2 0.1 1.9 0.9 0.6 27
Smaller solar elliptic R2 11.98359564 30.0410667 2 2 -1 274.555 0.9 0.2 0.1 0.1 28
Shallow water semidiurnal 2SM2 11.60695157 31.0158958 2 4 -4 291.555 0.5 31
Smaller lunar elliptic semidiurnal L2 12.19162085 29.5284789 2 1 -1 265.455 13.5 0.1 0.5 2.4 1.6 0.5 33
Lunisolar semidiurnal K2 11.96723606 30.0821373 2 2 275.555 11.6 0.9 0.6 9.0 4.0 2.8 35

#### Diurnal

Darwin

Period

Phase

Doodson coefs

Doodson

Amplitude at example location (cm)

NOAA

Species

Symbol

(hr)

(°/hr)

n1 (L) n2 (m) n3 (y) n4 (mp)

number

ME

MS

PR

AK

CA

HI

order

Lunar diurnal K1 23.93447213 15.0410686 1 1 165.555 15.6 16.2 9.0 39.8 36.8 16.7 '4
Lunar diurnal O1 25.81933871 13.9430356 1 -1 145.555 11.9 16.9 7.7 25.9 23.0 9.2 6
Lunar diurnal OO1 22.30608083 16.1391017 1 3 185.555 0.5 0.7 0.4 1.2 1.1 0.7 15
Solar diurnal S1 24 15 1 1 -1 164.555 1.0 0.5 1.2 0.7 0.3 17
Smaller lunar elliptic diurnal M1 24.84120241 14.4920521 1 155.555 0.6 1.2 0.5 1.4 1.1 0.5 18
Smaller lunar elliptic diurnal J1 23.09848146 15.5854433 1 2 -1 175.455 0.9 1.3 0.6 2.3 1.9 1.1 19
Larger lunar evectional diurnal ρ 26.72305326 13.4715145 1 -2 2 -1 137.455 0.3 0.6 0.3 0.9 0.9 0.3 25
Larger lunar elliptic diurnal Q1 26.868350 13.3986609 1 -2 1 135.655 2.0 3.3 1.4 4.7 4.0 1.6 26
Larger elliptic diurnal 2Q1 28.00621204 12.8542862 1 -3 2 125.755 0.3 0.4 0.2 0.7 0.4 0.2 29
Solar diurnal P1 24.06588766 14.9589314 1 1 -2 163.555 5.2 5.4 2.9 12.6 11.6 5.1 30

#### Long period

Darwin

Period

Phase

Doodson coefs

Doodson

Amplitude at example location (cm)

NOAA

Species

Symbol

(hr)

(°/hr)

n1 (L) n2 (m) n3 (y) n4 (mp)

number

ME

MS

PR

AK

CA

HI

order

Lunar monthly Mm 661.3111655 0.5443747 0 1 -1 65.455 0.7 1.9 20
Solar semiannual Ssa 4383.076325 0.0821373 0 2 57.555 1.6 2.1 1.5 3.9 21
Solar annual Sa 8766.15265 0.0410686 0 1 56.555 5.5 7.8 3.8 4.3 22
Lunisolar synodic fortnightly Msf 354.3670666 1.0158958 0 2 -2 73.555 1.5 23
Lunisolar fortnightly Mf 327.8599387 1.0980331 0 2 75.555 1.4 2.0 0.7 24

## References

1. Rice University - Galileo's Theory of the Tides - by Rossella Gigli, retrieved 10 March 2010
2. http://kiwi.atmos.colostate.edu/group/dave/pdf/LTE.frame.pdf
3. http://siam.org/pdf/news/621.pdf
4. A T Doodson (1921), "The Harmonic Development of the Tide-Generating Potential", Proceedings of the Royal Society of London. Series A, Vol. 100, No. 704 (Dec. 1, 1921), pp. 305-329.
5. S Casotto, F Biscani, "A fully analytical approach to the harmonic development of the tide-generating potential accounting for precession, nutation, and perturbations due to figure and planetary terms", AAS Division on Dynamical Astronomy, April 2004, vol.36(2), 67.
6. D E Cartwright, "Tides: a scientific history", Cambridge University Press 2001, at pages 163-4.
7. See e.g. T D Moyer (2003), "Formulation for observed and computed values of Deep Space Network data types for navigation", vol.3 in Deep-space communications and navigation series, Wiley (2003), e.g. at pp.126-8.
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