1. Spring & neap tides
Tidal range often changes regularly, i.e. every fortnight (14 day period)
We see:
Spring tides - times of greater tidal range; coincide with full moon or new moon
Tidal range during spring tides is usually 20% higher than mean tidal range
Neap tides - times of lower tidal range; coincide with first or third quarter
Tidal range during neap tides is usually 20% lower than the mean tidal range

2. Pliny the Elder (23-79 A.D.): tides ‘follow’ the moon
If that were strictly true, would have diurnal (daily) tides everywhere

3. Origin of tides - equilibrium model
Newton proposed the first explanation for semidiurnal tides with his equilibrium model
In Newton’s equilibrium model:
Earth & moon exist in isolation
Earth is a non-rotating sphere
Have a single ocean that encircles the globe
Ocean is static, i.e. has no currents
The only forces acting on ocean result from the movement of Earth & moon about their common center of mass

4. Equilibrium means ‘no net force’
Earth & moon persist in their relative motion, so there are no net forces acting on their centers of mass - forces must sum to zero
Earth & moon exert gravitational attraction on each other, & so fall toward each other
Earth & moon never collide because they are moving past each other - each has inertia
Gravitational attraction is exactly sufficient to produce the centripetal acceleration required to move each body in a circular path about their common center of mass

5. Explain this situation by saying that the two bodies are forced outward by an apparent (or fictitious) force - the centrifugal force

6. But have local inequities in forces
At center of Earth, moon’s gravitational attraction balances exactly the centripetal acceleration for Earth’s circular path
Elsewhere on Earth, gravitational attraction need not balance centripetal acceleration exactly
Because all points on Earth trace out circles of equal radii, all points experience equal centripetal acceleration
Gravitational attraction exerted by moon diminishes with increasing distance according to the inverse square law
If gravitational & centripetal accelerations have different magnitudes at different points, forces cannot be equal

7. Net forces produce accelerations
On moonward side of Earth, gravitational attraction is greater, so particles are accelerated toward moon
On side of Earth facing away from moon, inertial effects related to earth’s circular path exceed gravitational attraction, so particles are accelerated away from the moon
Accelerations perpendicular to earth’s surface are miniscule - on the order of one millionth of Earth’s gravitational attraction
Accelerations parallel to Earth’s surface are more effective – they displace water into two bulges, one on side facing moon & one on side away from moon

8. Equilibrium tide
If Earth rotates beneath the two bulges, points experience semidiurnal tides
Call this the equilibrium tide
Refinements to equilibrium theory include:
Plane of moon’s orbit is inclined 61.5° to Earth’s rotation axis - expect daily inequity that varies with a monthly cycle
Moon’s orbit is elliptical - expect variation in magnitude of bulge with a day cycle

9. Solar equilibrium tide
Earth-sun interactions produce ocean tides
Earth-moon distance = 59 x RE; Mass of moon 1/82 mass of earth
Earth-sun distance = 23,000 x RE; Mass of sun 330,00 x mass of earth
Solar tide is 47% of that generated by earth moon interactions
Solar tide has 12 hr period; see effects of tilt of rotation axis & elliptical orbit, too
Add lunar & solar equilibrium tides together to get spring & neap tide

10. Shortcomings of the equilibrium model
It predicts semidiurnal tides at all locations - not observed
It predicts that high tides should occur when moon passes overhead or 12 hours 25 minutes later - rarely observed
Calculations suggest that tidal ranges should be cm - observed tidal ranges are often much larger
It predicts values for daily inequities that are rarely observed

11. What is wrong with equilibrium model for the tides?
It ignores the fact that ocean basins have irregular shapes, that bulges must respond to friction in moving through basins, & that water itself has inertia once it is moving

12. Dynamic theory for tides
Begins where equilibrium model ends, i.e. with two bulges created by gravitational/inertial interactions of Earth & moon
Explains real tides by envisioning tidal bulges as tidal wave
Tidal wave has small wave height - about 50 cm in open ocean
Tidal wave has very long wavelength, about one half earth’s circumference = 20,000 km

13. The tidal wave
Wavelength (L) of tidal wave = 1/2 Earth’s circumference = 20,000 km
Water depth of oceans = 4 km <<< L/2
Tidal wave is a shallow water wave everywhere, i.e. it interacts continually with the ocean bottom
As a shallow water wave, tidal wave feels the bottom, slows, steepens, & sometimes breaks
Tidal wave reflects, refracts, & interferes with other waves or with reflections of itself

14. Standing waves
Have a wave with long wavelength (L) in a rectangular basin of uniform depth (D)
Wave advances across basin, reflects off boundary, & travels back across basin
Given enough time, a regular wave will interfere to produce a standing or stationary wave
Where water level does not change = nodal line (node)
Where water level changes the most = antinodal lines (antinodes)
Oscillation period for standing wave with one node = [2 x L]/ [g x D]1/2 (depends on basin length & depth)
Length & depth dependence holds for all standing waves

15. In a rectangular basin of uniform depth, perturbing wave advances across basin, reflects off boundary, & travels back across basin. In many cases, interference between inducing wave & reflected wave generates a standing wave.
•Nodal line or node = where water level does not change
•Antinodal lines or antinodes = where water level changes the most
•Oscillation period depends on basin length & depth
•May see resonance if natural period of basin is close to that of
perturbation

16. Tidal standing waves
Tidal standing waves are not free waves, where generating force acts only at the outset
Tidal standing waves are forced waves, where the wave generating force(s) continue(s) to act as the system responds
Each passage of the moon overhead is another disturbing force
Complicated wave motion in any basin is the sum of latest disturbance interacting with waves generated by numerous previous passes overhead
Depending on the shape of a basin, the net result may be a standing wave that corresponds to a diurnal tide, a semidiurnal tide, or a mixed tide

17. Kelvin waves
Special kind of standing wave seen in large basins where masses of water experience the Coriolis effect
Wave crest swings about the basin margin like water swirling in a glass
Nodal point, called an amphidromic point, occurs near the center of the basin
Co-tidal lines - connect points that have high tides at the same time, typically measured in hours after the moon crosses the Greenwich meridian
Co-range lines connect points that have equal tidal ranges

19. Amphidromic point (node) occurs near center of the basin
Co-tidal lines connect points that have high tides at the same time (denoted by hours after the moon crossed the Greenwich meridian)
Co-range lines connect points that have equal tidal ranges
Co-tidal lines are spokes; co-range lines are closed loops about node

20. Tidal currents
Can understand tidal currents by envisioning tide as one of three types of wave
Progressive wave tide - where coast is too irregular to give pronounced reflection, so no standing wave
High tide = wave crest; low tide = wave trough
Flood current coincides with high tide; ebb current coincides with low tide
Standing wave tide - in confined rectangular basin
Slack water coincides with high or low water level
Flood and ebb currents coincide with mean water level
Kelvin wave tide - azimuth & magnitude of currents may vary