1. The Form factor F = [ K1 + O1 ] / [ M2 + S2 ] is customarily used to characterize the tide.
When 0.25 < F < 1.25 the tide is mixed - mainly semidiurnal
When 1.25 < F < 3.00 the tide is mixed - mainly diurnal
F > 3 the tide is diurnal
F < 0.25 the tide is semidiurnal

2. F > 3 the tide is diurnal
F < 0.25 the tide is semidiurnal
When 1.25 < F < 3.00 the tide is mixed - mainly diurnal
When 0.25 < F < 1.25 the tide is mixed - mainly semidiurnal
Superposition of constituents generates modulation - e.g. fortnightly, monthly
This applies for both sea level and velocity

4. Subtidal modulation by two tidal constituents

5. In Ponce de León Inlet: M2 = 0. 41 m; N2 = 0. 09 m; O1: 0. 06 m; S2: 0
In Ponce de León Inlet: M2 = 0.41 m; N2 = 0.09 m; O1: 0.06 m; S2: 0.06 m; K1= 0.08 m
F = [K1 + 01] / [S2 + M2 ] = 0.30
GNV

6. Panama
City
In Panama City, FL: M2 = m; N2 = m; O1: m; S2: m; K1= m
F = [K1 + 01] / [S2 + M2 ] = 7.52

7. The momentum balance is then:
Progressive wave
Assume linear, frictionless motion in the x direction only, under homogeneous conditions.
The momentum balance is then:
linear, partial differential equation (hyperbolic)
And the continuity equation is:
The solution is d’Alembert’s solution, which can be studied with the sinusoidal wave form:: a
time

9. This indicates that the flow is in phase with the elevation
time
This indicates that the flow is in phase
with the elevation
a C/H
time

10. Standing wave
The momentum balance is also:
And the continuity equation is:
The solution is:

11. This indicates that the flow is out of
time
This indicates that the flow is out of
phase with the elevation by 90 degrees
a C/H
time

12. Resonance L
At the mouth x = L,
Substituting
into
at x = L
For resonance to exist, the denominator should tend to zero, i.e.,
and
The natural period of oscillation is then:

13. H (m) L (km) C (m/s) TN (h) Long Island Sound 20 180 14 Chesapeake Bay10
250 28
Bay of Fundy 70 26
10.7
Mode 1
(n =1)
Merion’s Formula

14. , u
, u
, u , , , u u u
(kLw)-1
r/
Perillo (2012)

15. Tidal Waves With Friction
Momentum balance for a progressive wave:
Integrating vertically:
The bottom stress becomes:
Expanding this in a Fourier cosine series, to lowest order:
With continuity:
These are the governing equations for progressive tidal motion with friction.

16. If we let
and
We obtain the solution:
where
maximum U precedes maximum eta

18. Effects of Friction on a Standing Tidal Wave in an Estuary
For resonance, we have again U =0 at the head of the estuary, i.e.,

20. Winant (2007, JPO)

21. Effects of Rotation on a Progressive Tidal Wave in a Semi-enclosed basin
Solution:
R = C / f
KELVIN WAVE

22. Effects of Rotation on a Standing Tidal Wave in an Estuary
Two Kelvin waves of equal amplitude progressing in opposite directions.
Instead of having lines of no motion, we are now reduced to a central region --
amphidromic region-- of no motion at the origin. The interference of two
geostrophically controlled simple harmonic waves produces a change from a linear
standing wave to a rotary wave.

26. Effects of Bottom Friction on an amphydromic
system
Parker (1990)

27. Virtual Amphidromes
Parker (1990)

28. Virtual amphidromes
in Chesapeake Bay
Fisher (1986)

29. A EFFECTS OF CHANNELS ON TIDAL FLOWS B H (y) Euler’s formula x y
Huijts, et al.(2006, J. Geophys. Res., 111, C12016, doi: /2006JC003615)

31. Example of bathymetry effects on tidal flows in the James River

32. (Looking into the estuary)
Strongest tidal flows in channels relative to shoals
(Looking into the estuary)

33. (Looking into the estuary)
Tidal flows in channels lag behind tidal flows over shoals
Surface flows lag behind bottom flows
(Looking into the estuary)

34. Stronger tidal flows in channels than over shoals
Tidal currents lead over shoals relative to channels