Earth–moon orbit, in which the distance between the
moon and Earth varies over a 27.5546-day period, from
perigee (the moon closest to the Earth, and so a stronger
tidal force) to apogee (the moon farthest from the Earth,
and so a weaker tidal force) and back to perigee. This modulates the main lunar tidal force (M 2 ), and that modulation
is represented by N 2 combined M 2 . The diurnal lunar tidal
forces resulting from lunar declination are represented by
two diurnal tidal constituents, O 1 and K 1 , with periods of
25.8193 and 23.9345 h (and frequencies of 0.9295 cpd
and 1.0027 cpd). The minimum combined effect of these
two constituents occurs every 13.66 days, at the times
when the moon is over the equator. Their maximum combined effect occurs at maximum lunar declination, when
the moon is either farthest north of the equator or farthest
south of the equator. Because of these well-known astronomical frequencies, some form of tide prediction has
existed since even the earliest civilizations. For a history
of tide prediction, see Parker (2012).
Hydrodynamic effects on tide ranges and tidal
current speeds in estuaries
When the very long tide wave generated in the deep ocean
reaches the shallower water of the continental shelf and
the even shallower water of estuaries, it is slowed up,
amplified, modulated, and distorted by a number of hydrodynamic mechanisms. To understand what happens to
a tide wave in an estuary, it is helpful to first look at two
opposite extremes, the tide wave in a long river and the
tide wave in a closed-off bay, both basins being deep
enough for a minimal damping effect from bottom
friction.
Progressive tide waves
The tide wave propagates up the river as a progressive
wave (see Figure 1, top panel), which means that the crest
of the wave (high water) moves progressively up the
river, as does the trough of the wave (low water). In such
a progressive tide wave, the maximum flood current
(namely, when the current is flowing the fastest up the
river) occurs at the same time as high water, and the maximum ebb current (namely, when the current is flowing
the fastest down the river) occurs at the same time as
low water. Slack water (when the current speed is zero)
occurs exactly halfway between high water and low
water. If the river is of constant width, the amplitude
(tide range) of this progressive tide wave will not change
as it moves up the river. However, if the width decreases
as going upriver, then the amplitude (tide range) will
increase, because the same amount of water is being
forced through a smaller basin cross section. If the depth
of the river decreases, there is a similar though less dramatic amplifying effect (which is generally outweighed
by the increased energy loss from bottom friction due to
the shallower depths).
Standing tide waves
In a closed-off bay, the tide wave progressing up the bay
will be reflected at the closed end and will travel back
down the bay. This reflected wave is not observable by
someone on the shore because it is superimposed on the
incoming tide wave propagating up the bay, and it is
the combination of the two waves that is observed. The
resulting combined wave is called a standing wave,
because the high and low waters do not progress up the
bay (see Figure 1, bottom panel). The water surface simply
moves up and down everywhere at the same time, with the
largest tide range at the head of the bay. With a standing
tide wave, the tide range increases as one moves from
the ocean entrance toward the closed end of the bay,
assuming the length of the bay is less than or equal to
one fourth of a tidal wavelength from the head of the
bay. If the bay is longer than that, there will be a location
with minimum tide range (at one fourth of a tidal wavelength from the head of the bay), so from the entrance
moving up the bay the tide range would first decrease
and then increase. This location is explained by looking
at the incident and reflected progressive waves. In a
progressive wave, high water comes one half a
wavelength before low water, so if the high water of
a progressive wave travels a distance equal to one fourth
of a tidal wavelength up the bay to the head, where it is
reflected, and then travels one fourth of a wavelength back
down the bay, it will have gone one half a wavelength and
so coincide with low water of the incoming progressive
wave, and the two will roughly cancel each other out at
that location, producing the minimum tide range (if there
was no friction and the incident and reflected wave were
equal in amplitude, the minimum range would be zero
and it would be called a node).
For a standing wave, high waters occur at the same time
everywhere on one side of the minimum (node), which is
the same time that low waters occur on the other side. The
strongest tidal currents occur when water level is near
mean tide level, halfway between the times of low water
and high water. At the times of high water and low water,
there is no current flow (slack water). The water flows into
the bay, stopping the inward flow at high water; reverses
direction; and flows out of the bay until low water, at
which time it reverses again and starts flowing into the
bay again.
Amplification of tides in an estuary
The tidal wavelength is determined by the depth of the
estuary. If friction is ignored, the formula for the tidal
wavelength, l, is l ¼ T(gD)
½ , where D is the depth of
the water, T is the tidal period, and g is the acceleration
due to gravity. The shallower the bay, the shorter is the
wavelength. When the length of the bay equals one fourth
of a tidal wavelength, then the bay’s natural period of
oscillation will be the same as the tidal period. One finds
the largest tide ranges in bays that are exactly one fourth
of a tidal wavelength long, due to what is called resonance.
TIDAL HYDRODYNAMICS
687
moon and Earth varies over a 27.5546-day period, from
perigee (the moon closest to the Earth, and so a stronger
tidal force) to apogee (the moon farthest from the Earth,
and so a weaker tidal force) and back to perigee. This modulates the main lunar tidal force (M 2 ), and that modulation
is represented by N 2 combined M 2 . The diurnal lunar tidal
forces resulting from lunar declination are represented by
two diurnal tidal constituents, O 1 and K 1 , with periods of
25.8193 and 23.9345 h (and frequencies of 0.9295 cpd
and 1.0027 cpd). The minimum combined effect of these
two constituents occurs every 13.66 days, at the times
when the moon is over the equator. Their maximum combined effect occurs at maximum lunar declination, when
the moon is either farthest north of the equator or farthest
south of the equator. Because of these well-known astronomical frequencies, some form of tide prediction has
existed since even the earliest civilizations. For a history
of tide prediction, see Parker (2012).
Hydrodynamic effects on tide ranges and tidal
current speeds in estuaries
When the very long tide wave generated in the deep ocean
reaches the shallower water of the continental shelf and
the even shallower water of estuaries, it is slowed up,
amplified, modulated, and distorted by a number of hydrodynamic mechanisms. To understand what happens to
a tide wave in an estuary, it is helpful to first look at two
opposite extremes, the tide wave in a long river and the
tide wave in a closed-off bay, both basins being deep
enough for a minimal damping effect from bottom
friction.
Progressive tide waves
The tide wave propagates up the river as a progressive
wave (see Figure 1, top panel), which means that the crest
of the wave (high water) moves progressively up the
river, as does the trough of the wave (low water). In such
a progressive tide wave, the maximum flood current
(namely, when the current is flowing the fastest up the
river) occurs at the same time as high water, and the maximum ebb current (namely, when the current is flowing
the fastest down the river) occurs at the same time as
low water. Slack water (when the current speed is zero)
occurs exactly halfway between high water and low
water. If the river is of constant width, the amplitude
(tide range) of this progressive tide wave will not change
as it moves up the river. However, if the width decreases
as going upriver, then the amplitude (tide range) will
increase, because the same amount of water is being
forced through a smaller basin cross section. If the depth
of the river decreases, there is a similar though less dramatic amplifying effect (which is generally outweighed
by the increased energy loss from bottom friction due to
the shallower depths).
Standing tide waves
In a closed-off bay, the tide wave progressing up the bay
will be reflected at the closed end and will travel back
down the bay. This reflected wave is not observable by
someone on the shore because it is superimposed on the
incoming tide wave propagating up the bay, and it is
the combination of the two waves that is observed. The
resulting combined wave is called a standing wave,
because the high and low waters do not progress up the
bay (see Figure 1, bottom panel). The water surface simply
moves up and down everywhere at the same time, with the
largest tide range at the head of the bay. With a standing
tide wave, the tide range increases as one moves from
the ocean entrance toward the closed end of the bay,
assuming the length of the bay is less than or equal to
one fourth of a tidal wavelength from the head of the
bay. If the bay is longer than that, there will be a location
with minimum tide range (at one fourth of a tidal wavelength from the head of the bay), so from the entrance
moving up the bay the tide range would first decrease
and then increase. This location is explained by looking
at the incident and reflected progressive waves. In a
progressive wave, high water comes one half a
wavelength before low water, so if the high water of
a progressive wave travels a distance equal to one fourth
of a tidal wavelength up the bay to the head, where it is
reflected, and then travels one fourth of a wavelength back
down the bay, it will have gone one half a wavelength and
so coincide with low water of the incoming progressive
wave, and the two will roughly cancel each other out at
that location, producing the minimum tide range (if there
was no friction and the incident and reflected wave were
equal in amplitude, the minimum range would be zero
and it would be called a node).
For a standing wave, high waters occur at the same time
everywhere on one side of the minimum (node), which is
the same time that low waters occur on the other side. The
strongest tidal currents occur when water level is near
mean tide level, halfway between the times of low water
and high water. At the times of high water and low water,
there is no current flow (slack water). The water flows into
the bay, stopping the inward flow at high water; reverses
direction; and flows out of the bay until low water, at
which time it reverses again and starts flowing into the
bay again.
Amplification of tides in an estuary
The tidal wavelength is determined by the depth of the
estuary. If friction is ignored, the formula for the tidal
wavelength, l, is l ¼ T(gD)
½ , where D is the depth of
the water, T is the tidal period, and g is the acceleration
due to gravity. The shallower the bay, the shorter is the
wavelength. When the length of the bay equals one fourth
of a tidal wavelength, then the bay’s natural period of
oscillation will be the same as the tidal period. One finds
the largest tide ranges in bays that are exactly one fourth
of a tidal wavelength long, due to what is called resonance.
TIDAL HYDRODYNAMICS
687