at a point on the earth is the difference between (1) and
(2), which simplifies to
F t ¼ Ga
M e M m
R
3
ð3Þ
where a is the radius of the earth.
The local variation in the tide-generating force does not
cause the water on the earth’s surface to be drawn into two
bulges on opposite sides of the earth. Instead it is the small
horizontal component of the force which causes water
movement (Figure 1b). This force, although small, has
nothing to oppose it; whereas the vertical component of
the tide-generating force acts against the much larger
gravitational attraction acting upon the ocean.
The solar tidal force can be thought of in a similar
manner. The mass of the sun is much greater than the mass
of the moon, but this is offset by the greater distance from
the earth. Therefore, the solar tidal forces are a factor of
0.46 weaker than the lunar forces.
Equilibrium tidal theory
To improve understanding of tides, Newton developed the
equilibrium tidal theory, in which three main assumptions
are made: (1) the earth is covered by an ocean of uniform
depth with no land masses; (2) the ocean responds immediately to the tide-generating force; and (3) the effects of
rotation and friction can be ignored (Masselink et al.,
2003). Let us assume a stationary moon, aligned with the
earth’s equator. The lunar tide-generating force would
cause bulges on opposite sides of the earth in our uniformly deep ocean (Figure 2a). The earth rotates in an anticlockwise direction on its polar axis, beneath these bulges,
taking 24 h (a solar day) to complete one rotation. Hence,
anywhere on the earth’s surface you would experience two
high tides each day, but with decreased height as you
moved north or south of the equator (Figure 2b). In reality
by the time the earth has completed one full rotation, the
moon has moved on in its rotation around the barycenter.
It takes about 50 min longer than a solar day for the point
on the equator to reach the maximum part of the first bulge
again. This is why high semidiurnal tides occur every 12 h
and 25 min, not every 12 h.
At any point on the earth, the change in the height of the
tide can be described by a simple harmonic curve of the
form (Figure 2b, d)
H cos ot À g
ð
Þ
ð4Þ
where H is the amplitude, o is the frequency, g is the phase
lag relative to a defined time zero, and t is time. Assuming
that the moon rotates around the earth in a purely circular
path aligned with the equator, we term this harmonic M 2 .
If we were to ignore the moon and just consider the sun,
then the same features would be observed, except the time
between high waters would be exactly 12 h and the height
of high water would be just less than half of that observed
for the moon. This harmonic is termed S 2 .
Now consider the combined effect of the moon and
sun. When the earth, moon, and sun are aligned during
full or new moon, the equilibrium tidal bulges associated
with the moon and sun are in phase and combine to
create a spring tide with a larger than average tidal range
(Figure 3). This occurs about every 15 days. When the
moon is positioned half way between the new and
full phases, the two tidal bulges are out of phase and
combine to produce a neap tide with smaller than average
tidal range.
Tidal patterns are complicated by the fact that for much
of the year the moon and sun are not aligned with the equator. The earth rotates around the sun on a plane called the
ecliptic, which is inclined at 23
27
0 to the equator. The
moon orbits the earth on a plane that is included by 5
9
0
to the ecliptic. As the moon revolves, its position above
the earth varies between latitudes 28
36
0 (23
27
0 + 5
9
0 )
north and south of the equator, and the tidal bulges are
titled relative to this angle (Figure 2c). An observer on
the earth’s surface would experience two high tides
a day, but the relative heights of each would be different,
with the relative differences increasing moving farther
north or south (Figure 2d). This so-called diurnal inequality reaches a maximum when the moon is at its maximum
northern or southern declination about every 2 weeks.
The sun’s declination varies over a 365.25-day cycle,
by 23
27
0 north or south of the equator, causing the seasons. The largest diurnal tides occur in June and December
during the solstices and the smallest diurnal tides in March
and September during the equinoxes. As the strength of
the diurnal force increases, the semidiurnal force reduces,
and vice versa (Pugh, 2004). Hence, the semidiurnal tidal
forces reach their maximum at the equinoxes.
Tidal patterns are further complicated because the
orbits of the earth and moon are elliptical, not circular.
As a result the distance between the moon and sun with
the earth varies, altering the strength of the tide-generating
force. Over a period of 27.6 days, the moon moves
from perigee (its closest approach to the earth) to apogee
(its farthest approach to the earth) and back. At perigee
and apogee the lunar tidal forces are 15 % greater or less
than average, respectively. The orbit of the earth around
the sun is also elliptical, moving from perihelion (closest
to the sun) to aphelion (farthest from the sun) over
a period of 365.25 days. The difference in distance
between these two positions is only about 4 % so tides
are only marginally larger at perihelion compared to
aphelion.
Over longer time scales, variations in tidal forces arise
as a result of the 8.85-year cycle of lunar perigee (which
influences tides as a quasi 4.4-year cycle) and the 18.61year lunar nodal cycle (Haigh et al., 2011).
Harmonic analysis and tidal prediction
Predicting times and heights of tides has many practical
applications, and there has been a long history of development from simple techniques that related the time of high
708
TIDES
(2), which simplifies to
F t ¼ Ga
M e M m
R
3
ð3Þ
where a is the radius of the earth.
The local variation in the tide-generating force does not
cause the water on the earth’s surface to be drawn into two
bulges on opposite sides of the earth. Instead it is the small
horizontal component of the force which causes water
movement (Figure 1b). This force, although small, has
nothing to oppose it; whereas the vertical component of
the tide-generating force acts against the much larger
gravitational attraction acting upon the ocean.
The solar tidal force can be thought of in a similar
manner. The mass of the sun is much greater than the mass
of the moon, but this is offset by the greater distance from
the earth. Therefore, the solar tidal forces are a factor of
0.46 weaker than the lunar forces.
Equilibrium tidal theory
To improve understanding of tides, Newton developed the
equilibrium tidal theory, in which three main assumptions
are made: (1) the earth is covered by an ocean of uniform
depth with no land masses; (2) the ocean responds immediately to the tide-generating force; and (3) the effects of
rotation and friction can be ignored (Masselink et al.,
2003). Let us assume a stationary moon, aligned with the
earth’s equator. The lunar tide-generating force would
cause bulges on opposite sides of the earth in our uniformly deep ocean (Figure 2a). The earth rotates in an anticlockwise direction on its polar axis, beneath these bulges,
taking 24 h (a solar day) to complete one rotation. Hence,
anywhere on the earth’s surface you would experience two
high tides each day, but with decreased height as you
moved north or south of the equator (Figure 2b). In reality
by the time the earth has completed one full rotation, the
moon has moved on in its rotation around the barycenter.
It takes about 50 min longer than a solar day for the point
on the equator to reach the maximum part of the first bulge
again. This is why high semidiurnal tides occur every 12 h
and 25 min, not every 12 h.
At any point on the earth, the change in the height of the
tide can be described by a simple harmonic curve of the
form (Figure 2b, d)
H cos ot À g
ð
Þ
ð4Þ
where H is the amplitude, o is the frequency, g is the phase
lag relative to a defined time zero, and t is time. Assuming
that the moon rotates around the earth in a purely circular
path aligned with the equator, we term this harmonic M 2 .
If we were to ignore the moon and just consider the sun,
then the same features would be observed, except the time
between high waters would be exactly 12 h and the height
of high water would be just less than half of that observed
for the moon. This harmonic is termed S 2 .
Now consider the combined effect of the moon and
sun. When the earth, moon, and sun are aligned during
full or new moon, the equilibrium tidal bulges associated
with the moon and sun are in phase and combine to
create a spring tide with a larger than average tidal range
(Figure 3). This occurs about every 15 days. When the
moon is positioned half way between the new and
full phases, the two tidal bulges are out of phase and
combine to produce a neap tide with smaller than average
tidal range.
Tidal patterns are complicated by the fact that for much
of the year the moon and sun are not aligned with the equator. The earth rotates around the sun on a plane called the
ecliptic, which is inclined at 23
27
0 to the equator. The
moon orbits the earth on a plane that is included by 5
9
0
to the ecliptic. As the moon revolves, its position above
the earth varies between latitudes 28
36
0 (23
27
0 + 5
9
0 )
north and south of the equator, and the tidal bulges are
titled relative to this angle (Figure 2c). An observer on
the earth’s surface would experience two high tides
a day, but the relative heights of each would be different,
with the relative differences increasing moving farther
north or south (Figure 2d). This so-called diurnal inequality reaches a maximum when the moon is at its maximum
northern or southern declination about every 2 weeks.
The sun’s declination varies over a 365.25-day cycle,
by 23
27
0 north or south of the equator, causing the seasons. The largest diurnal tides occur in June and December
during the solstices and the smallest diurnal tides in March
and September during the equinoxes. As the strength of
the diurnal force increases, the semidiurnal force reduces,
and vice versa (Pugh, 2004). Hence, the semidiurnal tidal
forces reach their maximum at the equinoxes.
Tidal patterns are further complicated because the
orbits of the earth and moon are elliptical, not circular.
As a result the distance between the moon and sun with
the earth varies, altering the strength of the tide-generating
force. Over a period of 27.6 days, the moon moves
from perigee (its closest approach to the earth) to apogee
(its farthest approach to the earth) and back. At perigee
and apogee the lunar tidal forces are 15 % greater or less
than average, respectively. The orbit of the earth around
the sun is also elliptical, moving from perihelion (closest
to the sun) to aphelion (farthest from the sun) over
a period of 365.25 days. The difference in distance
between these two positions is only about 4 % so tides
are only marginally larger at perihelion compared to
aphelion.
Over longer time scales, variations in tidal forces arise
as a result of the 8.85-year cycle of lunar perigee (which
influences tides as a quasi 4.4-year cycle) and the 18.61year lunar nodal cycle (Haigh et al., 2011).
Harmonic analysis and tidal prediction
Predicting times and heights of tides has many practical
applications, and there has been a long history of development from simple techniques that related the time of high
708
TIDES