Predicted tide for Bridgeport CT, September 1-28, 1991

The tide prediction method used today in the
United States is called ``harmonic analysis.''
The ideal tide curve for any given port is represented as an
average height *Z0* plus a
sum of terms (``constituents'') each of which is of the form
*f(t) = H* cos*(at + \phi).*
The time *t* is measured in hours, and *f* comes out in feet. The
numbers * H,a,\phi* are the amplitude, the speed and the
phase of the constituent. The speed is given in degrees/hour, and
the phase in degrees (so this is the cosine function that accepts
input in degrees!).
Different constituents have different speeds, which are
sums and differences of small integral multiples
of 5 basic astronomical speeds, which are:

*T* the rotation of the Earth on its axis,
with respect to the Sun, 15 degrees/hour

*h* the rotation of the Earth about the sun, .04106864 degrees/hour

*s* the rotation of the Moon about the Earth, .54901653 degrees/hour

*p* the precession of the Moon's perigee, .00464183 degrees/hour

*N* the precession of the plane of the Moon's orbit,
-.00220641 degrees/hour.

Thus the rotation of the Earth with respect to the fixed stars
is *T+h* = 15.04106864 degrees/hour, and the change in the moon's
longitude per hour is *T+h-s*= 14.49205211 degrees/hour.

The amplitude *H* and phase *\phi*
for each constituent are determined from
the tidal record by Fourier analysis. The
number of constituents needed for accurate prediction
varies from port to port. In making
up the tide tables for Long
Island Sound, the National Oceanic and Atmospheric Administration
uses 23 constituents. The eleven whose amplitude is greater than
.1 foot are:

Symbol speed amplitude phase at midnight, 9/1/91 M2 2T-2s+2h = 28.984 3.185 -127.24 N2 2T-3s+2h+p = 28.439 0.696 263.60 S2 2T = 30.000 0.538 -343.66 K1 T+h = 15.041 0.295 142.02 L2 2T-s+2h-p = 29.528 0.277 -4.72 O1 T-2s+h = 13.943 0.212 505.93 Sa h = 0.041 0.192 301.50 \nu2 2T-3s+4h-p = 28.512 0.159 45.70 K2 2T+2h = 30.082 0.144 -2.55 Mm s-p = 0.544 0.108 86.82 P1 T-h = 14.958 0.102 340.11

The M2 constituent has by far the largest coefficient. This term alone would give the tide if the sun could be neglected, and if the moon orbited in a perfect circle in the plane of the earth's equator. For Bridgeport, starting at midnight on Sept 1, 1991, this term is (from the table above)

M2 = 3.185 cos (28.984 t -127.24)

M2 for Bridgeport CT, September 1-28, 1991

The constituent S2 would give the solar tide if the sun was always in the earth's equatorial plane and the earth's orbit was a perfect circle. The plot of M2 + S2 shows the combined tidal effect of this ideal sun and the ideal moon of the previous graph.

M2+S2 for Bridgeport CT, September 1-28, 1991

This shows the sun and moon working together at full and new moon, and at
cross purposes at first and third quarters. An additional
significant effect, too subtle to be read from this graph,
is a 2-week cycle of advancement and retardation of the
*times* of high and low tides called the
priming/lagging of the tide.

The N2 constituent involves the non-circularity of the moon's orbit.
During the month the moon describes an ellipse, and the tides are
higher when it is near its perigee (nearest the earth) and
lower when it is near its apogee (farthest). The perigee itself
precesses with angular velocity *p*, so this pattern shifts
slightly from month to month. The next graph
shows the sum of the M2 and N2 components.

M2+N2 for Bridgeport CT, September 1-28, 1991

The diurnal constituents K1, O1, P1 take into account
(among other things) the
inclination of the earth's equatorial plane with respect to the
plane of the moon's orbit. During a month, the moon spends about two
weeks above the equator, and then two weeks below. When it is above,
for us in the Northern Hemisphere the high tide when the moon is
visible will be higher than the next tide (when the moon is below
the earth from our point of view), and vice-versa when it is
below the equator. This effect is called the * diurnal inequality*.
Twice a month, when the moon crosses the equator, the two tides
are roughly equal. This is shown in the next graph,
which gives the plot
of M2+K1. (The corresponding inequality for the sun
reverses itself twice a year, when the sun crosses the equator
at the equinoxes on Sept. 21 and March 21).

M2+K1 for Bridgeport CT, September 1-28, 1991

When all 23 components are added together, and added to the average height (3.35 at Bridgeport), we get the curve shown in the graph at the top of the page. This represents the predicted tide for Bridgeport for September 1-28, 1991. The actual height of the water depends also on meteorological factors such as barometric pressure and the direction and force of the wind.

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