EM 1110-2-1100 (Part II)
30 Apr 02
tide-producing forces. However, the actual tide does not conform to this theoretical value because of friction
and inertia as well as differences in the depth and distribution of land masses of the earth.
(b) Because of the above complexities, it is impossible to exactly predict the tide at any place on the earth
based on a purely theoretical approach. However, the tide-producing forces (and their expansion component
terms) are harmonic; i.e., they can be expressed as a cosine function whose argument increases linearly with
time according to known speed criteria. If the expansion terms of the tide-producing forces are combined
according to terms of identical period (speed), then the tide can be represented as a sum of a relatively small
number of harmonic constituents. Each set of constituents of common period are in the form of a product of
an amplitude coefficient and the cosine of an argument of known period with phase adjustments based on
time of observation and location. Observational data at a specific time and location are then used to
determine the coefficient multipliers and phase arguments for each constituent, the sum of which are used to
reconstruct the tide at that location for any time. This concept represents the basis of the harmonic analysis,
i.e., to use observational data to develop site-specific coefficients that can be used to reconstruct a tidal series
as a linear sum of individual terms of known speed. The following presentation briefly describes the use of
harmonic constants to predict tides.
(c) Tidal height at any location and time can be written as a function of harmonic constituents according
the following general relationship
H(t) ' H0 % j fn Hn cos [ant % (V0 % u)n & κn]
N
(II-5-16)
n'1
where
H(t) = height of the tide at any time t
H0 = mean water level above some defined datum such as mean sea level
Hn = mean amplitude of tidal constituent n
fn = factor for adjusting mean amplitude Hn values for specific times
an = speed of constituent n in degrees/unit time
t = time measured from some initial epoch or time, i.e., t = 0 at t0
κn = epoch of constituent n, i.e., phase shift from tide-producing force to high tide from t0
(d) In the above formula, tide is represented as the sum of a coefficient multiplied by the cosine of its
respective arguments. A finite number of constituents are used in the reconstruction of a tidal signal. Values
for the site-specific arguments (H0, Hn, and κn) are computed from observed tidal time series data, usually
from a least squares analysis. The National Oceanic and Atmospheric Administration's (NOAA) National
Ocean Survey (NOS) generally provides 37 constituents in their published harmonic analyses (generally
based on an analysis of a minimum of 1 year of prototype data). The NOS constituents, along with the
corresponding period and speed of each, are listed in Table II-5-3. The time-specific arguments (fn and
V0 + u) are determined from formulas or tables as will be discussed below or through application of programs
II-5-14
Water Levels and Long Waves
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