2 U. S. COAST AND GEODETIC SURVEY 
GENERAL EXPLANATION OF TIDAL MOVEMENT 
4, That the tidal movement results from the gravitational attraction 
of the moon and sun acting upon the rotating earth is now a well- 
_ established scientific fact. 'The movement includes both the vertical 
rise and fall of the tide and the horizontal flow of the tidal currents. 
It will be shown later that the tide-producing force due to this attrac- 
tion, when taken in connection with the attraction between the par- 
ticles of matter which constitute the earth, can be expressed by mathe- 
matical formulas based upon the well-known laws of gravitation. 
5. Although the acting forces are well understood, the resultant 
tidal movement is exceedingly complicated because of the irregular 
distribution of land and water on the earth and the retarding effects 
of friction and inertia. Contrary to the popular idea of a progressive 
tidal wave following the moon around the earth, the basic tidal 
movement asevidenced by observations at numerous points along the 
shores of the oceans consists of a number of oscillating areas, the move- 
ment being somewhat similar to that in a pan of water that has been 
tilted. Such oscillations are technically known as stationary waves. 
The complex nature of the movement can be appreciated when con- 
sideration is given to the fact that such stationary waves may overlap 
or be superimposed upon each other and may be accompanied by a 
progressive wave movement. 
6. Any basin of water has its natural free period of oscillation de- 
pending upon its size and depth. The usual formula for the period 
of oscillation in a rectangular tank of uniform depth is 2Z/-/gd, in 
which L is the length and d the depth of the tank and g is the accelera- 
tion of gravity. When a disturbing force is applied periodically at 
intervals corresponding to the free period of a body of water, it tends 
to build up an oscillation of much greater magnitude than would be 
possible with a single application of the force. The major tidal 
Geallations have periods approximating the half and the whole lunar 
we HARMONIC TREATMENT OF TIDAL DATA 
7. The harmonic analysis of tides is based upon an assumption that 
the rise and fall of the tide in any locality can be expressed mathe- 
matically by the sum of a series of harmonic terms having certain 
relations to astronomical conditions. A simple harmonic function is 
a quantity that varies as the cosine of an angle that increases uniformly 
with time. In the equation y=A cos at, y is an harmonic function of 
the angle at in which a is a constant and ¢ represents time as measured 
from some initial epoch. The general equation for the height (h) of 
the tide at any time (¢) may be written 
h=H)+A cos (att+a)+B cos (bt+6)+C cos (ct+y)+ ete. (1) 
in which H is the height of the mean water level above the datum 
used. Other symbols are explained in the following paragraph. 
8. Each cosine term in equation (1) is known as a constituent or 
component tide. The coefficients A, B, O, etc. are the amplitudes of 
the constituents and are derived from observed tidal data in each 
locality. The expression in parentheses is a uniformly-varying angle 
and its value at any time is called its phase. Any constituent term 
has its maximum positive value when the phase of the angle is zero 
and a maximum negative value when the phase equals 180°, and the 
=> 
