HARMONIC ANALYSIS’ AND PREDICTION OF TIDES 3 
term becomes zero when the phase equals 90° or 270°. The coefficient 
of ¢ represents the rate of change in the phase and is called the speed 
of the constituent and is usually expressed in degrees per hour. The 
time required for a constituent to pass through a complete cycle is 
known as its period and may be obtained by dividing 360° by its 
speed. The periods and corresponding speeds of the constituents are 
derived from astronomical data and are independent of the locality 
of the tide station. The symbols a, 8, y, etc. refer to the initial phases 
of the constituent angles at the time when ¢ equals zero. The initial 
phases depend upon locality as well as the instant from which the 
time is reckoned and their values are derived from tidal observations. 
Harmonic analysis as applied to tides is the process by which the 
observed tidal data at any place are separated into a number of 
harmonic constituents. The quantities sought are known as harmonic 
constants and consist of the amplitudes and certain phase relations 
which will be more fully explained later. Harmonic prediction is 
accomplished by reuniting the elementary constituents in accordance 
with astronomical relations prevailing at the time for which the 
predictions are being made. 
ASTRONOMICAL DATA 
9. In tidal work the only celestial bodies that need be considered 
are the moon and sun. Although every other celestial body whose 
gravitational influence reaches the earth creates a theoretical tide- 
producing force, the greater distance or smaller size of such body 
renders negligible any effect of this force upon the tides of the earth. 
In deriving mathematical expressions for the tide-producing forces of 
the moon and sun, the principal factors to be taken into consideration 
are the rotation of the earth, the revolution of the moon around the 
earth, the revolution of the earth around the sun, the inclination of 
the moon’s orbit to the earth’s equator, and the obliquity of the 
ecliptic. Numerical values pertaining to these factors will be found 
in table 1. 
10. The earth rotates on its axis once each day. There are, how- 
ever, several kinds of days—the sidereal day, the solar day, the lunar 
day, and the constituent day—depending upon the object used as a 
reference for the rotation. The sidereal day is defined by astronomers 
as the time required for the rotation of the earth with respect to the 
vernal equinox. Because of the precession of the equinox, this day 
differs slightly from the time of rotation with respect to a fixed star, 
the difference being less than the hundredth part of a second. The 
solar day and lunar day are respectively the times required for rotation 
with respect to the sun and moon. Since the motions of the earth 
and moon in their orbits are not uniform, the solar and lunar days 
vary a little in length and their average or mean values are taken as 
standard units of time. A constituent day is the time of the rotation 
of the earth with respect to a fictitious satellite representing one of 
the periodic elements in the tidal forces. It approximates in length 
the lunar or solar day and corresponds to the period of a diurnal 
constituent or twice the period of a semidiurnal constituent. 
11. A calendar day is a mean solar day commencing at midnight. 
Such a calendar day is known also as a civil day to distinguish it from 
the astronomical day which commences at noon of the same date. 
