100 U. S. COAST AND GEODETIC SURVEY 



The mathematical development of these forces shows that the tide-producing 

 force of a heavenly body varies directly as its mass and inversely as the cube 

 of its distance from the "earth. The sun has a mass about 26,000,000 times as 

 great as that of the moon; but it is 389 times as far aw^ay from the earth. Its 

 tide-producing force is therefore to that of the moon as 26,000,000 is to (389)^ 

 or somewhat less than one-half. 



When the relative motions of the earth, moon, and sun are introduced into the 

 equations of the tide-producing forces, it is found that the tide-producing forces 

 of both sun and moon group themselves into three classes: (a) Those having a 

 period of approximately half a day, known as the semidiurnal forces; (b) those 

 having a period of approximately one day, known as diurnal forces; (c) those 

 having a period of half a month or more, known as long-period forces. 



The distribution of the tidal forces over the earth takes place in a regular 

 manner, varying with the latitude. But the response of the various seas to 

 these forces is very profoundly modified by terrestrial features. As a result 

 we find the tides, as they actually occur, differing markedly at various places, 

 but apparently with no regard to latitude. 



The principal tide-producing forces are the semidiurnal forces. These forces 

 go through two conjplete cycles in a tidal day, and it is because of the pre- 

 dominance of these semidaily forces that there are at most places two complete 

 tidal cycles, and therefore two high and two low waters in a tidal day. 



VARIATIONS IN RANGE 



The range of the tide at any given place is not constant but varies from day 

 to day; indeed, it is exceptional to find consecutive ranges equal. Obviously 

 changing meteorological conditions will find reflection in variations of range, 

 but the principal variations are due to astronomic causes, being brought about 

 by variations in the position of the moon relative to earth and sun. 



At times of new moon and full moon the tidal forces of moon and sun are 

 acting in the same direction. High water then rises higher and low water falls 

 lower than usual, so that the range of the tide at such times is greater than 

 the average. The tides at such times are called "spring tides," and the range 

 of the tide is then known as the "spring range." 



When the moon is in its first and third quarters, the tidal forces of sun and 

 moon are opposed and the tide does not rise as high nor fall as low as the 

 average. At such times the tides are called "neap tides," and the range of 

 the tide then is known as the "neap range." 



It is to be noted, however, that at most places there is a lag of a day or two 

 between the occurrence of spring or neap tides and the corresponding phases 

 of the moon; that is, spring tides do not occur on the days of full and new moon, 

 but a day or two later. Likewise neap tides follow the moon's first and third 

 quarters after an interval of a day or two. This lag in the response of the tide 

 is known as the "age of phase inequality" or "phase age" and is generally 

 ascribed to the effects of friction. 



The varying distance of the moon from the earth likewise affects the range of 

 the tide. In its movement around the earth the moon describes an ellipse in a 

 period of approximately 27J^ days. When the moon is in perigee, or nearest the 

 earth, its tide-producing power is increased, resulting in an increased rise and 

 fall of the tide. These tides are known as "perigean tides," and the range at 

 such times is called the "perigean range." When the moon is farthest from the 

 earth, its tide-producing power is diminished, the tides at such times exhibiting 

 a decreased rise and fall. These tides are called "apogean tides" and the 

 corresponding range the "apogean range." 



In the response to the moon's change in position from perigee to apogee, it is 

 found that, like the responses in the case of spring and neap tides, there is a 

 lag in the occurrence of perigean and apogean tides. The greatest rise and 

 fall does not come on the day when the moon is in perigee, but a day or two 

 later. Likewise, the least rise and fall does not occur on the day of the moon's 

 apogee, but a day or two later. This interval varies somewhat from place to 

 place, and in some regions it may have a negative value. This lag is known 

 as the "age of parallax inequality" or "parallax age." 



The moon does not move in the plane of the Equator but in an orbit making 

 an angle with that plane of approximately 23J/^°. Daring the month, there- 

 fore, the moon's declination is constantly changing, and this change in the posi- 

 tion of the moon produces a variation in the consecutive ranges of the tide. 

 When the moon is on or close to the Equator — ^that is, when its declination is 

 small — consecutive ranges do not differ much, morning and afternoon tides be- 



