radius is almost unopposed by other forces and produces an acceleration toward tlie 

 sublunar point or tlie subsolar point; i.e., toward the point on the surface of tlie Eartli at 

 wliich the Moon or Sun is directly overhead. A similar analysis, carried out for a point on 

 the opposite side of the Earth from the Moon or Sun would reveal a force which causes an 

 acceleration toward the nadir of tlie Moon or Sun. 



Schureman (1941) shows that this horizontal component of tlie tide-generating force per 

 unit mass, Fr, is given by 



F - Mai 

 F« - g E (i2 



l-2(|)cosz. (^)J" 



(8) 



Equation (8) applies to particles of the solid Earth and the atmosphere as well as to the 

 waters of the sea. The magnitude of the Earth tides, as the tides in the soUd Earth are called, 

 are several orders of magnitude smaller than the sea tides, but they can be measured in 

 subterranean observatories (Melchior, 1966). Atmospheric tides are more readily measured 

 tlian Earth tides, but are about an order of magnitude smaller than the periodic changes in 

 pressure due to solar heating and very much smaller than the changes due to traveling storm 

 systems. Earth tides and atmospheric tides rarely, if ever, have any importance to coastal 

 engineers and are not discussed here in any greater detail. In equation (8), r is replaced 

 by a, the mean radius of the Earth, in all subsequent developments. 



The binomial theorem can be used to convert the fraction in equation (8) into a power 

 series in a/d. Since a/d is approximately 1.67 X 10" for the Earth-Moon system, and 

 4.26 X 10"^ for the Earth-Sun system, only the first few terms of the series are needed. 

 Thus, 



[l-2(|)cosz. (I)']"''' = 1.3 (I) cos z 



^ (l)(lT(5cos2-l)+... (9) 



Substitution from equation (9) into equation (8) yields: 



Fn = gg (|-) 3cosz-H2-|-(5cos2 z- l) 



Equation (9) is derived from equation (8) by: 



(a) rewriting equation (8) in the form 



(10) 



Ma2 

 where a = a/d and c = cos z. 



F„ - gf I [(1- 2ca+ a2)-3/2 _ {j ^^^ ^^^^ 



28 



