21 



The expression for the equilibrium tide, u, in terms of cos 6 is given 

 in equation (16) : 



u=y2{Ma?IEm) (3 cos^ d-l)a=^a{Ma^lER^) (cos^ d-]i) 



Substituting the value of cos Q given in equation (21) : 



■u=^a{MayER^) (cos^ X cos^ 5 cos^ H+2 cos X cos 8 sin X sin 5 cos H 



+sin2 Xsin^S-Zs) (22) 



Since cos^ i?=K (1+cos 2IT), sin X cos X=K sin 2X etc., this equation 

 reduces to: 



^=% a{Ma?IER^) (cos^ X cos^ 5 cos 2iJ+sin 2 X sin 2 5 cos H 



+ cos^ X cos^ 5+2 sin^ X sin^ 8-)i) (23) 



Substituting for cos^ X and cos^ 5 in the third term their equivalents, 

 1— sin^ X and 1— sin- 5 respectively, and reducing, the equation for u 

 becomes: 



u=%a {Ma^lER^) cos^ X cos^ 5 cos 2H+ % aiMa^ER^) sin 2 X sin 2 5 cosH 

 + % a{Mo?IER^) (1-3 sin^ X) (1-3 sin^ 5) (24) 



38. Semidiurnal and diurnal parts of the lunar tide. — Equation (24) 

 shows that the lunar equilibrium tide at any tidal station is composed 

 of the following parts: 



(a) That represented by the term % a{Ma^/ER^) cos^ X cos- 5 cos 2H. 

 Since the angle 2H obviously goes through two complete cycles from 

 to 360° while H is making one cycle in a lunar day, tliis part goes 

 through two cycles every lunar day and is therefore called the semi- 

 diurnal part of the tide. 



(b) That represented by the term % a{Ma^/ER^) sin 2 X sin 5 cos H. 

 This part goes through one cycle each lunar day and is the diurnal part 

 of the tide. 



(c) That represented by the term : 



Yi a{MayER^){l-3 sin^X) (1-3 sin^ 8). 



Since this term is independent of the angle H, it imdergoes no change 

 because of the rotation of the earth. It is therefore the height of the 

 daily mean sea level above that of a sea undisturbed by tidal forces. 

 Its variation due to the changing declination of the moon will be 

 later discussed (par. 42). 



39. A typical example of the diurnal and semidiurnal fluctuations 

 of the lunar equilibrium tide, and of the total tidal fluctuation re- 

 sulting therefrom (disregarding the variation due to the changing 

 distance between the moon and the earth) is illustrated in figure 13, 

 which shows these fluctuations at a station at 40° north latitude during 

 £L luLtiar day in which the declination of the moon increases from 7° 

 to 13°. 



