282 



Theory of the Tides 



for the free oscillations. In the case of the fortnightly lunar tide, / is the ratio 

 of a sidereal day to a lunar month and is 00365. Thus,/ 2 becomes = 000133 

 and we obtain a very good approximation in putting / = 0, which shortens 

 the computation considerably. The same applies for the other tides with 

 a long period. The result can be found in the first lines of Table 29 under 



Table 29. Ratio of the polar and equatorial tides to 1 he ir- 

 respective equilibrium values 



40 



20 



10 



qIqo 



Depth (ft). 

 (m). 



7260 

 (2210) 



14,520 

 (4430) 



29,040 

 (8850) 



58,080 

 (17,700) 



Long-period tides ■ 



Equator 



Pole 



Laplace 

 Hough 



Laplace 

 Hough 



0-455 

 0-426 



0154 

 01 40 



0-551 



0-266 



0-708 i 0-817 

 0-681 0-796 ; 0181 



0-470 

 0-443 



0-651 

 0-628 0181 



Semi-diurnal 



tides 



(uniform 



depth) 



S 2 Equator 



\ Laplace 

 [ Hough 



[a — 2a>] 



M 2 Equator Hough 

 [ff/2o> = 0-96350] 



Laplace. They give the ratios of the range of these partial tides to the range 

 of the corresponding tides of the equilibrium tide at the equator and at the 

 pole, for four different depths of the ocean. These ratios show that the long- 

 period tides, at the considered depths, are everywhere direct, i.e. high water 

 always occurs simultaneously with that of the equilibrium tide though; the 

 position of the nodal lines will, of course, shift somewhat from the positions 

 given by the equilibrium theory. For a depth corresponding to the actual 

 mean depth of the oceans this tide is less than half that of the equilibrium value 

 and it approximates this equilibrium value more and more when the water 

 depth becomes very large, with consequent decrease of /?. 



The forced oscillations of the second species (s — 1) which include the 

 lunar and solar diurnal tides, have a disturbing potential corresponding to 

 a tesseral harmonic of the second order. This gives 



V] = A 2 s'mftcos&cos(at + A + e) , 



in which a does not differ very greatly from oj. Here also the computation is 

 considerably simplified if we select a = to, which means that the proper 

 motion of the disturbing body is neglected and therefore / = \. With this 



