Theory of the Tides 



285 



Table. 30. Periods of oscillations of "the first class"' on a rotating earth 



in sidereal time 



second class has in each case a period of over a day and the periods of the 

 remainder are very much longer (Lamb, 1932, p. 350). 



As regards the forced oscillation of the second species (s = 1), Laplace's 

 conclusion that when a = to, the diurnal tide vanishes in the case of uniform 

 depth, still holds. The computation for the most important lunar diurnal 

 tide, for which ojco = 0-92700, shows that with such depths as we have con- 

 sidered the tides are small compared with the equilibrium heights and are in 

 the main inverted. 



Of the forced oscillations of the third species (s = 2) we note first the solar 

 semi-diurnal tide, for which a = 2co with sufficient accuracy. In Table 29 

 we have listed under S 2 the ratio of the dynamical tide height to the equi- 

 librium tide height at the equator. A comparison with Laplace's values in 

 this table shows that the influence of the attractive forces of the flood pro- 

 tuberance is to increase the ratio of the tide to the equilibrium tide. The very 

 large coefficient for = 4m 2 R 2 /hg = 10 indicates that for this depth the 

 period of the free oscillation of semidiurnal type differs but slightly from half 

 a day, which means we have nearly resonance. Table 30 shows, in fact, that 

 when /5 = 10, the period is 12 h 1 min. We see then that though, when the 

 period of forced oscillations differs from that of one of the types of free oscilla- 

 tions by as little as 1 min, the forced tide S 2 may be nearly 250 times as great 

 as the corresponding equilibrium tide; whereas a difference of 5 min between 

 these periods will be sufficient to reduce the tide to less than 10 times the 

 corresponding equilibrium tide. It seems then that the tides will not tend to 

 become excessively large, unless there is very close agreement with the period 

 of one of the free oscillations. The critical depths for which the forced tides 

 become infinite are those in which a period of free oscillations coincides exactly 

 with 12 h. They may be computed by putting a = 2co in the period equation 

 for the free oscillations and solving this equation for the determination of h. 



