Theory of the Tides 291 



f = (A s cossk+B s smsX)cos{a Q t + e) (IX. 10) 



in which s is an integer and a = S{c/R). These normal oscillations are 

 simple harmonic with the period T = 2nR/sc. The free waves all travel 

 with the velocity c and travel around the circumference of the earth in 

 a time T . The most important free wave for the theory of the tides is the 

 one which needs 24 h to travel around the equator. The necessary velocity 

 is 900 nm/h or 1670 km/h, and to it corresponds an ocean depth of 72,160 ft 

 (22 km). For smaller ocean depths the waves travel slow (e.g. for h = 1 6,400 ft 

 (5000 m), c = 432 nm/h or 800 km/h, and T = 50 h), for greater depths more 

 rapidly. The free wave can only keep pace with the sun when the depth is 

 13 nm or 22 km; for the free wave moving with the moon the critical depth 

 is 12-5 nm or 205 km. 



For the oscillations forced by a periodic horizontal tidal force, the force 

 K h has to be added to the right term of the first equation in (IX. 9). This force 

 is K h = rasin2#, according to equation (VIII. 11), if ft is the zenith distance 

 of the tide-producing body. 



The simplest case is, when the disturbing body (moon) describes a circular 

 orbit in the equatorial plane of the earth with the apparent angular velocity 

 n = co — n x when co = angular velocity of the earth's rotation and n x = angular 

 velocity of the moon in her orbit. Then, according to (VIII. 12), d = nt + X + e 

 and the equation of motion can be written 



d 2 £ c 2 d 2 i 



W = ^2^-wsrn2(n/+A+£) . (IX. 11) 



For the tides or forced waves we have, if we put 



nR 



mR 1 1 . ., 



sm2{nt J r A + e) , 



4c 2 \-p 



V = cos2(nt + A + e) = — -cos2(Hf + A+e). (IX.12) 



2 l-p 2 2 1— (of/crS) 



According to (VIII. 18), we have H = mR/g (for the moon 55 cm, for the 

 sun 24 cm) and a = 2n is the frequency of the forced oscillation, a = 2c/R 

 the frequency of the free oscillation (s = 2). \H is the amplitude of the tide 

 producing force, so that J//cos2(«/ + A + e) represents the tide according to 

 the equilibrium theory. 



The tide is therefore semi-diurnal (the lunar day being of course under- 

 stood), and, when/; > 1 or c < nR and a > a it is inverted; on the contrary 

 when p < 1 or c > nR and a < a it is direct. The second form of the equa- 

 tion for the amplitude of r\ corresponds to the equation (1.9) and one sees 



19* 



