Theory of the Tides 279 



2. The Dynamical Theory of the Tides 



A century after Newton (1774), Laplace substituted for the equilibrium 

 theory, which considers the tides hydrostatically, a dynamical theory. Accord- 

 ing to this theory, the tides are considered as waves induced by rhythmical for- 

 ces and, therefore, have the same periods as the forces. The problem of the tides 

 thus becomes a problem of the motion of fluids. In the development of these 

 forced tide waves, various factors other than the periodic forces play a de- 

 cisive part, like the depth and the configuration of the ocean basin (its mor- 

 phologic configuration), the Coriolis force and frictional influences of various 

 kinds. The tide generating forces are known with great accuracy, so that 

 there is no difficulty in establishing the hydrodynamical equations for the 

 corresponding motion of the water particles. The first equations of this kind 

 were derived by Laplace. The general equations of the dynamical theory 

 have not been solved yet insofar as the tides of the oceans are concerned: 

 we need simplifications to solve the equations. These simplifications are 

 based in the first place on the fact that the tide waves belong to the group 

 of the "long waves", for which reason the latter are also designated as 

 tidal waves. 



The tidal motion belongs, insofar as it is generated and maintained by 

 external periodical forces, to the forced waves; however, it is obvious that, 

 especially when fulfilling the boundary conditions, "free waves" will also 

 appear. The importance of these free waves, which fix the natural periods of 

 the oscillatory system, becomes particularly evident if one considers that the 

 amplitude and the phase of the forced oscillation are determined (see p. 8) 

 by the difference between the free oscillation period of the system and the 

 period of the force. However, perfect resonance occurs only very seldom and 

 when it does, a closer consideration of the frictional influences is necessary. 



The dynamical theory of the tides is extremely intricate in its details and 

 requires, even if ideal conditions are assumed for the depths and the contours 

 of the oceans, considerable mathematical work. Laplace (1775-76, 1799) 

 succeeded in computing the theoretical tides of a homogenous ocean covering 

 uniformly the entire earth, but the results are, in general, very poor and 

 without much importance for the comprehension of the terrestrial tides of 

 the seas. Only in recent times have they gained importance in connection 

 with atmospheric tides. It appears that only lately, methods have been de- 

 veloped which to some extent take into account the complicated contours 

 of the oceans and which, therefore, promise more success. 

 (a) The Theory of Laplace 



The hydrodynamical equations of motion in polar co-ordinates (R radius 

 of the earth, & pole distance, I geographical longitude) are: 



-£— loovcosft+lcowsind = — -=-— -^(.Q+- 

 at R + z d& 1 



