296 Theory of the Tides 



4. Remarks Concerning the Dynamic Theory of Tides 



The dynamic theory of the tides of Laplace starts out from the general 

 dynamic principle of the wave theory, according to which in an oscillating 

 system the motions forced by an external disturbing force have the same 

 period as this force, but that the amplitudes and phases of these motions 

 are also influenced by the possible free oscillations of this system. This in- 

 fluence is the stronger, the more the periods of the free oscillations approach 

 those of the disturbing force. For the completeness of such a dynamic theory, 

 it is necessary to know and to consider all the free oscillations of the system. 

 According to V. Bjerknes, however, this has not always been done. He has 

 proven that a kind of possible free oscillations, which he calls "gravoid- 

 elastoidic" oscillations, have been disregarded, which, if taken into account, 

 will possibly modify the results. He, therefore, designates the Laplacian 

 theory as a "semi-dynamic" method, to explain the theory of ocean tides. 

 However, the complete theory in the exact form has not yet been applied 

 hitherto. 



The elastoidic waves occur in rotating fluids as oscillating motions under 

 the influence of forces which originates by the variation of the centrifugal 

 force in the case of radial displacements of the water particles. These are 

 forces acting like the elastic forces on a solid body, for which reason they 

 are called "elastoidic". This kind of waves can appear both as standing and 

 as progressive waves, and it is preferable to start from the elastoidic free 

 oscillations in a tube bent to return in itself. It can be shown that, in case 

 of a slightly bent tube of rectangular cross-section, the length of the period 

 of the free oscillations depends only upon the ratio of the lateral side to the 

 radial side of the cross-section. Such elastoidic waves must, of course, also 

 occur, for instance, in a canal along the equator of the rotating earth; only 

 in this case the action of gravity must be added, which is also directed towards 

 the vortex axis of the rotating water ring. These gravoid-elastoidic waves have 

 the same character as those of the purely elastoid oscillations. If the water 

 ring has a free surface, as in the case of an equatorial canal, and if V is 

 the period of the orbit of the oscillating motion, L the width of the tube 

 (canal) and h the thickness of the layer in which the wave disturbance occurs, 

 the equation h = 4L 2 /gT' 2 applies both for standing and for progressive waves 

 of a gravoid-elastoidic character. 



The wave disturbance appears as an oscillatory motion between the side 

 walls of the open canal. When all particles move in the same phase, the dis- 

 turbance has the character of a standing oscillation ; if the phase varies along 

 the cross-section, the disturbance propagates along the canal .* 



* It is noteworthy that the above formula is identical with the simple Merian formula for 

 oscillations for a rectangular basin of the length L and the depth /;. The oscillations would then 

 only be gravitational waves between the walls of the canal. Has the "elastoid" of the oscillations 

 not disappeared by neglecting? 



