Theory of the Tides 297 



If we select for V =T Q — \ sidereal day = 86,164 sec, it appears that the 

 "gravoid-elastoidic" surface waves only occur in an extremely thin surface 

 layer; if we imagine an ocean of a constant depth of 5 km or divided in tubes 

 (canals) of 8-5 km width, the top layer influenced by the waves would be 

 only 4 mm thick. This shows that the inner elastoid oscillations of the sea 

 are without any importance for the variations of the surface. This changes 

 however, if we consider the "gravoid-elastoidic" surface disturbances in thicker 

 layers. The equation gives the following correlated values of the depth h and 

 the width L of the canal, when V = T , hence for diurnal waves: 



For a canal extending in width between the northern and the southern tropics 

 for instance, the water layer occupied by the gravoid-elastoidic waves would 

 be 2000 m deep, like oceanic depths. With these diurnal waves, the water, 

 in case of a standing wave, flows everwhere in the canal during 12 h to the 

 north, then during 12 h to the south, or in case of a wave with a wave length 

 equal to the circumference of the earth, the current will be directed northward 

 on one hemisphere and southward on the other hemisphere. 



There is no doubt that "gravoid-elastoidic" surface waves may possibly 

 occur in the oceans, also in the preference of diurnal tide generating forces. 

 Inasmuch as in the existing dynamic theory this kind of free oscillations 

 has not been considered, it might easily happen that the diurnal tidal phe- 

 nomena mentioned therein are not complete, and particularly that the theory 

 of Laplace concerning the disappearance of the diurnal tide in a sea of con- 

 stant depth is not tenable, because the possible "gravoid-elastoidic 1 ' inertia 

 waves have not been considered. V. Bjerknes is of the opinion that the great 

 success encountered by the existing "semi-dynamic" method can only be 

 attributed to the favourable circumstances that the "gravoid-elastoidic" dis- 

 turbances can only in an extreme case have a period approaching the semi- 

 diurnal period which is most important for the tides. Bjerknes also proves 

 that the inaccuracy of the prevailing presentation lies in the fact that in the 

 equations of motion the vertical acceleration of the water particles and the 

 vertical component of the Coriolis force have been neglected. Only when 

 they are taken into account we will have together all possible free oscillations 

 of the systems necessary for the comprehension of the phenomenon. However, 

 only the application of the exact theory in all its details will show in how 

 far the objections of Bjerknes mean a serious improvement of the theory 

 of the tides. 



Solberg (1936, p. 237) has made a thorough investigation regarding the 

 free oscillations of a homogeneous water layer on a rotating earth in con- 

 nection with the dynamic theory of the ocean tides, and has shown that 

 their character changes completely when 1— 4co 2 /a 2 passes through zero. For 



