290 Theory of the Tides 



approaching the real depths of the oceans, there remain in each quadrant two 

 positively rotating amphidromies. The tidal range is largest at the boundary 

 with the maximum occurring at the equator at the western and the eastern 

 points. 



These investigations of Proudman and Doodson are undoubtedly of great 

 importance for the future tidal research. Surveying the theory of the tides, 

 we recognize how, starting from the simple equilibrium theory and progressing 

 by way of the Laplacian dynamic theory of the tides in an ocean covering 

 uniformly the whole earth, it arrives at the study of the tides of bounded seas. 

 These studies lead to results that are comparable to the tides in the actual 

 seas bounded by the continents. So far, the most important knowledge is 

 the occurrence of systems of amphidromies, which sometimes may comprise 

 the whole or parts of an ocean. When a large number of such theoretical 

 tidal pictures have been computed for different depths and limits of the 

 oceans, they will form a good basis for the understanding of the observed 

 tidal features. 



3. Canal Theory of the Tides 



The Laplacian theory started from the assumption of an ocean covering 

 the entire earth and considered only recently the land masses bounding the 

 seas. Airy (1845) has attempted to solve the phenomena of the tides by 

 a study of the oscillatory processes in narrow canals covering the whole earth 

 or parts of it (see Lamb, 1932, p. 267). He thus became the founder of the 

 "canal theory of the tides"; he has treated this theory exhaustively in his 

 famous paper "Tides and waves". Although, in many respects, it can hardly 

 claim today the same significance it had previously, it still offers many 

 interesting features, especially for the explanation of the tides in narrow 

 sections of the oceans, in straits and estuaries. 

 (a) Canal of Uniform Depth Extending all over the Earth 



In chapter VI, paragraph 1 we discovered the tide waves in a canal of 

 constant rectangular cross-section. The pertaining equations of motion (VI. 8) 

 were expressed in the horizontal and vertical displacements of the water 

 particles £ and r\. On account of the assumed narrowness of the canal, the 

 transverse motions and the influence of the Coriolis force are neglected. 

 For a continuous canal of a uniform depth h coincident with the earth's 

 equator, it is appropriate to introduce the geographical longitude X (counted 

 from a fixed meridian eastward) instead of the .r-co-ordinate. The equations 

 then take the form (c 2 = gh, R the radius of the earth) : 



U~*mP and n = -hf fx . (IX. 9) 



The free oscillations of such a canal can be easily derived therefrom : one 

 obtains 



