The Tides in Relation to Geophysical and Cosmic Problems 507 



waters, whose bottom configuration must be known exactly, means that 

 transverse currents may be neglected. On the other hand, the partial tides 

 with a short period can also be considered, which is of great advantage. 

 Proudman has developed two methods for the computation of y, which will 

 be explained summarily. 



If x is taken in the direction of the longitudinal axis of the narrow sea 

 in the form of a canal, and u represents the average value of the tidal current 

 on a cross-section perpendicular to .v, while there is no transverse current, 

 then u will only be a function of x and t. The differential equations of the 

 water motion can be written into the form 



(XV. 3) 



du c _ 



dt" -gtoto+i*-*-^> 



„ . d , - -x 



whereas the equation of continuity takes the form 



l x (Su) + b^ = (XV. 4) 



in which S represents the area and h the width of the cross-section at the 

 point x. If u is known the second equation of (XV. 3) helps to reduce the 

 tidal ranges, observed along the coasts to the central axis of the canal. 



The first method determines the difference between the tides computed 

 without assuming a tide of the solid earth and the observed tides. If we put, 

 as usual, 



?] = Hcos(at—x) = fjxCOsat + rj.iSinot , 



u — Ucos(at—e) = w 1 cosc? + u 2 s'mat , 

 we obtain from (XV. 4) 



ASu x = —a //sin x- bAx and ASu 2 = -\-aHcosx- bAx . 



If the tidal range along the entire canal is known, one can compute from 

 these relations u^ and u 2 , i.e. the velocity of the current for each cross-section. 

 With these values we have for all sections the variation of the current velocity 

 during a unit time; because 



o P P 



Tr-tn-V) or Mrix-ih) and ^-(%-^) 



OX s C'A OX 



are known from the observations, and the theory of the tides' generating 

 forces, in equation (XV. 3) only 



^:0?o-^o)i and ^-(^-^0)2 

 are still unknown and can be computed in this way. The method gives, 



