Tides and Tidal Currents in the Proximity of Land 361 



essentially a process of integration along certain lines from coast to coast, 

 right across the North Sea. 



(a) Defanfs Method 



Defant uses the simple equation of continuity in the form (VI. 100). If we 

 introduce therein 



n =■-- >/ 1 coso'^+ >/ 2 sino7 , 



u = UiCOsat + UoSinot , (XI. 36) 



v = Vicosot + Visincrt , 



it is split up into the two equations: 



1 IdiUo (hr,\ 



1 uhi h dhvA 

 a\ dx oy J 



If the area of an adjacent sea is divided into a great number of equal area 

 squares by means of orthogonal lines and if the length of the sides of these 

 squares is taken so small that, in first approximation, all quantities along it 

 can be considered to vary linearly then the two values r} x and rj 2 can be com- 

 puted for the centre of each square, when the velocities of the current u x , u> 

 and v 1} v 2 ar e known at all corner points. Within such a small area the depth 

 can be taken as constant. The determination of these quantities which, 

 combined, give the range and the phase of the tide at this point, is independent 

 of the values in the vinicity and likewise independent of the coastal values. 



Thorade (1924, p. 63) has given a generalization of this method, in case the currents at the 

 surface and also in greater depths are sufficiently well known. The usual assumptions of a simple 

 harmonic function for the currents and for the tide, as well as for the friction must not be con- 

 sidered. One has only to compute for each hour the water-masses going in and out of each layer 

 in a prismatic space. From these values it is then possible to calculate the rise and fall of the surface. 

 In this way one can obtain, by a somewhat laborious but not difficult calculation, step by step 

 the tidal curve at a certain point. However, at the present time, our knowledge of the currents is 

 limited to only a few stations, so that, for the time being, an application of this refined procedure 

 cannot be considered. 



(b) The Method of Proudman — Doodson 



We introduce in the equations of motion (XI. 30) the values of (XI. 36) 

 for //, //, and r; for the frictional forces F and G we take Taylor's assumption 

 that both are proportional to the square of the velocity 



ku\ (u 2 +v 2 ) and kr\ (//- /•-) 



respectively. If u t . u. 2 , i\. r, are known, these equations can then be subjected 

 to a harmonic analysis. 



F F, cos ot—F* sin at , I 

 G = GiCOsat -—Go sin or/ . | 



