362 Tides and Tidal Currents in the Proximity of Land 



If, furthermore, we put 



a = 10 7 - = 1-432, 

 g 



b = \0 7 ^ = 1-487 sin</>, 



c = 10' 



g _ 1019 



A ~~ h : 10 4 ' 



(XI. 39) 



in which the numerical values apply for the M 2 — tide, the equations (XI. 30) 

 take the form 



10* Jb = -au t +.ht> l -cF l , 



ox 



(XI. 40) 



10^ = -av. 2 -bu 1 -cG 1 , 



10' |^- 2 = a Ul + bv 2 -cF 2 , 



dx 



\0 7 ^ = av 1 -bu 2 -cG 2 . 



If good and reliable tidal current measurements are available for a certain 

 stretch, we can compute the gradients of % and r\ 2 for the single intervals 

 of this stretch. If these gradients are connected to known values of rj x and % 

 at the coastal points where the section starts or where it ends, then we have 

 the tide along the entire curve. The accuracy of the method depends upon 

 the number of the current measurements used and when these become in- 

 sufficient to make reliable interpolations, or if the current values vary too 

 strongly, interpolations should not be attempted. 



Both methods have given good results in determining the tides of the North 

 Sea and at present they are the only methods which permit the computation 

 of the tides of an adjacent <e\ based on shore values. 



Hansen (1940, p. 41; 1942, p. 65; 1943, p. 135; 1948, p. 157) has shown 

 recently that, when the tides and tidal currents are given for any arbitrary 

 line of demarcation of an area, the tide in the whole region can be clearly 

 defined by solving a system of linear equations. If we introduce in the equa- 

 tions (XI. 30) for the frictional forces /iw resp. /to and eliminate the time by 

 introducing e~ iat we get with io + fi = X and using the equation of continuity 

 the following equations (the derivation with respect to x and y is indicated 

 by their indices) 



to—fv-\-gr} x = , 



fu+to+grjy = 0, 



iari + (hu) x +(ko), = 0. 



(XI.41) 



