324 Tides and Tidal Currents in the Proximity of Land 



The forces which cause the tidal motions are, on the one hand, the tide-gene- 

 rating forces, and on the other hand, the gradient forces; their components 

 along the axes of the co-ordinates are given by 



+*|S and + g |, 



drj , dri 



- g 8x and -*&> 



(XI. 4) 



to which correspond the frictionless tides and frictionless tidal currents of 

 the form 



u = u^cosat -\- u 2 s\nat = Ucos(at—a) , 

 v = v^osat +v 2 sinat = Vs'm(at—(?) , 

 r\ = r) 1 cos at -{-r] 2 smat = rj cos(at—e) . 



The two forces (XI. 4) can be combined into one single force of the form 



G x = Gjc.iCOSoY+G^sina? , | 



G y = G y , icos at +G y ,2smat . 



(XI. 5) 



(XI. 6) 



Since the tide-generating forces in small seas are small compared to the 

 gradient force, G xl G x2 and, similarly, G y are represented by —giptjjdx) 

 and —g(dr) 2 /dx), respectively and similarly by —gidrjjdy), —g(dr} 2 /8y) respec- 

 tively. The end-point of the vectors representing the frictionless tidal current 

 (u, v) and the total force (G x , G y ) lie on an ellipse (current ellipse and diagram 

 of forces). According to the theory, frictionless tidal current and tidal force 

 are in a mutual adjustment to each other: To the force (XI. 6) belongs a fric- 

 tionless tidal current (XL 5) and vice versa. 



The Coriolis force is directed perpendicularly to the direction of the motion. 

 If all other forces, acting upon a body that was moving with a horizontal 

 velocity c, referred to the rotating co-ordinate system, suddenly ceased to 

 act, the body, under the influence of this force of inertia, would continue 

 in an orbit which in the rotating co-ordinate system would be a circle with the 

 radius of r = elf. If the rotation is positive (contra solem), the direction of 

 revolution in the circle of inertia is negative (cum sole) and vice versa. 

 Consequently, rotations cum sole are favoured on the rotating earth. If 

 a water particle completes a circle with the radius / and the angular velo- 

 city a, its orbital velocity is s = al and, when the rotation is cum sole, u = 

 = s cosat and v = s s'mat. The current diagram is then a circle. The cor- 

 responding centrifugal force is a 2 l = as , the Coriolis force acting inward 

 is fs . If these forces are in equilibrium (a =/), the water particle moves 

 on the circle of inertia. For tide waves 2oosin0 =f<a (the most impor- 

 tant case), so that a tidal force S must be present to compensate for the 

 difference (a—f)s = S. Relative to the motion of the particle, the com- 

 ponents of the force will then be: G x — — Ss'mat G y = — Scosat and it is 



