Tides and Tidal Currents in the Proximity of Land 323 



/e / 1 / 1-r 2 



u = l/-l/ - - i^oe'^^sinCff/- jcc) with c = \/gh'\/ - - , (XI. 3) 



f hi \-r l } (l — srf 



v = y- 1/ -I—rj e-( mlc )ycos(ot-xx) . 



These equations have for it and v the same form as (XI. 2), except that 5 is replaced by r. According 

 to (XI. 3) the configuration of the motion is an ellipse with the ratio r between the axes. When r = 0, 

 we have m = /and c = ygh and this corresponds to a tide wave of the Kelvin type (narrow canal); 

 if, on the contrary, m = 0, r = fja = s, we obtain the conditions prevailing on the rotating disk 



S--06 



Fig. 131. Tide wave in a rotating unlimited frictionless layer. A, B and C as in Fig. 130. 



of infinite dimensions, equation (XI. 2). Generally < m < f, and the wave is of an intermediate 

 character; the motion is rotary, but the ratio between minimum and maximum velocity is smaller 

 than on an infinite disk. The amplitude decreases along the wave-front, but less rapidly than in 

 a channel and the velocity of propagation is greater than in a channel but smaller than on an infinite 

 disk. When r = 1, a = f, the wave degenerates into a simple inertia motion. 



(b) Relations Between the Tide and Tidal Currents 



Thorade (1928-29, p. 290) has studied the relation between the current 

 and tide level variation, not by solving the corresponding differential equa- 

 tions, but in a different manner. If friction is disregarded, the tides can 

 be designated, after Thorade, as "frictionless tides" ("Nulltiden"), and their 

 corresponding tidal currents as "frictionless currents" ("Nullstrome"). The 

 equations (XI. 3) for instance, give the frictionless tides and currents for a very 

 large body of water. According to the theory of ocean currents of Ekman 

 (see vol. II, Chapter VI), the effect of the bottom friction makes itself 

 felt only up to a relatively small height above the ocean bottom (lower 

 frictional depth); it can be assumed that the frictionless tidal current is 

 practically similar to conditions existing in the upper layers of deep oceans. 



21 1 



