322 Tides and Tidal Currents in the Proximity of Land 



r\ = r) sm(ot— xx) , 



/k / 1 

 U = V hV F-T 2 foSmOrf-**) , 



*=yVi 



5 2 



— 3 ^ cos(<T/— *x) , (XI. 2) 



In contrast to the Kelvin waves, the amplitude of the wave is constant along 

 the wave-front. To the motion in the direction of progress, which may be 

 called the longitudinal, is now added a motion along the wave-front which 

 may be called transversal, and there is a phase difference of quarter period 

 length between the longitudinal and the transversal motion. If the velocities 

 of the fluid particles are represented by a central vector diagram, the end- 

 points of the vectors lie on an ellipse, or we can say with Sverdrup that the 

 configuration of the motion is an ellipse. The direction in which the velocities 

 rotate is negative or cum sole, if the rotation of the disk is positive (contra 

 solem) or vice versa. The direction of the maximum velocity of the current 

 (major axis of the current ellipse) coincides with the direction of progress of 

 the waves and is reached when the wave reaches its maximum height. The 

 ratio between the axes of the ellipse, i.e. the ratio between the minimum 

 and maximum velocity, is s =f/cr. The current ellipse is the same for all 

 depths and maximum velocity occurs at the time of high or low water as was 

 mentioned before. The velocity of propagation is c = \/gh\/\j{\ — s 2 ), which is 

 greater compared with the velocity of a wave of the same period length on 

 a non-rotating disk. One can see that, when s = 1 or a = /, the velocities 

 become infinite, i.e. the wave degenerates into a simple circular motion 

 at r\ = (inertia motion ; see vol. 1/2, Chapter VI/6). Consequently, tide 

 waves are only possible in this case if s < 1 or a > f. This means that the 

 period of the tide wave must be larger than the period of the inertia wave. 

 Figure 131 shows conditions when s = 0-6. 



In a very wide rotating canal a tide wave might perhaps be expected to be characterized by 

 a rotary motion of the fluid particles in the middle of the canal and alternating motion along 

 the walls. A wave of this kind, however, could not exist in an infinitely long canal, because the 

 velocity of progress would vary with the distance from the walls but it seems possible that such 

 a wave could exist on a short stretch. A solution of the fundamental equation which satisfies the 

 boundary conditions in this case seems impossible, but a formal solution which does not agree 

 with any fixed boundary conditions may be of value in future applications, because it represents 

 a wave of an intermediate character compared with the two kinds of waves treated previously. 

 According to Sverdrup, this wave has the form: 



i i \ f—m 



rj = r) e (mlc ' y sm(ot—xx), r = a 



fm 



