204 Long Waves in Canals and Standing Waves in Closed Basins 



Finland one has to count with velocities of the seiches of 15-20 cm/sec, 

 whereas in the centre of the Baltic the values are being reduced to the order 

 of magnitude of 5 cm/sec. 



Neumann has used the current observations made on lightships in the 

 Gulf of Finland, in order to prove the variations in the current which ac- 

 company the uni-nodal free oscillation. The current observations made by 

 the lightship "Tallinn" show the same period as the free oscillation (27-5 h) 

 and a difference in phase with the free oscillation of approximately 7 h, which 

 corresponds to the theoretical value IT. These periodically varying currents, 

 which in one instance reached an amplitude of 45 cm/sec, do cause a periodic- 

 ally varying transversal slope of the sea surface in a line Helsingfors-Reval. 



When studying drift and wind currents, attention should be paid to possible 

 periodic currents resulting from the free oscillations of the water-masses. 

 (b) Progressive Waves in a Rectangular Canal Considering the Rotation of 

 the Earth 



Like any moving object on the earth, horizontal water displacements are 

 subjected to the Coriolis force. Fully developed, this force is capable of 

 modifying the currents considerably. In most cases it should not be neglected, 

 as it is of the same order of magnitude as the pressure forces. If the amplitude 

 of the periodical horizontal velocity of the wave is U the Coriolis force will 

 be 2cosin(pU, and its direction is perpendicular to the direction of propagation 

 of the wave. Since U changes its sign the Coriolis force during one half of 

 the wave period is directed to the left, during the other half to the right. 

 Gravity acts in the vertical direction on the moving water particle. Together 

 they give a resultant, which is inclined by a small angle y towards the vertical. 

 Its tangent is given by 



2<x>£/sin<p 



tany 



g 



If the surface is always perpendicular to the resulting force, it should also 

 make an angle y with the horizontal plane. The order of magnitude of 

 this angle is very small. At y = 45° we have tany = 1027x \0~ 7 U and with 

 U = 20 cm/sec y = 0-4". The slope of the sea surface is, therefore, extra- 

 ordinarily small. However, considering that the slope of the sea surface is 

 equally small when long waves pass by, we cannot neglect the transverse 

 slope caused by the earth's rotation. In a tidal wave of 200 km length 

 and an amplitude of 25 cm the slope of the waves is on the average 

 5 x 10 -6 , the transverse slope caused by Coriolis force is approximately 

 1 x 10~ 6 , thus of the same order of magnitude. In a canal of a width b 

 the rise and fall of the surface caused by the action of the Coriolis force 

 is \ b tan y, which can become important, provided that b is sufficiently large. 



The horizontal velocity u varies at one point with the period of the wave T 

 and, therefore, the Coriolis force will cause corresponding variations in the 



