Long Waves in Canals and Standing Waves in Closed Basins 203 



Progressive wave 

 £ = a cos (at — xx) 

 rj = ahxs'm(at — xx) 

 u = —aas\n(at — xx) 



Standing wave 

 £ = a cos xx cos at 

 rj = aim sin xx cos at 

 u = —aa cos xx sin at 



(VI. 99) 



The horizontal velocity in the wave results from the equation u = d£/dt. 

 Equations (VI. 99) show that with progressive waves the greatest velocities 

 coincide with the greatest rise and fall of the free surface. The free surface 

 rises and falls because of the convergence and divergence of the horizontal 

 motion of the water particles. Within a progressive wave the horizontal flow 

 at the wave crest is in the direction of progress of the wave, and at the trough 

 it is opposite to the direction of progress. Within a standing wave, on the 

 contrary, the horizontal velocity is zero at every point, at the time when 

 the wave reaches its greatest height, and is strongest at the nodes; at the 

 antinodes (crest and trough) it is constantly zero (p. 6). The vertical 

 velocity is always zero at the nodes halfway between the trough and the crest. 



Because it is possible, for standing waves in seas and bays, to compute 

 the horizontal and vertical displacements £ and rj for each cross-section, 

 it is easy to compute the horizontal velocities and to compare them with 

 the results of current measurements. There are only very few of these meas- 

 urements, even though they constitute an important indicative feature as 

 to the oscillatory form of completely or partly closed basins. Neumann 

 (1942, p. 1) has analysed the currents connected with the seiches in the Baltic. 

 From £ computed according to the residual method he derived the distribu- 

 tion of the horizontal current velocities in the direction of the "Talweg", 

 assuming an amplitude of 50 cm at the head of the Gulf of Finland. These 

 values are shown in Fig. 88 by the broken line; the dotted curve indicates 



Trovemunde Bomholm 



Gotland 



Dago FS Tallinn" Petersburg 



50 



20 



8 



12 



16 



20 24 28 32 36 40 44 48 52 



Fig. 88. Current velocities computed by the stepwise integration along the "Talweg" of 



the Baltic assuming an amplitude of the seiches of 50 cm at the closed end of the Gulf of 



Finland (the dotted line is for a rectangular basin of constant depth). 



the distribution of the current computed by equations (VI. 99) for a rec- 

 tangular basin (/ = 1473 km, h = 55 m). The distribution of the current 

 velocities is changed by the complicated basin configuration compared to 

 a rectangular basin. At the Darser Sill and at the opening into the Gulf of 



