Long Waves in Canals and Standing Waves in Closed Basins 155 



.x k =^(2k+l), 



in which k will be 0,1, ... to (w— 1). The antinodes, on the contrary, are 

 situated at x B = (l/n)k and k = 0, 1 , ... to n. 



The streamlines of a two-nodal standing wave are represented in Fig. 70. 

 It shows that, at the antinodes the horizontal motion of the particles dis- 

 appears completely, and the vertical motion is at a maximum, whereas at 

 the nodal points the movements are reversed. 



Fig. 70. Streamlines of a binodal standing oscillation. 



When the depth is not negligible compared with the wavelength A, we obtain with equation 

 T = 2//c as a more exact equation for the period of the free oscillation with one node (see von 

 DER Muehl, 1886, p. 575). 



T= 2 



]/(t 



Jih 



coth — 

 / 



If nh\l is small, we obtain as a second approximation instead of (VI. 34) 



2/ 



n\\sh) 



We also wish to point out that in the equation for the period of oscill- 

 ation (VI. 34) the density of the oscillating medium does not appear. Forel 

 has proven this by means of laboratory tests in a trough with water, mercury 

 and alcohol. 

 (b) Effect of Friction on Oscillations in a Rectangular Basin 



The motions related to the oscillations of a water-mass in a basin are 

 subject to frictional influences which are due, on one hand to molecular 

 friction and, on the other hand to frictional stresses at boundary surfaces, 

 namely at the bottom and the walls of the basin. The free oscillations of 

 a water-mass in a basin of a given cross-section would last indefinitely, once 

 they are generated, were it not that the energy is gradually dissipated through 

 friction. When there are no other disturbing influences, the amplitude of 

 the oscillations decreases gradually, whereas the period of the oscillations 

 increases slightly through friction. The influence of the friction can be 

 illustrated by adding to the right side of the equation of motion for the 

 horizontal displacement of the water-mass (VI. 8) another term of the form 



