160 Long Waves in Canals and Standing Waves in Closed Basins 



basin, the less the period will be affected by friction; the same applies for 

 the logarithmic decrement, which is inversely proportional to the depth of 

 the water-mass. 



An example will illustrate the computation. Let us assume a closed basin of / = 100 km and 

 h = 20 m. Let the frictional coefficient bz v = lOOcm^/sec; then T will correspond to 3-96 h. 

 From (VI. 42) it follows that a = 0-00322 and from a graph of f— r\ and 2£//j/a over j/a we get 

 the values I = 2-994 and r\ = 2-692. We thus obtain T r = 4-35 h or an increase through friction 

 of 9-9% of the period of the free oscillation. The damping coefficient is y = 4-2 9x 10 -5 sec -1 = 

 2-57 x lO^min -1 with a logarithmic decrement X = 0-336. We can compare these values with 

 the Vattern lake in Sweden with a length / = 124 km, greatest depth 120 m, mean depth approx. 

 54 m. Since for this lake T= 2-99 h one obtains the values y = 0-7 x 10~ 3 min -1 and I = 0-06 

 (see p. 184). 



(c) Variable Cross-section. Theories of Seiches 



Oblong lakes can be considered as basins of variable width and depth. 

 It is to be expected that periodic fluctuations of the water-level will be observed 

 when the water-masses in the lake basin, once the equilibrium is disturbed, 

 oscillate back into equilibrium position. Already in the eighteenth century 

 such oscillations were reported to have been observed in the Lake of Geneva. 

 According to an old chronicle by Schulthaiss, oscillations of this kind were 

 note at the Lake of Constance as early as in 1549. Forel (1895) was first 

 in making systematical and methodical observations of the oscillations of 

 the Lake of Geneva in 1869. Since then, these oscillations have been observed 

 in other lakes; they are known as "Seiches , \ These seiches are free oscill- 

 ations of a period that depends upon the horizontal dimensions, the depth 

 of the lake and upon the number of nodes in the standing wave. Forel com- 

 pared the observed periods of the free oscillations of the lake of Geneva with 

 the period derived from Merian's equation: T = 21/ \ (gh) which only applies 

 to a rectangular basin of constant depth. He substituted for the depth h the 

 mean depth of the lake, but the result was not very satisfactory; later on, 

 Du Boys proposed that, if the travel time of the wave is T = 2l/c for a dis- 

 tance 21, when the depth h is a constant, the period of a lake with varying 

 depth should be 



i 1 (gh) 



in which h represents the depths along a line connecting the lowest points of 

 the bottom of the lake called "Talweg" (valley-way). Although a little better, 

 this relation still left many discrepancies unexplained, especially for the harmo- 

 nics (multinodal seiches). Forel rightly wondered why the depths of the "Tal- 

 weg" are taken for h, instead of the mean depths of the single cross-sections, 

 as it is these cross-sections which determine the oscillating process in the 

 whole lake. 



The problems connected with free oscillations (seiches) in closed basins 

 of an oblong from, but of variable width and depth, have been the subject 



