166 Long Waves in Canals and Standing Waves in Closed Basins 



4n 2 1 C 



Ar\ = -=^%Ax and £ = — ~ I n]bdx 



(VI. 54) 



Furthermore, we have to add the boundary conditions, which for both ends 

 of the lake x = and x = / require 1=0. The first approximation for the 

 period of the free oscillation with one node T x gives Merian's formula (VI. 34), 

 if we substitute for h the mean depth of the lake. Then a = (4ji 2 /gT 2 )Ax is 

 known approximately. If the cross-sections drawn along the "Talweg" are 

 closely spaced, it can be assumed in a first approximation that the changes 

 of the displacement between two successive cross-sections are linear. For 

 the practical computation, the equations (VI. 54) then take the easier form: 



S2 



S 2 1 + 



av 2 

 4S 2 



, >h + % 



Qi 



>h 



«!m 



(VI. 55) 



The quantities with the subscripts 1, 2 represent respectively the value 

 for two successive cross-sections, whereas v t is the surface area of the sea 

 between section (i— 1) and the i. The quantity q is equal to zero. First, an 

 approximate value of T is found by means of Merian's formula (VI. 34), 

 using the average depth of the basin. The result is an approximate value of a. 

 The computation is then started at one end of the basin x = 0, £ x = 0, and 

 where rj x = 100 cm (an arbitrary value). The second equation (VI. 55) then 

 gives | 2 and introducing this value in the first equation of (VI. 55) we get 

 the vertical displacement rj 2 at the second cross-section. The third equation 

 then gives q 2 . Now all quantities are given at the second cross-section, and 

 from there to the next cross-section the computation can then be continued, 

 repeating the previous computation with the new values of £ and r\. If the 

 approximate period which was derived from Merian's formula (VI. 34) is 

 correct for the simplest seiche, the computation must arrive at a value q and 

 I = at the other end of the lake, in order to fulfil the boundary condition. 

 The computed value will usually differ from zero and, hence, it will be necess- 

 ary to select another value of the period, i.e. another value of a and to repeat 

 the entire computation. If this second value of the period does not lead to 

 a correct result, one has to select a third one, which can then usually be deter- 

 mined by suitable interpolation. The final result will give relative values of the 

 displacements and its related currents, and the exact locations of the nodal line, 

 and finally also the currents related with the oscillations within the lake. Ac- 

 cording to (VI. 4), u = d£/dt, the horizontal component of the current 



