168 Long Waves in Canals and Standing Waves in Closed Basins 



As the normal curve a(v) is known, the integrals can be easily computed 

 by simple summations, but not for the ends when v = and v = a, as for 

 these a(v) = and the expressions under the integrals are indefinite. At the 

 first place, however, where v = 0, all J n = 0, so that only the J n (a) are to be 

 computed. In order to overcome this difficulty, Doodson divides the lake 

 into two equal parts at v = \a and determines the /„ by two integrations 

 first, starting from one end (east end) J e n , and then the J™ from the other end 

 (west end) up to the middle of the lake. As, due to the unsymmetry of the 

 lake, the two parts are not identical, the /„ values obtained for the middle 

 of the lake will not agree. As the equations of (VI. 57) are linear, it is possible 

 to adjust the two values to each other by a multiplication factor. For the 

 other end of the basin the theory then gives the conditional equation 



n 

 r = 



in which the argument \ a has to be taken for the functions J and J in the 

 left-hand term of the expression in parentheses. As these values are known 

 for the middle of the lake, the /„ can also be determined for the other end. 

 Doodson and his collaborators determine these J n for 10 values of v of 

 the western and the eastern half. When v = 0, the J n (0) = and, therefore 

 there are in all, till v = a, 18 values of /„. At both ends of the lake, A must 

 be equal to 0. This condition is automatically fulfilled for the end v = 0, 

 as here all J n = 0; for the other end, however, v = a must be 



2(r-XfJJLa) = J&a)-U x (a)+mAa)-... + 2PJJfi) = . 



n 



This is the equation for the determination of the values of A, which accord- 

 ing to (VI. 57) define the periods of the free oscillations of the lake. This 

 equation is solved by Doodson according to the method of Horner. 



Figure 72 shows the normal curve of the lake of Geneva and Fig. 73 the 

 distribution of the vertical displacement along the lake for the uni- and two- 

 nodal Seiches. According to the method of Proudman, the periods are 



7\ J 2 T 



74-45 min 35 1 min 28 min, 



whereas Forel has observed the following values: 



740 min 355 min 



The position of the nodal lines are also agreeable with those derived from 

 the limnographic recordings. Fig. 73, moreover, shows very clearly how much 

 the profile of the seiches can deviate from a cos-line in irregularly shaped 

 lakes. The method of Proudman has not yet been tested for other lakes. 



