Long Waves in Canals and Standing Waves in Closed Basins 167 



u = — y £ (X)sin —/-£ 



2jc . 



= y| COS 



2jzI T 

 tV + 4 



(VI. 56) 



The largest current intensity corresponding to the seiches therefore is 

 (2njT)£ and its phase differs by one quarter period from that of high tide. 



This method is not limited by the exterior form of the oscillating water- 

 mass; however, it naturally gives the unequivocal results, especially when 

 the main direction of the oscillation is determined by the direction of the 

 longitudinal extent of the lake. For a correct application, it requires a good 

 knowledge of the orographic nature of the lake. An example of the application 

 of the method will be given on p. 198. 



(iv) Proudmari's Method. This method starts with the equation of mo- 

 tion (VI. 50) of Chrystal, which if 



is transformed into 



u = S$ 

 d 2 A 



dv* + a{v) A 



v4cos(cof + e) 

 X 



0, 



(VI. 57) 



which is similar to equation (VI. 52) and in which I = a> 2 /g the function 

 a(v) = b(x). S(x) is given by the "normal curve" of the lake. (VI. 57) is 

 to be solved with the boundary conditions that A = for v = o and v = a, 

 that is to say the entire surface of the lake. Proudman (1914) has given an 

 exact theoretical analysis of this differential equation, and also the procedure 

 of computing the free oscillations of a basin of any shape. Doodson, Carey 

 and Baldwin (1920) have applied this method to the lake of Geneva with 

 excellent results. This method is better than the one by Chrystal, in so far 

 as it is not necessary to adapt the "normal curve" to fragments of analytical 

 curves, which is always very difficult with an irregular "normal curve". 

 We will only discuss here the characteristic features of this method. 



As a solution of (VI. 57) we can establish an infinite series of the form: 



oo 



A = £ {-K) n J n {v) = J -J l k+JA^-J 3 X 3 + ... 



n=o 



The determination of the functions J n (v) considering the boundary condition v = gives 



M 



and 



and generally 



Cj x r V 



(V) = V, — = A= — 



ov J aU 



V 



J I = J JydV 



(v) 



dv 



' •> n — l 



Jn(v)= -^—^dv and JJv) = \ J' n dv . 

 J a(v) J 



