172 Long Waves in Canals and Standing Waves in Closed Basins 



(vi) The method of Ertel. Ertel (1933) has developed a new and an elegant 

 method which follows the wave mechanical perturbation theory. The only 

 restriction to its application is that the lake should extend preponderantly 

 in the x-direction, so that transversal oscillations can be disregarded. It starts 

 out from the Chrystal equation, to which we can give the form 



d 2 A m 2 



w+^-°- ,VL65 > 



It is identical with equation (VI. 58) if we put 



co 2 I 



in which 



<p{z) = (g/a 2 )<*(z) and a(z) 



v 

 is the normal curve of the lake, referred to the abscissa z = - 



a 



(0 < z < a , if < x < /) . 



If we assume a rectangular cross-section of constant depth of the width 

 b and an area So = b h , then b l = a and y = gh /l 2 . Considering here 

 only the longest free oscillation, we have for this rectangular basin 



with ^o(O) and A (\) =0. Its solution is 



A = ]/2sin(jrz) , (VI. 67) 



and its proper frequency is 



eo =n\/<p , (VI. 68) 



or as period of the free oscillation Merian's formula T Q = 2//j/(g/7 ). This 

 amplitude in (VI. 67) has been selected so that 



i 



| Aldz = 1 (normalized eigenfunction). (VI. 69) 







In order to obtain a general solution of (VI. 65) we make the usual 

 assumption in the perturbation theory 



A = A +AA and -^ = ^ +A l-^-X (VI.70) 



and substituting in (VI. 65), we obtain 



»' AA + 4 AA ^_ A (^\ A<1 , (VI.71) 



fe 2 f \<p(z) 



