Long Waves in Canals and Standing Waves in Closed Basins 173 



/co 2 \ Aw 2 Aw 



Ertel transforms the perturbation term —A\ — A into M o-r^- > 



\ <P I <f Aw- 



Aw 

 neglecting quantities of second order, which can be done only if -< 1, 



Aw 



as was shown by Fr. Defant (1953). This will not always be so; therefore, 



one must use equation (VI. 71). 



For the frequency of uni-nodal seiches, we then obtain with (VI. 67) 



2 i 



ft» 



sin-.-rr 



' [1+(#W] 



dz. (VI. 72) 



For Aw = (rectangular basin w = w ) the integral becomes 1/2 and « = ft> . 

 With the two-nodal or multi-nodal seiches, nz in (VI. 72) must be substituted 

 by 2nz or nnz. This method was first applied by Fr. Defant in the determination 

 of the free oscillations of Lake Michigan. The periods obtained deviate by 

 less than 1 % from those computed by means of other methods. It can be 

 shown that the "Japanese method" (see p. 164) is a special case of the method 

 given by Ertel. It does not give especially good results as compared to other 

 procedures, because Ertel did not consider terms of higher order in AS/S 

 and Ab/b . 



As the homogeneous equation (VI. 66) has the solution (VI. 67), the in- 

 homogeneous equation (VI. 70) can only be solved according to the theory 

 of differential equations, if the condition (VI. 73) is satisfied: 



i 



f A (jU Ald2=o - (VL73) 



(d) Standing Waves in Partially Open Basins 



A water-mass in a basin not enclosed on all sides, but communicating 

 in one or several points with a considerably larges body of water, can also 

 have standing waves. We have then to deal with the free oscillations of bays 

 and canals. In such oscillations the water-masses will be drawn in a horizontal 

 direction from the great body of water of the ocean; there must always be 

 a nodal line across the opening of such bays and canals. The longest free 

 oscillation of a bay must, therefore, be identical with that of a basin consist- 

 ing of two identical sections which are mirror images in respect to the opening 

 which joins the two parts. If the length of the bay is / its cross-section 

 rectangular and its depth constant (canal closed on one side of rectangular 

 cross-section and of a depth h) then 



4/ 

 T = -r^— . (VI. 74) 



}(gh) 



