Long Waves in Canals and Standing Waves in Closed Basins 163 



If T = 2//i (gh ), lakes with such longitudinal profiles will have the 

 following periods for the uninodal and multi-nodal (harmonic) oscillations 

 (Table 21). This compilation shows that convex forms shorten the time of 

 all oscillations compared with the simple, horizontal profile, whereas concave 

 forms increase it. Furthermore, it is shown that the nodes of the harmonics 

 tend to move towards the shallow ends of the lake. The shallower the extreme 

 ends of the lake, the stronger this movement is. This is clearly shown in 

 Fig. 71, which indicates the position of the nodes for the uninodal and multi- 



Fig. 71 . Positions of nodal lines of oscillations having 1-4 nodes in basins of different shapes. 

 The numbers at the different curves (bottom shape) give the positions of the nodes for 



the 1-4 nodal oscillations. 



nodal oscillation of the different longitudinal profiles. A number of theoretical 

 investigations of free oscillations in water basins of different "Talweg", is 

 also given by Hidaka in "Problems of water oscillations in various types 

 of basins and canals", Parts I to X, mostly in Mem. Imp. Mar. Obs. Kobe, 

 1931-1936. 



The application of Chrystal's method to certain basins requires that the 

 "normal curve" be replaced by carefully selected sections of the above 

 mentioned longitudinal profiles. Where these sections join, the horizontal 

 and vertical displacements I and r\ must tend to the same values, from both 

 sides, and at the end of the lake I must vanish. The general solutions for 

 the simple longitudinal profiles contain sufficient number of constants to 

 satisfy these requirements. As a final result of the elimination of these free 

 constants, there remains a transcendental equation for the determination of 

 the period of the free oscillation after a computation which grows more 

 intricate with the number of sections. The computation to find the position 

 of the nodes is very long, and therefore Chrystal's method has not been 

 applied often in practice; but wherever it has been used it has given very 

 satisfactory results. (In the following paragraph an example of this is being 

 given p. 182). . 



