162 Long Waves in Canals and Standing Waves in Closed Basins 



the oscillations is geometrically similar in both cases, so that it is sufficient 

 to deal with the simpler case. This "normal curve" alone is decisive for the 

 forms of oscillations of the lake under consideration. 

 If we select, according to the oscillation 



u = A (x) cos (cot -he) when co 

 then the first of the equations (VI. 50) gives 



d 2 A 



IjijT, 



dx 2 gh (x) 



A =0, 



(VI. 52) 



which is to be solved. The problem, therefore, lies in the determination of 

 the form of the oscillations of a basin of constant width but of variable 

 rectangular cross-sections. A requirement by the theory is to have the 

 "normal curve" of a lake substituted by one or several mathematical curves. 

 The "normal curve" can be approximated with sufficient accuracy by a curve 

 or parts of several simple analytical curves. 



Chrystal has given the solution for various longitudinal profiles h(x), 

 in which 



horizontal 



two sloping straight lines 



convex parabolic 



concave parabolic 



convex "quartic" 



concave "quartic" 



h=h 



1± 



x 



h=hA\ + 



h = hA\ 



h=hAl + 



h=ho[l~ 



Table 21. Period of free oscillation of basins 

 with various longitudinal profiles 



