Long Waves in Canals and Standing Waves in Closed Basins 165 



and to greatly, the double in these equations are small compared with the first term and can 

 be neglected. The period of oscillation of the lake can then be computed in first approxima- 

 tion from 



llnV C n + AC n 



J n J A n +AA n 



After a few transformations, we finally obtain 



T = 2/ 



n | (gh n ) 



1 t (IS Ab\ Inn , 

 - c °s xdx 



n \ 



,V Of I 



(VI. 53) 



in which V is the volume and O the surface area of the whole lake and AS 

 and Ab are the deviations from the mean values S and b and S /b = h , 

 where S is the mean area and b the mean width of the mean cross-sections. 

 The term in parentheses is the correction to be applied to Merian's formula 

 for the period of free oscillation of an irregularly shaped lake. The first part 

 of this correction represents the effect of the variations in cross-section, the 

 second one that of the variations in width. It is easy to see that there will 

 be an increase of the period when the lake narrows in the centre whereas 

 the period will decrease when the lake narrows at the end. 



To apply equation (VI. 53), one makes a numerical integration by dividing 

 the lake in a sufficient number of cross sections perpendicular to the "Talweg", 

 which are carefully measured. This method, which is limited to lakes with 

 a fairly regular shape, gives only the period, and does not give any information, 

 either on the position of the nodal lines, nor on the relative magnitude of 

 the horizontal and vertical water displacements, caused by the oscillations. 

 In this respect this method is inferior to the other ones. 



(iii) Defanfs method (1918). It finds its origin in the equations of motion 

 and of continuity transformed by Sterneck (1915). Their solution can be 

 adapted accurately to the most complicated shape of basins by stepwise 

 integration, using finite difference-method. Besides the period, Defant's method 

 gives at the same time the relative magnitude of the vertical and horizontal 

 displacements and, consequently, also the position of the nodal lines along 

 the entire basin. In this it is superior to other methods, even though it 

 generally requires considerable work (see Defant, 1918, p. 78). 



As a solution of the equation of motion (VI. 5) and of the equation of 

 continuity (VI. 47), one assumes, periodic functions of t in the form 



£ = £o(*)cos(-=-f — el and r\ = ^ (x)cos(^r — el 



in which | and r] represents the horizontal and vertical displacements and 

 are functions of x only. If we substitute these expressions into the equa- 

 tions (VI. 5) and (VI. 47), we obtain for the small variations zlf and Arj which 

 occur along a small distance Ax of the "Talweg", the relations 



