Long Waves in Canals and Standing Waxes in Closed Basins 111 



in its respective basin. Let their surface be O = bl, their cross-section S = bh, 

 their longitudual section F = //?, and their volume Q = Ihb. Let the horizontal 

 and vertical displacements of the water particles be £ and r\ (see Fig. 75; 

 everywhere in this figure £ stands in place of rj). 



Suppose basin I oscillates regularly (seiches); then there will be in the 

 connecting canal II periodical pressure variations and a corresponding per- 

 iodically alternating current, whereas in basin III the oscillations are caused 

 through the filling and emptying of the basin in the same rhythm as the 

 free period of the complete system. The resulting period of the connected 

 system to depends, of course, first of all on the dimensions of the system, 

 which will be expressed in the boundary conditions. These boundary con- 

 ditions have been added to Fig. 75 underneath the three points jc = 0, / x 

 and l x +h an d do not need any further explanation. L and R as upper indices 

 mean left and right of the section in question. 



The oscillations in III are simple variations in level caused by water 

 flowing in or out at x = l x + k and the condition for continuity gives 



&& = O a %- (VI. 78) 



In II the equation of motion (VI. 5) applies 



d 2 £ 2 dr J2 



dt 2 = 8 dx 



ffof-irf), (VI. 79) 



because this is a simple pressure current. 



All | and r\ in the oscillatory system are proportional to a periodical 

 function of the time, and therefore I, ?;~e"°' Considering (VI. 78) and the 

 boundary condition at x = / x , it follows from (VI. 79) 



in which 



l-^)^|rf + frf=0, (VI. 80) 



or) o 2 lo 



555-*<£— (VL8I) 



So 



c 3 = Y(gh 3 ) and a, - -y . 



In analogy to the theory of the resonators of Helmholtz, a can be designated 

 as the "conductivity" of the canal and 



2ji _ / a 2 



is then in agreement with Rayieigh's equation for the natural frequency of 

 such a resonator. 



The equation (VI. 80) represents a relation between If and »/f of basin I. 

 A second equation can be derived from the equation for the oscillations 



12 



