Long Waves in Canals and Standing Waves in Closed Basins 111 



To this transversal current belong longitudinal currents of the form 



oo oo 



m, = 2l A n cos\\S n xcosny and u 2 = 2j A^s'mhS n xs'mny . (VI. 127) 



« = i 



We obtain for the ratios A n : A' n exactly the same expressions as in (VI. 123). The condition that 

 for x = ±1 the difference between the two longitudinal currents (VI. 125) and (VI. 127) always 

 vanishes, requires for all y: 



al 



cos — cosh cry = y /l„cosh5„/cos/;>' 

 c 



and 



al 



sin — sinhay = > A n s\n\\S n l$\nny 



(IV. 128) 



If we restrict ourselves to a certain number of ^-points, we obtain a sufficient number of equa- 

 tions (with VI. 123) to compute ajc, which will give the natural period of the basin. 



The assumption that the differences between the longitudinal currents vanish for y = and 

 y == ± k n * give? already a good first approximation. We obtain the simple relation 



al al Is, l + coshictTi 



- tan - = — tan*/ - — . (VI. 129) 



c c a sinh&ajr 



Taylor has found in another way that for a basin which is twice as long 

 as it is wide (/ = n) and whose rotation period is equal to the longest free 

 oscillation of the non-rotating system (which means ffc = a = 1), a/c must 

 be = 0-429. From (VI. 129) we obtain by repeated tries (as a/c is also hidden 

 in Sx) a/c = 0-433, which comes very close to Taylor's value. 



We find [hat for long, narrow basins (/ a multiple of n) the term on the 

 right-hand side in (VI. 128) increases rapidly and", therefore, al/c nears \n. 

 This means that for this kind of basin the period of oscillation differs 

 slightly from the period of a similar basin at rest (see later the case of the 

 Baltic, p. 219). 



In the example given above a/c = 0-5 for the basin without rotation, and 

 we find that the period in case of rotation is being increased at the ratio: 

 0-50:0-429 = 114, i.e. by 14%. 



The wider the canal in proportion to its length, the greater the increase 

 of the period. The standing wave of the free oscillation is changed into an 

 amphidromy (rotatory wave) with its centre in the middle of the basin. 



A special investigation of free tidal oscillations in a rotating square sea 

 was made by Corkan and Doodson (1952). 

 (d) Influence of the Earth's Rotation on the Seiches in Closed Basins 



The reason for not considering the influence of the earth's rotation on 

 the seiches of the lakes is that, in view of the small expansion of the oscil- 

 lating water-masses this influence is hardly noticeable. In dealing with the 

 free oscillations of large lakes and oceans, this can no longer be neglected. 

 The disturbance caused by the earth's rotation can be summarized by com- 

 puting the transverse oscillations caused by the variations in velocity of the 



