Long Waves in Canals and Standing Waves in Closed Basins 205 



level of the surface in the transverse direction (transverse oscillations). If at 

 one point I = Acos(ati-e), then 



u = — oA sin (at+e) = oA cos (<r/+£ + \tz) = aAcos[a(t-\- IT) + e] . 



This means that the phase of the transverse oscillations has shifted against 

 the phase of the longitudinal oscillations by one quarter period. It is here assu- 

 med that the surface adjusts itself immediately to the resultant between gravity 

 and Coriolis force, i.e. that there be always equilibrium between all acting 

 forces. It remains to be seen whether this applies for all cases. 



Let us now examine more closely the effect of the earth's rotation on 



a wave progressing in a canal of constant, rectangular cross-section. The 



equations of motion are the same which we used in the theory of stationary 



ocean currents. 



du - dr> 



fv = — g — L 



dt J *dx 



dv 

 dt 



drj 

 dt 



+ fu = ~g 



dr\ 



dy 



dhu dhv 



dx dy 



(f= 2cosin«^). 



(VI. 100) 



Likewise, the equation of continuity. Only the Coriolis force has to be added 

 to the equations (VI. 16). 



If we assume that the periodical disturbance r\ has the form e iat , then u 

 and v become also proportional to e iat and, eliminating this time factor, 

 e iat , (VI. 100) becomes 



iau-fo = —g 



dx 



■ i r dr l 



wv + fu= -g~ 



and 



iorj 



dhu dhv 



dx dy 



(VI. 101) 



In the case of uniform depth where h is constant, the elimination of u and v 

 leads to the oscillatory equation in a rotating system in the form: 



when 



dH\_ d 2 r] <r 2 -f 2 

 dx 2 dy 2 gh 



= or (V 2 + k 2 )7]=0 

 a 2 -f 2 



(VI. 102) 



gh 



and V 2 is the Laplacian operator. The velocities u and v must obey the same 

 differential equations. 



We can write the equation for wave motion in a rotating canal the (longitu- 

 dinal axis of the canal is chosen to be the .v-direction) of constant width a 



