Long Waves in Canals and Standing Waves in Closed Basins 225 



variations in level in Trieste exceed by far those to be expected by the change in pressure. The 

 seiches waves succeeding the large wave are oscillations of the Gulf of Trieste. 



An important paper by Proudman and Doodson considers friction but 

 neglects the earth's rotation. The basic equations are identical with those 

 in (VI. 40), except that, instead of >/, we have rj—rj, according to (VI. 130). 

 In accordance with the boundary conditions, r\ must be proportional to 

 cos xx, and u proportional to sin** and k = (nn)/!, in which n is an integer. 

 If we put 



xrj = —Pcosxx and -j — = Wsinxx , (VI. 136) 



' gh dz 



the effect of a stationary change in atmospheric pressure and wind over the 

 basin is determined by the constants P and W. To these stationary influences 

 corresponds, in the basin, a stationary level deformation and a stationary 

 current, which, as can easily be shown, is given by 



x n =-[p+\w}cos*x and ^ W = ^ + l)(! + i)^sin*x. (VI.137) 



We have to add to the statical action of the atmospheric pressure P on the 

 surface deformation a piling up effect by wind in the amount of \ W. The 

 current moves in the upper third part of the water in the direction of the wind, 

 in the lower two-thirds as a compensating current in the opposite direction. 

 The strength of the current is inversely proportional to the water depth and, 

 therefore, the piling up effect of the wind is greater in shallow water than in 

 deep water. 



The condition given by (VI.137) is stationary and starts only when the 

 acting forces, atmospheric pressure and wind have acted upon the water- 

 mass steadily for a certain length of time. The final state is attained by oscilla- 

 tions. The simplest case is as follows: Let an atmospheric disturbance (atmos- 

 pheric pressure and wind) occur suddenly at the time t = in the form of 

 (VI. 136). At the start, the equations of motion are only satisfied when at 

 the right side of the equations (VI.137) we add 



_^ C s e s and 2j C s v(z)e s 



s s 



(p. 157) respectively. We get C s and the function v(z) from the general solu- 

 tion of the differential equations, considering the boundary conditions. These 

 terms have a certain similarity with those on p. 158, which were derived 

 for the free oscillations of the basin. After a certain length of time, the sta- 

 tionary state (VI. 137) will set in. We have now free oscillations superimposed 

 on the stationary state, and these free oscillations are damped by the friction 

 and in time become imperceptible, so that only the stationary state remains. 

 A similar treatment can be applied to a sudden or gradual appearance of 



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