156 Long Waves in Canals and Standing Waves in Closed Basins 



—p(d£/dt). The frictional force is thus assumed to be proportional to the 

 horizontal velocity u, an assumption which can indicate only roughly the 

 real frictional influences but which permits to make numerical computations. 

 This gives some idea of the effects of friction. The quantity /? has the di- 

 mension [sec -1 ] and characterizes the boundary friction at the bottom and 

 on the walls of the basin. 



The equation of motion 



— = -/? — +c 2 — (VI.36) 



8t 2 ' dt 8x 2 



has a solution in the form £ = ae yt+iHX , and by substituting we find that 

 ■f + fiy+ c*x 2 = or y = %p±i p/(c 2 * 2 - ii? 2 ) ■ 



If with x = tm/l, £ = for x = and x = I and when, moreover, we 

 put for the free constants 



a x = \Ae+ iE and a 2 = — iAe +k 



we obtain as the final equation of oscillation for I 



| = Ae~ y 'cos W(c*-x\-lp*)(t-e)\swxx . (VI.37) 



The damping factor is represented by e~^^' and the logarithmic decrement 

 X can be computed from the decay of the amplitude (see p. 9). The coefficient 

 of friction j3 can be computed from the logarithmic decrement; when A x is 

 the amplitude at the start and T the period, the amplitude after n half waves is 



A. = Al e~^> , 

 and from the equation (VI.37) results 



A = i0T. (VI. 38) 



If we designate by T n = 2ljn\/{gh) the period without any effect of 

 friction, the period of the damped oscillation T r will be from (VI.37) 



T r = 77 2/ — :-- = T n (l + Sg + ...) . (VI. 39) 



V(-S) 



32tt 2 



For water depths of 50-100 m £ is of the order of magnitude of 10~ 5 sec -1 

 and the factor of T 2 in the brackets is of the order of magnitude of 3 x 10" 13 . 

 This leads to belief that the influence of the friction on the period of the 

 oscillations of water-masses in closed basins is small under normal circum- 

 stances. 



The influence of the molecular friction on standing waves can be determined 

 more correctly by starting from the complete equations of motion with the 

 usual frictional terms. Its solution leads to simple results, assuming that 



