Long Waves in Canals and Standing Waves in Closed Basins 157 



the kinematic coefficient of viscosity v is small, which is the case for water. 

 For a standing wave in a rectangular basin of the length / Hidaka (1932) 

 finds in this manner that in a first approximation the period of the free oscill- 

 ation is not different from Merian's formula, but that the amplitude of the 

 nth nodal oscillation decreases with the damping factor r = l 2 /(2n 2 jc 2 v). This 

 is the same damping factor found previously for the decrease in amplitude 

 of waves of the wave length 2l/n through friction (see p. 79). 



Using greater values of v, far more correct solutions of the equations of 

 motion are obtained where the frictional terms and the corresponding 

 boundary conditions are taken into consideration. Solutions of this kind 

 have been given by Defant (1932); they are to be found also in a paper 

 of Proudman and Doodson (1924, p. 140), to which we will refer later. 

 However, they do not show the frictional influences as clearly as the simple 

 assumption stated previously. 



The stresses in different fluids under similar circumstances of motion will 

 be proportional to the corresponding values of /j,; but if we wish to compare 

 their effects in modifying the existing motion we have to take account of 

 the ratio of these stresses to the inertia of the fluid. From this point of view 

 the determinining quantity is the ratio /h/q = v, the kinematic coefficient of 

 viscosity. 



The equations of motion and of continuity now takes the form 







du du du 2 , d f , dr] 



» = ~*ta H "V and ai J udz+ »7 = ° ' (VI ' 40) 



-h 



when v is the kinematic coefficient of viscosity (cm 2 sec -1 ) and when the 

 origin of the co-ordinates lies in the undisturbed water surface and the 

 depth is —h. 



We have to add, the boundary conditions that u = for x = and x = /, 

 as well as for z = —h; also no wind on the water surface and du/dz = for 

 z = 0. Part of the condition is fulfilled at once, if we start with u propor- 

 tional to sin^.v, where xl/n =n (n = 1, 2, 3, ...). 



For the general solution one can select the form 



xrj = cos xx • e~ Uvlh2)t , 



-^= u = sin xx • v (z) e~( Evlh2)t 

 gh 2 



« is a numerical parameter and v(z) a certain function of z. If we put 



(VI.41) 



a= , (VI. 42) 



gx 2 h b 



the equation of continuity gives 



u 



I v(z)dz = as , 



-h 



