with a simple harmonic motion in the x direction prescribed by a and U). 

 The size of the flume can then be kept within reasonable limits since 

 fl(y) is a rapidly decaying function of y. 



b. Experimental Work 



Velocity measurements were made in an experimental flume in which 

 a horizontal plate near the bottom was oscillated in its own plane with 

 a simple harmonic motion with respect to the water above. A detailed 

 description of the apparatus and the measuring device has been given 

 elsewhere (Kalkanis , 1957). In that previous study experimenting with 

 a smooth plate it has been found that the expression describing the 

 vertical distribution of the horizontal velocity is of the form 



u 2 = u Q k 1 (0y) C sin(U)t-a in k 2 3y) (3-22) 



1 

 where (3 = (-4^-) 2 ; the values of the exponent c and of the constants as the 



results of these experiments indicated were the following 



c = -0.65, k 1 = 0.342, a = 1.05, k 2 = 2.883 



The above expression describes the velocity distribution relative to a 

 fixed boundary; the distribution relative to the oscillatory boundary 

 according to equation (3-20) is given by the expression 



u = u Q Jsin (Wt-k 1 Oy) C sin(UJt-o' in k 2 gy)| (3-23) 



At the time of the conclusion of the experimental work of this first 

 phase of the study the lack of sufficient data led to the adoption of the 

 parameter 1/13 as the proper length scale. The data collected in subse- 

 quent tests, however, indicated that more suitable characteristic lengths 

 were the amplitude of the oscillation a for the case of the hydraulically 

 smooth wall and the parameter aBD for the rough wall, where D is the 

 roughness diameter (either two or three dimensional). 



The analysis of the new experimental data which can be found in 

 Appendix B led to the following expression: 



l 

 -H. = ("l + f x 2 (y) - 2f x (y) cos f 2 (y) ! 2 sin(ujt + e) (3-24) 



-1 fi(y) sin f2(y) 



where q = tan x _ 



l-f L (y) cos f 2 (y) 



The form of the functions f^ (y) and f 2 (y) that seemed to best fit the 

 experimental data is the following 



- 75| 

 (i) Smooth plate fi<y) = 0.3 e a 



1/3 

 f 2 (y) = 1.55 (0y) 



