The practical application of equation (4-27) necessitates the determina- 

 tion of the constants A* and B* and of the standard deviation T] of the 

 turbulent lift force. The procedure used toward this end will be de- 

 scribed in the following section 



5. DETERMINATION OF A*., B* and T\ Q 



Suppose that a rather large set of corresponding values of the para- 

 meters Y and $ is available. These values were calculated from equations 

 (4-10) and (4-26) respectively. The data used in these calculations may 

 have been obtained in the field or in the laboratory but in either case we 

 are confident that they are reasonably accurate. Our task then consists 

 in selecting a set of values of the parameters to be determined so that Y 

 and $ calculated from equations (4-10) and (4-26) will satisfy as closely 

 as possible equation (4-27). The determination of the constants in the 

 present study was based on values of Y and § calculated from experimental 

 data. The description of the equipment and the procedure followed in this 

 phase of the study is given in Appendix C. In these experiments three 

 different sizes of sand were used and for each size nine runs were made 

 with different amplitudes of oscillation and frequencies. The values of 

 q were directly obtained from the experiment and they were next introduced 

 in equation (4-26) to obtain § while equation (4-10) was used to calculate 

 the corresponding Y. The latter was then plotted against the former on a 

 log-log paper as shown in Figure 8. On the same paper a family of curves 

 was plotted also representing graphical solutions of equation (4-27) with 

 T| as parameter. The four curves shown on the figure were calculated with 

 1/T] equal to 1.0, 1.5, 2.0, and 2.5. Both A* and B* in these calculations 

 were taken equal to unity. The left hand side of equation (4-27) was in- 

 tegrated numerically with the simultaneous use of normal error tables. The 

 three steps of the procedure used to determine the points of the theoret- 

 ical curve were the following. First an arbitrary value of Y was chosen 

 which was divided by (cos j6±) where jz$£ is the midpoint of one of the 

 nine equal intervals dividing the quarter of the cycle, so that j6± = 5°, 

 &2 = 15° etc. The probability p which is approximately equal to the 



average value of p^ where 

 P, - ^ 



?/2 



Y 1 



dz (5-1) 



(cos i6 L ) 2 % 



was next obtained. The rearranged form of equation (4-25) was finally 

 used to calculate the corresponding $; so that 



(5-2) 



1-p 



By examining the shape and relative position of the graphs on Figure 8, 

 one may conclude that the translation of the experimental $, Y curve along 

 the two axes would bring it to a close approximation with the theoretical 

 one constructed on the basis of 1/T| = 1.5. The coefficients by which the 



19 



