U = 5.75 U^ Log -^ (5) 



^o 



(8) 

 Concerning the roughness factor, Z^, Zingg proposed the equation 



Z^ = 0.081 Log _2 (6) 



0.18 



where Z and the sand grain diameter, d, are expressed in mm. Once the 

 wind velocity is great enough to move sand particles, the velocity pro- 

 files for different wind speeds seem to meet at a certain point, which 

 he called a "focus." The height of the focus, Z' , appears to be 

 associated with the height of the ripples which form on the surface. 

 Studies made by Zingg allow one to predict the focus by means of the 

 formula, 



Z' = lOd millimeters (7) 



U' = 20d miles/hour ^8) 



where the grain diameter, d, is expressed in millimeters. Thus, using 

 the component of the focus, Z' , U', the wind-velocity distribution can 

 be expressed by 



U = C Log —+ U« (9) 



Z 



Bagnold assumed a coefficient C of 5.75 U^ , which corresponds to the 

 value of 0.40 for the Karman constant. But the experiments by Zingg 

 yielded the equation 



U = 6.13 U^ Log _L+ U' (10) 



Z' 



which indicates values of 0.375 for the Karman constant. 



APPLICATION TO NATURAL BEACHES 



An illustration of the application of the methods of calculating 

 sand transport by wind was made for Salmon Beach near Bodega Head in 

 northern California (Figure 1). Sand samples were taken at the mid-tide 

 level, or reference point, for eight localities along the coast from 

 Salmon Creek to Mussel Point, a distance of more than 2 miles. Figure 2 



