SECT. 3] BEACH AND NEARSHORE PROCESSES 527 



Under wholly laminar flow, 6t appears to have the same high order of value 

 as is found (curve C) for a grain size of 0.2 mm under turbulent flow. 



Provided no other grains are moving in suspension at the time the threshold 

 stress Tt for a particular grain size D is considered, the value of rt may be 

 calculated from the measured velocity gradient of the fluid above the bed by 

 the Karman-Prandtl relation which gives : 



If velocity measurements are made at distances such that z-2=\0zi, 

 loge (22/31) = 2.3, and for sediment-free flow of a fully turbulent fluid, k = 0.4, 

 whence u^ = 0.174{w(i02) —U(z)}, where Wi and U2 are measured flow velocities at 

 distances Zi and Z2 from the bed, and k is the Karman constant, approximately 

 0.4 for a homogeneous fluid. When suspended sediment is present the fluid is 

 not homogeneous, and the value of k is found to be apf>reciably reduced. Stress 

 determinations from measured velocity gradients are, therefore, unreliable. 



In the case of a laboratory flume, n can of coiu-se be measured directly as 

 pgh tan ^. 



B. Oscillatory Water Motion under Waves 



There appears to be no theoretical or experimental justification for assuming 

 that the effective threshold stress, t<, bears any similar relation to the orbital 

 velocity near the bed, in the case of oscillatory water motion, as rt bears to 

 the velocity gradient in the steady flow case. The Karman-Prandtl relation in 

 the latter case presupposes the establishment of a constant and j^ei'sisting 

 turbulent boundary layer, and such a layer camiot become established when 

 the flow is continually changing in direction, unless the amplitude of the water 

 oscillation is very large, as in tidal currents. 



Experimental evidence (Bagnold, 1946) has shown that for semi-amplitudes 

 of up to 30 cm the water osciUated over flat, unrippled, grain beds without 

 forming any appreciable turbulent boundary layer. The distortion of the 

 water close to the bed occurs as simple viscous shear. 



It is clear from the evidence that the threshold of grain movement on an 

 unrippled bed is determined not only by the orbital velocity but also by the 

 relative value of the end-of-stroke acceleration which, for the same orbital 

 velocity, increases with decreasing amplitude. 



In Fig. 6 the experimental data from Bagnold (1946) are plotted as threshold 

 orbital velocity, Uot, against the semi-amplitude, r. It will be seen that it is 

 impossible to extrapolate with any certainty to larger amplitudes at which Uot 

 might be expected to become constant and independent of r. Quantitative 

 experiments will have to be made on a considerably larger scale. 



There appears to be no similar information relative to conditions where the 

 bed sm'face is already rippled ; and it seems that, at present, we have but little 

 reliable knowledge about the threshold of bed movement under full-scale sea 

 conditions. 



