SECT. 3] BEACH AND NEARSHORE PROCESSES 513 



P per unit area of shear plane which has the effect of making each grain layer 

 repel its neighbours. Thus, if this stress were not resisted by an equal and 

 opposite stress applied either between two parallel boundaries impenetrable 

 by the grains or by the normal component of the gravity body-force pulling 

 the grains towards the bed boundary, the grains would continue to disperse 

 ever further through the fluid. When restrained from expansion by a normal 

 gravity component, the grain dispersion may be likened to the Earth's atmos- 

 phere, whose dispersive pressure is in equilibrium with its weight. The stress P 

 pertains, of course, only to the solid phase. 



The diffusion of the components of grain collision momenta parallel to the 

 planes of shear constitutes a shear stress T which is likewise transmitted from 

 grain to grain and pertains only to the solid phase. It is additional to the shear 

 stress T maintained by the shearing of the pervading fluid. The total apphed 

 stress necessary to maintain the shearing of the combined substances may 

 therefore be written : 



^ = T + T (1) 



It was found experimentally that at high grain concentrations such as occur 

 within the bed-load zone close to the bed surface the value of the solid-phase 

 element T may exceed 100 times that of the fluid element r, even when the 

 densities of solid and fluid are the same. At such concentrations, therefore, 

 the whole of the resistance to motion may be considered as exerted by the 

 shearing of the sohd phase. The fluid element did not begin to dominate until 

 the concentration had been reduced to less than 9%. 



Since the same grain collision gives rise simultaneously to impulsive com- 

 ponents in both directions, the dynamic grain stresses, T and P, are propor- 

 tional to one another, the proportionality depending on the mean angle (f)' 

 of colhsion. Therefore, the ratio TjP is to be considered as a friction coefficient, 

 denotable by tan (f)', just as in the static case of continuous contact. Moreover, 

 the mean angle ^' was found to be of the same order as the limiting angle given 

 by the angle of repose of the granular material. 



These results open a new approach to the sediment -transport problem. They 

 render the conventional approach of the hydi'aulic engineer untenable. For the 

 latter, being largely based on the kinematics of fluid flow, has been handicapped 

 by the assumption that the sediment grains move frictionlessly, and without 

 appreciable effect on the internal motion of the fluid. 



We can now write down the condition of tangential stress equilibrium at the 

 bed surface (suffix 0) independently of the degree of grain dispersion above as : 



^p = f-^ — L gmb cos ^ (tan ^ —tan j8) -I- to, (-) 



Ps 



where .^f is the stress externally applied via the agency of the fluid flow, m^ is 

 the whole mass of the bed load and to is the residual fluid stress exerted directly 

 on the bed smface and may be neglected when any considerable grain move- 

 ment is taking place. If the bed inclination, ^ (measured downwards in the 



