530 INMAN AND BAGNOLD [CHAP. 21 



made to travel in a narrow water tank over a bed of specified sediment material 

 inclined n})wards at an arbitrary initial slope to the water sm-face. The action 

 is contimied until progressive change of the bed by wave action has ceased. 



The inherent limitations of such experiments made in narrow tanks are that 

 the mean mass transport of water over the bed is zero everywhere, and that all 

 motions both of water and of bed material are constrained to take place only 

 in the plane of wave motion. 



A further, human limitation, applicable to all model experiments in this 

 field, is imposed by the time available for experiment. The rate of sediment 

 displacement in the offshore zone decreases with increasing depth, till it 

 becomes very slow. In order, therefore, to achieve a mature littoral profile in 

 the time available, the initial littoral slope is usually made to extend steeply 

 to the depth at which bed movement ceases. This limits the quantity of sedi- 

 ment amenable to movement, so that no progressive building out or recession 

 of the profile is possible. 



Nevertheless, in general, the mature model profiles reproduce the main 

 characteristics of natural littoral profiles to a degree which is surprising in 

 view of the great discrepancy in the sediment-scale ratio in terms of wave 

 height to grain size. 



It is surprising, too, that the well known and marked difference in profile 

 form and slope that occurs between cobble and sand beaches in nature is 

 reproduced in smaU models using materials of natural size. This fact is of 

 sufficient interest to demand a physical explanation. 



B. Energy Profile 



The mean gradient of wave energy due to energy losses by bed friction, and 

 by surface turbulence within the surf zone, is always onshore. But since the 

 energy gradient due to bed friction is always in the direction of local flow 

 relative to the bed, we have, superimposed, local fluctuations onshore and 

 offshore due to the wave oscillations. 



Consider any unit area of a littoral slope, locally inclined at an angle /S to the 

 horizontal. The energy loss, AEi, over this unit area due to bed friction, in- 

 volving appreciable sediment displacement during the time of onshore water 

 displacement, may be assumed proportional to the mean force exerted by the 

 water on the displaced sediment times the distance x that it is displaced. If 

 the coefficient of intergranular friction is tan (f), and the displaced sediment 

 mass is wi per unit area, we have 



AEi = a — — - gniiXi cos ^ (tan ^-Htan /S), 



Ps 



where, following the notation of Fig. 3, the tan /3 term represents the gravity 

 component tangential to the slope, (p., -p)lps is the buoyancy factor, and a 

 is a coefficient of proportionality. 



Owing to losses of various kinds, including percolation into the beach, the 



