SECT. 3] BEACH AND NEARSHORE PROCESSES 545 



power per unit of wave crest, into the longshore component per unit of shore- 

 line, where a is the angle the breaking wave makes with the beach. Savage 

 (1959, fig. 4) gives a graph summarizing available data from field and labora- 

 tory experiments in terms of aS' and Pi. Inspection of this graph, excluding only 

 the data for model experiments with sand of low density, indicates that the 

 following relation, with a wave-power exponent of unity, fits the data equally 

 well : 



8 = 125Pi, 



where 8 and P are expressed in the same units as before. 



These relations, which summarize the available facts, are empirical only, 

 and are dimensionally incorrect. The ideas contained in Section 4 suggest that 

 the transport rate of the sediment should be converted from a volume to an 

 immersed weight basis by writing : 



Ii = [pg —p)ga'8 = constant x Pi, 



where Ii is the longshore dynamic transport rate and a' is the correction for 

 pore-space and may be taken as 0.6, approximately. The new constant is then 

 dimensionless, has a value of 2.0 x lO-i for natural beach sand, and is in- 

 dependent of the unit system used providing the units are consistent. 



Since this relation is empirical, it still fails to suggest what other relevant 

 factors may affect the value of the constant. A rational derivation is attempted 

 in the following section. 



D. Model for Longshore Transport of 8and 



Let E be the mean energy per unit surface area of a wave which is near the 

 breaking point and is approaching an infinitely long straight beach at an angle 

 a (Fig. lib). Then ECn is the rate of transport of energy (power) which is avail- 

 able for dissijjation over a unit crest length of beach, where C is the wave 

 phase velocity and Cn is the velocity at which energy is propagated forward. 

 Assume that the rate of energy dissipation by friction on the beach bed is 

 proportional to ECn, then the mean frictional force applied to the whole of the 

 beach bed per unit of crest length is proportional to ECnjUo, where Uo is the 

 mean frictional velocity relative to the bed within the surf zone and is assumed 

 to be proportional to the orbital velocity near the bottom just before the wave 

 breaks. The mean force applied to the beach bed per unit of beach length then 

 becomes proportional to {ECnfuo) cos a. The immersed weight of sediment in 

 motion should then be proportional to this applied force divided by the inter- 

 granular friction coefficient, tan <f). Once the sediment is in motion it becomes 

 available for transport by any current, such as the longshore current ui (see 

 Section 4-D). Thus the total immersed weight of sediment transported in unit 

 time past a section of beach A-A' becomes 



ps—p J ,^ ECn cos a _ ,-.,> 



Ii = - — -gJ = K — -ui, (21) 



Ps Uo tan 



