SECT. 3] BEACH AND NEAKSHORE PROCESSES 543 



to the steady state. Again, since longshore currents are essentially pulsating, 

 the remaining longshore component of momentum flux may be primarily 

 expended in accelerating the water-mass along the shore from near rest to 

 maximum current, which then turns to sea in the form of a rip current. It 

 seems likely that the frictional dissipation in the rip current mostly takes place 

 outside the surf zone. 



The following alternative approach takes the principle of continuity of flow 

 into consideration. Let q be the gross incoming wave discharge (volume of 

 water) in unit time per unit length of wave crest entering the surf zone at an 

 angle a to the beach normal (Fig. lib). The normal component is supposed to 

 be reduced to zero, as previously explained, by the superposition of an equal 

 and opposite outward discharge. The tangential component along the beach is 

 then q sin a per unit length of wave crest, or q sin a cos a per unit of beach 

 length, I. Hence continuity requires that the total longshore discharge, Qi, 

 shall increase progressively along the beach. 



Assuming that the longshore discharge Qi is zero at a point immediately 

 down-current from a previous outward-flowing rip current, its value through a 

 beach section A-A' distance I down-cm-rent from the previous rip current will 

 be given by 



Qi — ql sin a cos a. 



The average value of Qi wiU be given when l = lli, where li is the average 

 separation between rip currents (Fig. lib). Regarding the cross-section A-A' 

 of the longshore discharge as a triangular wedge whose base is the beach 

 which is inclined at an angle /S to the horizontal and whose outermost height is 

 h at the plunge line, its area is Ji'^f^ tan ^. Whence the mean longshore current 

 velocity, ui, over a long straight beach should be given by 



ui = q j-^ tan p sm a cos a. (20a) 



The power required to accelerate the surf water to the velocity ui and also 

 to maintain the flow against both bed drag and the internal resistance to flow 

 due to additional turbulence created by the breaking waves comes from the 

 longshore component of the entering flux of wave momentum. 



Both tliis applied power and that part of it which is used up in accelerating 

 the surf water may be assumed uniform, per unit beach length, along the 

 beach, but the power dissipated by internal turbulence would be expected to 

 increase progressively with the distance I as the velocity of flow increases. So 

 at some ultimate point, distant h, from a previous rip current, the acceleration 

 would be expected to cease and the longshore discharge, therefore, to break 

 outwards as another rip current because it can no longer be contained within 

 the surf zone cross-section. 



It remains to evaluate the mean longshore current velocity ui in terms of 

 the wave characteristics. For solitary waves (Munk, 1949a), the volume of 

 water per unit crest length brought forward with a breaking wave is 4:H ^ I \/ {Sy^) 



