because his results were put in dimensionless form, and further elaboration 

 was presented by Longuet-Higgins (1972a) . 



It is convenient to introduce a new x-direction coordinate system (after 

 Bowen, 1969a) where the origin is taken at the maximum setup line and a posi- 

 tive X is facing seaward (Fig. 23). This simply gives the longshore current 

 v=o at x=o, and eliminates the uncertainty as to where the x-direction coor- 

 dinate begins in deep water. The breaker position is x, . For the three terms 

 in the y-direction momentum balance equation (42) Longuet-Higgins (1970, 1972a) 

 derived the following. 



a. Driving Stress dS^y/dx. Differentiating equation (25) or equation 

 (45) in terms ot energy flux gave 



(1) Outside the Surf Zone 



dS 



— ^ = (47) 



dx 

 (2) Inside the Surf Zone 



xy 5 2/ ■u\ Z /Sina, dh , , ^^ 



-^ = YT- PY (gh) -^ (— :— ) — (^8) 



dx dx 



where in equation (47) a constant energy flux is assumed. In equation (48) it 

 is assumed that in the shallow-water surf zone, n=l; the wave angle is small, 

 cosa=l; and Snell's constant is still applicable. Longuet-Higgins (1970) also 

 neglected the effects of wave setdown and setup in further simplification of 

 equation (48) although he -recognized their effects. Wave setup is included in 

 later modifications of this original theory. , Therefore, within the surf zone 

 he simply derived for a plane sloping beach 



^\^_^lk^^_ (tan3)'/2x'/2tan3 

 dx dx 



so that equation (48) became 



xy . 5 % 2/.- o\%/Sin^.-% , . ^^ 



— -^ " 16 "^^ "^Y (tanB) ^( ^ )x ^ (49) 



dx 



From equation (46), therefore 



D =-Apg\2 (tan6)^''2-^/2 (50) 



meaning the local rate of energy dissipation D due to wave breaking varies as 

 x^^ from zero at the maximum setup line (on a plane beach with constant y) • 

 Other forms for equation (49) stem from using the celerity C=/gh in the surf 

 zone. 



83 



