reasonably reliable values provided that the frictional forces at the 

 surface or at the bed are not influenced hy each other. However, in 

 shallow water, the problem is more involved since the wind effects on 

 the water will influence the vertical velocity distribution over the 

 total depth. Consequently, the slope of the water surface due to the 

 winds is also influenced by bottom stresses. Moreover, in the vincinity 

 of the shore, the velocity at the surface may have a different direction 

 from that at the bottom. Friction models which take into account the 

 combined effects of surface and bottom stresses have been proposed by 

 Reid (1957) and Platzman (1963). 



Here, ,the simplified stress laws are used, and the influence of the 

 vertical velocity distribution is neglected. Moreover, similar formulas 

 are adopted for representing the surface stress and bottom stress and 

 this form is given by: 



2 



T = YpV (6) 



where x is the shear stress at the boundary; y is a dimensionless resist- 

 ance coefficient; p is the water density; and V is the fluid velocity. 

 More specifically, the shear stress at the bottom is divided by water 

 density, and consistent with the stress term required for Equation (5) 

 is: 



• -^ = K V^ (7) 



P 



Here K and V replace y and V, respectively, in Equation (6), and K is the 

 bed friction coefficient and V as defined previously is the y-component 

 of the water velocity. The bottom-stress relation given here is a dif- 

 ferent form than the one originally proposed by Freeman, Baer and Jung 

 (1957) and subsequently used by many investigators. As implied by the 

 above authors, the bed friction coefficient had dimensions of length to 

 the minus one-third power. This is in accordance with Manning's (1890) 

 relation which implies that the friction coefficient should be inversely 

 proportional to D'^/^. However, on the other hand, Prandtl-von Karman 

 boundary- layer theory implies that the friction coefficient is dimension- 

 less (Prandtl, 1935; von Karman, 1935). For simplicity in computation, 

 the latter theory is chosen. 



Equation (7) written in transport form is 



-^ = K U^ D"^ (8) 



P 



For typical seabed conditions, it has been found that K generally lies in 

 a range from 10-3 to 5 x 10-3. 



13 



