A number of qualitative findings have come out of these 

 measurements: the waves give added resistance to the mean flow; in the 

 region outside the wave boundary, layer, a logarithmic velocity 

 distribution is often found with a greater roughness than the physical 

 one; for a smooth bed there Is a distinct "overshooting" (Figure 4) In 

 the velocity profiles near the bed which increases with wave height 

 (over a rough bed there is a more complex change); there is an enhanced 

 upstream sediment transport when a weak current is superimposed on the 

 waves over a rough bottom; mean velocities near a smooth bed are 

 increased by the addition of waves whereas near the rough bed they are 

 reduced. 



It is difficult to assess properly the different theoretical 

 approaches since the number of very accurate measurements is limited, 

 and the range of wave and current parameters that have been investigated 

 is very limited. Where comparisons are made between experiments and 

 theories some agreement is found (Brevik, 1980; Brevik and Aas, 1980; 

 Freds^e, 1981; Christof fersen, In preparation, 1982). 



There is no doubt that the interaction of waves, current, and bottom 

 friction is nonlinear. The simplest nonlinear friction law is a 

 quadratic "Chezy" friction term, ku|ul, in shallow-water equations. A 

 detailed investigation of the effect of this type of terra in an 

 interaction problem has been made by Prandle and Wolf (1978). 



The problem considered by Prandle and Wolf is the interaction of two 

 long waves — the tide and a storm surge — both propagating in the North 

 Sea and up the Thames Estuary. They analyze terms such as u9u/3x and 

 ku I u I in a mathematical model. The model gives a good representation of 

 the observed behavior of the surge wave on top of the tide. The 

 nonlinear effects, which had been previously identified in statistical 

 analysis of surges, are quite appreciable. Prandle and Wolf find that 

 the friction term is a more important contribution to the interaction 

 than is the nonlinear advection term u9u/8x. 



If appropriate expressions for bed shear and dissipation are found, 

 it is straightforward to add appropriate terms to the momentum and 

 energy equations. But a more convenient equation is the wave action 

 conservation equation. Clearly, dissipation leads to a loss of wave 

 action. Christof fersen and Jonsson (1980) show that dissipation can be 

 included in the wave action equation (23) by adding a term 



Diss - T u • u ^11 ) 



to its left-hand side. Diss is the rate of dissipation per unit area 

 and T^ the mean bottom stress. 



41 



