W) 



is introduced, relating the 

 instantaneous bed shear stress to the square of an instantaneous near- 

 bed particle velocity, in analogy with Lundgren and Jonsson (1961) and 

 JONSSON (1966). The friction factor is assumed independent of time. 



The apparent roughness and the friction factor are found as 

 functions of a near-bed current speed, u^, over near-bed orbital speed, 

 Ul,, the near-bed orbital amplitude over Nikuradse roughness, and angle 

 between current and wave. It should be observed that u_ is not known a 

 priori but is itself a part of the solution to the problem. 



Numerous illustrative graphs are given. The reasonable magnitude 

 of the apparent roughness is demonstrated by comparison with field 

 observations of very large bottom roughnesses by previous investigators. 

 The paper ends with a discussion giving instructions for the use of the 

 proposed set of formulas. For many cases the ratio u^/u^^ is small. 

 This is convenient, since here the procedure to calculate the bed shear 

 stress and velocity profiles is greatly simplified. 



It should be noted that the quantity A^^ = uj,/(Og, where to is the 

 absolute angular frequency, is not the wave particle amplitude a^ at the 

 bottom (relative to the current). These quantities are related by A^ = 

 ((jJj./cOa)a|j, where tOj. is the relative frequency, given by the normal 

 dispersion relation for linear waves. 



There is an error in the first term on the right-hand side of 

 equation (54): in the numerator, a ' shuld be a ' . In this context 

 it should be observed that (54) is in fact a quadratic equation, and so 

 can be solved explicitly. 



I The paper has some weaknesses. Since the theory is a combination 

 of linear theory (eddy viscosity) and nonlinear theory (quadratic 

 friction), the near-bed current velocity u is somewhat fictitious, and 

 a physical interpretation is not feasible. Furthermore, since this 

 reference current velocity is not known before hand, it is of little use 

 as a parameter in the figures. Also the use of the near-bed orbital 

 speed as a reference velocity in the mean bed shear stress/friction 

 factor equation (15) is not very illuminating, since it forces the 

 dimensionless factor Vo to diverge for the case of a vanishing current. 

 If the reference velocity had been the average-over-depth current 

 velocity, then the corresponding dimensionless factor times f would 

 directly show the influence of the waves on the current bed shear 

 stress . 



The greatest weakness in the analytical model lies in the 

 estimation of the wave boundary layer thickness. According to equation 

 (38) the somewhat arbitrary choice of 6^ = 2£has been made, where 

 length scale Z is given by equation (29). A closer investigation 

 demonstrates that by choosing 6^ = £ instead, one can easily get results 

 that are 20 percent smaller for the maximum shear stress and 30 percent 

 smaller for the mean shear stress. Further inspection makes it likely 



24 



