Wave friction factor as related to the Shields parameter for steady currents J.P. Le Roux Departamento de Geologı ´a, Facultad de Ciencias Fı ´sicas y Matema ´ticas, Universidad de Chile, Casilla 13518, Correo 21, Santiago, Chile Received 2 October 2001; received in revised form 22 January 2002; accepted 14 March 2002 Abstract The wave friction factor f w is an important dimensionless parameter used to estimate wave-induced bed shear stress which, together with current-induced bed shear stress, controls sediment transport in the marine and lacustrine environment. However, f w is either overestimated or underestimated by existing equations. The matter is complicated by the fact that different equations must be used for hydrodynamically smooth or rough flows, but the limit between these two boundary conditions is still poorly defined. Some equations also require estimated values for the Nikuradse roughness length and the wave boundary layer thickness, neither of which can be derived accurately. It is, therefore, not surprising that f w values as calculated by existing equations differ by as much as a factor of 3. In the present paper, f w is related to the Shields parameter b for unidirectional currents, which not only facilitates the direct comparison and summation of bed shear stresses, but also yields a more accurate equation valid for grains of different density in any fluid irrespective of the boundary condition. D 2002 Elsevier Science B.V. All rights reserved. Keywords: Oscillatory waves; Sediment transport; Shields parameter 1. Introduction The prediction of sediment transport in the marine environment has many important applications, for example, in the control of siltation in harbours and beach erosion, the prevention of contamination by toxic minerals along coastlines and the modelling of depositional systems hosting placer deposits, oil, gas, geothermal energy or groundwater resources. Sediment transport is initiated when the threshold of grain motion, normally expressed as the critical bed shear stress (s cr ), is exceeded. In oceans and lakes, the total bed shear stress (s t ) is the sum of the bed shear stress related to unidirectional flows (s u ) such as tidal currents and the bed shear stress caused by oscillatory waves (s w ). As the water motion under waves is orbital, any current-induced bed shear stress will be augmented in one direction and diminished in the other during the wave cycle, which can result in sediment transport even where the current alone may be incapable of initiating grain motion. To calculate s t , it is imperative that s u and s w be computed in the same manner. For unidirectional currents, a commonly used method to determine s u is based on measured velocity profiles. However, in the case of waves, the form of the velocity distribution is uncertain (Teleki, 1972) and s w cannot be measured directly on a moving bed (Soulsby and Whitehouse, 1997). It is, therefore, usually derived from the peak 0037-0738/02/$ - see front matter D 2002 Elsevier Science B.V. All rights reserved. PII:S0037-0738(02)00157-4 E-mail address: jroux@cec.uchile.cl (J.P. Le Roux). www.elsevier.com/locate/sedgeo Sedimentary Geology 155 (2003) 37 – 43