Reply to comment by Cao and Hu on "Long waves in erodible channels and morphodynamic influence"

We thank Cao and Hu [ 2008] very much for their comment. We must honestly say that at first sight, we felt that their point, reproducing that raised by Cao and Carling [ 2003] to challenge the formulation of the 1-D governing equations proposed by Lisle et al. [ 2001], had already been proven to be erroneous by Cui et al. [ 2005]. However, after more careful examination of the matter, we have concluded that both the final forms of the continuity equation derived by Cao and Hu [ 2008] ( hereinafter referred to as CH) and Lanzoni et al. [ 2006] ( hereinafter denoted as LSFS) are not wholly correct. Indeed, while the main point raised by CH is definitely correct, they reach slightly incorrect conclusions, employing a framework which ignores the distinction between bed load and suspended load. On the other hand, the LSFS derivation is also incorrect in that it will be seen to ignore the defect of water flux due to the presence of particles transported as bed load, an effect which turns out to be of the same order of magnitude as that of the retained contributions. This effect modifies the final form of the continuity equation for the fluid phase which does indeed include a sediment transport correction, whose structure will be seen to be intermediate between the one derived by LSFS and that suggested by CH. While, as shown in section 4, this modification does not affect the treatment of one-dimensional long sediment waves presented by LSFS, the issue is of conceptual importance. We are therefore grateful to CH for motivating us to provide a hopefully more conclusive clarification of the matter. We are confident that the analysis presented below will be instructive for the reader, as it was for us.