Reply to comment by Jeffrey Olsen et al. on “Traveling wave solution of the Boussinesq equation for groundwater flow in horizontal aquifers”

Abstract

Olsen et al. [2014] are correct in pointing out the seemingly inconsistent assumption of a constant propagation speed ρ when the derived results show that ρ is in fact time-dependent. However, using a time-dependent ρ in the perturbation expansion does not alter the major results as Olsen et al. point out in the Comment. The reason is that ρ, or its new substitute , is just an intermediary variable that relates the flux rate with the head at the boundary. Therefore, the use of the more complicated definition of ρ is not warranted since it does not result in any improvement of the zero-order solution. On the other hand, a simplified definition of the traveling wave velocity ρ is intuitively more appealing and it leads to simple and elegant results. The fact that the first-order term is identically equal to zero is a consequence of the perturbation expansion used, as stated in sec. 3.3 in Basha [2013]. In any case, the zero-order solution provides a remarkably good approximation. The choice of would have improved the prediction slightly if a first-order correction were available. However, this is a particularly difficult task because the zero-order term is itself a nonlinear implicit solution that is not easily amenable to further manipulations. Instead, it is more worthwhile to consider tackling the Boussinesq equation via a refinement of its linearized solution in a perturbation series framework.