On re-examining the problem of mass transport due to partially standing waves in a domain that can be closed or open ended, it is pointed out that several aspects of the existing theories need to be clarified. Particular attention is paid to the free surface setup, whose expression is shown to be different when described by the Eulerian and Lagrangian approaches. In the Lagrangian system, it is the horizontal gradient of the setup, rather than the mean pressure gradient alone, that is implied by the condition of no net flux in a closed domain. In connection with this, one may determine for an unbounded domain the streamline value along the free surface, which cannot be fixed in the stream function formulation alone, by requiring the setup due to the unidirectional part of the mass transport to be zero. Three components of the free surface setup, which are of different orders of magnitude and arise owing to various mechanisms, are obtained in the process of deriving the solutions for the mass transport velocity in the boundary layers and the fluid core.