We consider an underwater networking scenario, and test the performance of two multihop routing paradigms, source routing and hop-by-hop relay selection, in the presence of different representations of the channel dynamics. We focus on a static channel case (obtained via empirical equations for path-loss), and on a sequence of channel realizations obtained using ray tracing, that vary both slowly and rapidly over time with respect to the expected reaction time of routing protocols; the two latter cases are also explored in the presence both of a flat bottom and of a rough bottom with several seamounts, to yield a total of five different channel models. Our results show that channel variations induced by environmental changes over time have an impact on routing performance metrics in connected topologies. A sea bottom with a rough shape adds a further impact to the routing performance, which is shown to be larger for source routing. We conclude that while empirical channel models yield a good first-order approximation, the time-variability of the channel and the shape of the network area boundaries are to be taken into account in order to achieve more realistic network performance estimates.