# An explanation of 2delta waves on non-staggered grids

While looking into the difference between staggered and collocated grids, I came across an effect called $$2\Delta x$$-oscillations, which happen on non-staggered grids, but not on staggered grids. This phenomenon is explained in an article by Stelling (1984) on pages 101-103, but I don't exactly follow the reasoning. These spurious oscillations happen at water-height, are of order $$\Delta x$$, and seem to have something to do with boundary conditions. Could someone who knows about this topic clarify a little bit? For example, what are the "normal modes" they are looking for in this article?

EDIT: I'm particularly interested in knowing if these $$2\Delta x$$-oscillations are the wiggles behind a wave solution of the shallow water equations, because you can see that they're worse for the non-staggered grid.