Inertia-gravity mode and Rossby mode dispersion properties are examined for
discretisations of the linearized rotating shallow-water equations using the
$P1_{DG}$-$P2$ finite element pair on arbitrary triangulations in planar
geometry. A discrete Helmholtz composition of the functions in the velocity
space based on potentials taken from the pressure space is used to provide a
complete description of the numerical wave propagation for the discretised
equations.