A type of coastally trapped wave motion where the velocity normal to the coast vanishes everywhere. The wave is nondispersive and propagates parallel to the shore with the speed of shallow water gravity waves, i.e. sqrt (gH). The profile perpendicular to shore either decays or grows exponentially seaward depending on whether the wave propagates with the coast to its right or left (in the northern hemisphere). For vanishing rotation, the decay or growth scale becomes infinite and the Kelvin wave reduces to an ordinary gravity wave propagating parallel to the coast. The dynamics of a Kelvin wave are such that it is exactly a linearized shallow water gravity wave in the longshore direction and exactly geostrophic in the cross-shore direction.