### Abstract:

A two-layer model is used to study the properties of free coastal trapped waves which propagate over an idealized continental shelf and continental slope bottom topography. With both stratification and depth variations that are typical of continental shelf and slope regions, barotropic shelf waves and baroclinic internal Kelvin waves axe coupled. The internal Kelvin waves have an offshore scale given by the internal “Rossby radius of deformation” δ′[subscript]R which is typically small (~15 km) compared with the width L of the shelf and slope region (~100 km), while the shelf waves have an offshore scale which, for the lower modes, is O(L). The strength of the coupling is represented by a parameter λ = δ′[subscript]R/δ′[subscript]B, where δ′[subscript]B = H′/H′[subscript]x′ is a characteristic length scale of the bottom topography and is essentially the distance over which the change in water depth H′, in the offshore direction x′, is the same magnitude as the depth itself. The nature of the interaction and modification of internal Kelvin waves and barotropic shelf waves is studied by a perturbation procedure for λ<<1. It is found that, for alongshore scales δ′y greater than L, O(1) internal Kelvin waves are accompanied by an O(λ) barotropic motion, which extends across the shelf and slope, and that O(1) shelf waves are accompanied by a weaker O(λ2) baroclinic motion. For short alongshore and onshore-offshore scales, i.e., for δ′[subscript]y<O(L½δ′[subscript]R½) and δ′[subscript]x<O(L½δ′[subscript]R½), the shelf wave solutions are coupled at the lowest order with baroclinic effects which alter the modal structure. For very short scales, δ′[subscript]y, δ′[subscript]x<δ′[subscript]R, the shelf wave motion is primarily confined to the bottom layer and the waves are “bottom trapped.”