The scattering of shelf waves at simultaneous changes in depth, direction and width is considered. In the low-frequency limit the scattering is shown to be determined by the connection of f/h contours. The description “simple” is introduced for regions in which no incident f/h contour terminates and then restarts. An explicit connection formula is derived for simple regions. It is shown that energy is transported without loss across a scattering region it no incident f/h contours terminate there. This subclass of simple regions is described as conservative. Particular examples are given for exponential shelves joined by both simple and nonsimple, conservative and nonconservative scattering regions, and for both incident shelf waves and irrotational flows driven across the region. In the latter case, energy is scattered out of the flow into a transmitted wave field. Finally it is noted that if the irrotational flow determined by a particular shelf geometry is geostrophic then even at arbitary frequencies no scattering of energy occurs from the flow or among shelf waves.