In a recent paper (Jansons & Johnson 1988) the authors discuss topographic Rossby waves over a random array of seamounts. It is noted that resonance is possible between a hill and an equal and opposite dale but such resonances are mentioned only briefly due to the small likelihood of correctly matched topography in the ocean. The present paper considers the resonances in detail showing how the normal modes formed by frequency splitting at resonance can be combined to give modes that slowly transfer energy from one region supporting topographic waves, across a region where such weaves are evanescent, to another region supporting waves. In addition to the simplest case of a hill—dale pair for which an exact energy-transferring mode is obtained, transferring modes are given for a three-hill system, for two hills near a coastal boundary, and for two-basin lakes. The analysis is simplified and the results generalized by extensive use of the invariance of the governing equation under conformal mappings. A transferring mode is given for a dale in a random array of seamounts showing energy leaking slowly from the dale to large distances and returning, with the rate of leakage depending on the area fraction of seamounts. It is concluded that although resonances and transferring modes are not likely to be important in random arrays on infinite planes, they are relevant to numerical models, which are necessarily restricted to finite domains, to coastal seamount chains, and to multi-basin lakes.