The behavior of an internal wave in a continuously stratified fluid over a sloping bottom is examined by finding approximate analytic solutions for the amplitude of waves in a coastal ocean with constant bottom slope, linear bottom friction, and barotropic mean flows. These solutions are valid for frequencies higher than the frequency of critical reflection from the sloping bottom. The solutions show that internal waves propagating toward the shore are refracted, so that their crests become parallel to shore as they approach the coast, and outward propagating waves are reflected back toward the coast from a caustic. Inviscid solutions predict that the amplitude of a wave goes to infinity at the coast, but these infinite amplitudes are removed by even infinitesimal bottom friction. These solutions for individual rays are then integrated for an ensemble of internal wave rays of random orientation that originate at the shelf break and propagate across the shelf. It is found that for much of the shelf the shape of the current ellipse caused by these waves is nearly independent of the waves' frequency. The orientation of the current ellipse relative to isobaths is controlled by the redness of the internal wave spectrum at the shelf break and the strength of mean currents. Friction is more important on broader shelves, and consequently, on broad shelves the internal wave climate is likely to be dominated by any internal waves generated on the shelf, not waves propagating in from the deep ocean.
