The dispersion equation for waves on an infinite uniform jet column of elliptic cross-section is derived, and approximated for large eccentricity with the aid of new asymptotics for the modified Mathieu functions. It is shown that the effect of appreciable eccentricity on lateral disturbances is greatly to reduce their growth rates below those for a circular jet, regardless of whether the disturbance grows spatially or temporally. For ‘vertical’ disturbances it is shown that the behaviour of waves of general length is qualitatively similar to that of long waves on a two-dimensional jet. Thus the mode symmetric about the major axis has small growth rate whether the mode grows temporally or spatially, while the mode antisymmetric about the major axis has small growth rate if temporally growing, but large growth rate if spatially growing. Comments are made as to the relevance of these results to the mode of action of jet silencers which squash a round jet into a flat ‘fish-tail’ shape.