A poleward shift of the mid-latitude jet-streams has been observed in both hemispheres during recent decades, and global circulation models indicate that this trend will continue throughout the 21st Century. This shift has already had significant climate impacts, and is the principal dynamical feature projected to affect the mid-latitudes under global warming. There is no proven theory for the mechanism of the shift. Here we show that the fundamental reason for the shift may be the accompanying transfer in the wave energy spectrum to waves with larger spatial extent. Longer waves are shown to propagate preferentially poleward, and are hence able to induce a corresponding shift in the mean flow that they maintain. This is verified using a simplified atmospheric general circulation model. The model is also used to address the cause of the shift to longer waves. Consistent with linear theory, an increase in the static stability, Equator-Pole temperature gradient, or tropopause height are all shown to induce longer wavelength eddies and a poleward shift of the eddy-driven jet stream. This insight serves to unify previous studies that have attributed the cause of the jet shift to different atmospheric processes. The extent to which such changes are consistent with increasing greenhouse gas concentrations is discussed. Other likely implications of an increase in the eddy wavelength include an increase in the stratospheric overturning, and intensity of individual storms, both of which are robust predictions of climate change associated with global warming.