Self-consistent electrodynamic coupling of the ionosphere and magnetosphere produces waves with clearly defined properties, described here for the first time. Large scale (ideal) disturbances to the equilibrium, for which electron inertia is unimportant, move in the direction of the electric field at a characteristic speed. This may be as fast as several hundred meters per second or approximately half the E x B drift speed. In contrast, narrow scale (strongly inertial) waves are nearly stationary and oscillate at a specific frequency. Estimates of this frequency suggest periods from several tenths of a second to several minutes may be typical. Both the advection speed and frequency of oscillation are derived for a simple model and depend on a combination of ionospheric and magnetospheric parameters. Advection of large scale waves is nonlinear: troughs in E-region number density move faster than crests and this causes waves to break on their trailing edge. Wavebreaking is a very efficient mechanism for producing narrow (inertial) scale waves in the coupled system, readily accessing scales of a few hundred meters in just a few minutes. All magnetosphere-ionosphere waves are damped by recombination in the E-region, suggesting that they are to be best observed at night and in regions of low ionospheric plasma density. Links with observations, previous numerical studies and ionospheric feedback instability are discussed, and we propose key features of experiments that would test the new theory.