Receivers were flown in rockets from Eglin Air Force Base, Florida, Natal, Brazil, and Fort Churchill, Canada, to study the fields of very low frequency (VLF) waves propagating in the earth-ionosphere waveguide, and leaking through the ionosphere to altitudes of 500-600 km. Amplitude-height, group delay, and polarization profiles are given for signals from a number of ground based transmitters. A coherent propagation model is proposed and partially tested by calculating Doppler shifts. The model, extended with a full-wave computer program, allows wave-field profiles to be approximated theoretically. The extent to which some sample calculations agree with the observations is discussed. (Author).