The characteristics of quasi-periodic (QP) VLF emission observed near the equatorial plane by the GEOS satellites are first compared with simultaneous ground data obtained at L≂6. The frequency structure of QP elements is shown to be the same in the magnetosphere and at ground, whereas their amplitude measurement gives evidence of a ≂20-dB attenuation between GEOS and the ground. Different events are described as a function of their more or less complex relationships with concomitant Pc 3--4 magnetic pulsations. The spectral analysis of the simultaneous QP envelopes and ULF waves permits us to deduce that QP emissions are due to the modulation of a VLF hiss by a compressional component of ULF waves. This component is a harmonic of an ULF standing wave, showing that the classification between type I and type II is not so obvious in space. When QP envelope and ULF periodicities are different, they are shown to correspond to different harmonics of a same fundamental oscillation of the field line. This set of new results is discussed in the light of the quasi-linear theory of wave-particle interactions. The modulation rate of nondispersive QP type can be roughly well explained, and some ideas are given about a relaxation mechanism which is able to interpret the frequency modulation of the QP type rising tone.