When electron waves converging on a specimen are coherent for a sufficiently large range of angles, interference effects are visible in shadow images of high magnification and in convergent beam electron diffraction patterns. A simple waveoptics theory provides explanations for the observed effects and shows the relationship to the transfer function of the probe-forming lens. For nonperiodic objects the theory reproduces the characteristic rings seen in underfocussed images which converge to circular patches as the axial in-focus, infinite magnification setting is approached. Periodic objects show patterns of fringes with well defined patches of nearly uniform intensity where the fringe spacing becomes large, corresponding to stable phase relationships between diffracted beams. These fringe patterns reflect the relative phases of the diffracted beams and so are very sensitive to the diffraction conditions in the specimen, to the instrumental parameters such as defocus and lens aberrations and to the position of the incident beam relative to the lattice planes of crystals. Calculated fringe patterns are compared with patterns observed using a scanning transmission electron microscope with a field emission gun.