The problem considered is that in which vertically polarized waves are propagated across a single straight-line discontinuity in the electrical properties of a smooth flat earth, the discontinuity being such as might occur at a (rather idealized) coastline. It is further specialized by the assumption that one of the media (e.g. sea) has infinite conductivity, and this is replaced by an infinitely thin, perfectly conducting half-plane lying in the interface of the air and the land medium, the latter now being taken to fill completely the region below the interface.With this model a method of solution is proposed which is a union of ray theory and rigorous diffraction theory. The presence of the land is accounted for by the introduction of a suitable image field and the problem thereby reduced to one in which two fields are incident upon a perfectly conducting sheet situated in free space. The technique is illustrated by application to the case of a three-dimensional plane wave, and this solution forms the basis for the later work.When the transmitter is at a finite distance from the coastline the solution can be obtained by expressing the incident field as an angular spectrum of plane waves the propagation of which has already been studied. This procedure is first applied to the two-dimensional problem of a line source and the results are shown to be in close agreement with previous work on the subject. On examination, however, the solution is found to be made up of the field of the actual transmitter diffracted at the coastline, together with the diffracted field arising from an image transmitter of strength equal to the Fresnel reflection coefficient appropriate to a particular angle of incidence, and this suggests an even more elementary way of tackling the problem. For any type of transmitter all that is now required is the solution of the corresponding Sommerfeld diffraction problem, and with this new approach the case of a point source is treated. The result is almost identical with that found by resolution into plane waves. Moreover, a comparison with the solution for a line source shows that the factor by which the free-space field must be multiplied to give the total field is of the same form for both transmitters, the factor for a point source being obtainable from that for a line source by replacing each two-dimensional distance by the corresponding three-dimensional one. In particular this justifies the application to each radial separately of the formulae for normal incidence on the coastline and enables the analytical difficulties associated with three-dimensional propagation to be avoided.The nature of the field is then studied with particular reference to the rapid changes of intensity and phase just beyond the coastline; the former is generally termed the ‘recovery effect’, whilst the latter gives rise to the phenomenon of coastal refraction. The use of height-gain factors is considered and it is shown that, for sufficiently small elevations of transmitter and receiver, the factors appropriate to homogeneous earths may be applied. Several numerical examples are given.