A new way of visualizing the Sommerfeld theory of propagation over a flat, imperfectly reflecting earth is presented. The Sommerfeld theory arises because the ray theory of propagation from a source in the presence of a flat, imperfectly reflecting earth is only an approximation. The ray theory involves the assumption that the Fresnel reflection coefficient of the earth does not vary rapidly with angle of incidence, and this assumption is not satisfied for glancing incidence of vertically polarized waves on the earth's surface at broadcasting wavelengths. The main object of the new presentation is to facilitate the solution of problems involving propagation near the surface of the earth partly over land and partly over sea, but these applications are not included in the paper.It is convenient to think of a two-dimensional problem in which the transmitter is a line source parallel to the earth's surface, having a vertical polar diagram of circular shape. Such a source may be Fourier analysed into plane waves whose directions are distributed in a vertical plane of propagation perpendicular to the line source; the amplitudes of all the plane waves are the same and they are in the same phase at the source. When these waves are reflected from the earth they produce an angular spectrum of reflected waves, the amplitudes and phases of which are determined by the Fresnel reflection coefficient. This angular spectrum could be thought of as arising, in the absence of the earth, from an aperture distribution on the vertical plane through the primary line source. The aperture distribution that produces the angular spectrum of reflected waves in this way is the exact image of the primary source in the imperfectly reflecting earth, and is given by the Fourier transform of the Fresnel reflection coefficient. For a perfectly conducting earth this aperture distribution reduces to a line source identical with the primary source and located at the optical image line. The correction required to this when the earth is not perfectly conducting is mainly the following. An aperture distribution extending indefinitely downwards from the image line must be introduced, and this consists essentially of the aperture distribution produced by diffraction of the Zenneck wave under a screen extending from the image line upwards. The field produced by the primary source in the presence of the imperfectly reflecting earth is thus the field that would be produced with an almost perfectly conducting earth, together with the field arising from diffraction of the Zenneck wave under the image line.If diffraction of the Zenneck wave under the image line is calculated by the edge-wave approximation we merely arrive at the ray theory of reflection from the earth of radiation from the primary source: the edge wave from the image line, together with the wave from an image in an almost perfectly conducting earth, makes up the wave from the Fresnel image for the imperfectly reflecting earth. But, at broadcasting wavelengths, points close to the earth are often too close to the shadow edge, formed by diffraction of the Zenneck wave under the image line, for application of the edge-wave approximation. We then have to apply the full theory of edge-diffraction based on the Cornu spiral, and this gives the Sommerfeld theory.