It has long been recognized that the simple ray theory provides only a very incomplete picture of the disturbance at a point due to a sudden localized movement in an elastic medium. In this paper an investigation is made of the disturbance created by a cylindrical pulse (of P- and S-type) emitted from a line source in a surface layer of elastic material overlying a semi-infinite medium of different elastic constants and density. An exact formal description of the motion is obtained in terms of a succession of pulses; the double integrals corresponding to each are evaluated by approximate methods. It is found that at a remote point (at or near the surface) there should be felt pulses corresponding to travel by each one of the minimum-time-paths predicted by the ray theory, and, in addition, a whole series of diffraction effects. Ray-path pulses are of the same type as the initial pulse, showing the same 'jerk' in the displacements (or in the rate-of-change of these); diffraction pulses are in general 'blunt', but certain of them become sharper as the surface is approached until, at the surface, they become part of a minimum-time-path disturbance. The apparent S- and Sg-anomalies are considered in the light of these results. At a certain range interference between pulses becomes important, and at very great range the dispersive Rayleigh wave-train becomes the dominant feature. A further study of the propagation of free Rayleigh waves shows that an infinite number of modes of vibration are possible. The degree to which each is excited and the resultant motion is determined in part II; the importance of the Airy phases is demonstrated. The pulse representation has a natural extension to systems of any number of layers; before the corresponding interference pattern at great range can be determined it will be essential to extend our knowledge of the dispersion of free surface waves to such multilayered systems.