With simple generalizations of the method due to Rosenbaum (1961) and Phinney (1961), single integral expressions may be written down for the long range pole contributions to the transient signal in a plane seismic waveguide. This method yields expressions for the leaking, or imperfectly trapped waves, and suffers from no restriction on the number of layers or the existence of coupling to one or two half-spaces. When it is applied to the simple interface wave problem of two halfspaces in contact, closed form expressions are obtained describing the propagation of pulses along the interface due to lower sheet poles. The theory is applied to the Lamb problem, the liquid/solid interface, and the solid/solid interface problems. The leaking wave generalizations of the Rayleigh and Stoneley waves are found and a new wave, coupled to the P-wave, is demonstrated. The physical importance of leaking interface pulses is shown to be in their coupling to the normal or leaking oscillations of layered structures.