Consider the three-dimensional scattering of a sound pulse generated by an impulsive point source and incident upon a penetrable wedge, identified by a density contrast. The wave speed is common to both regions and the radiation condition of only outgoing waves at infinity is applied in all directions. At the boundary of the wedge there is a pair of transmission conditions which ensure continuity of the acoustic pressure and normal velocity. By using Fourier transforms in time and parallel to the wedge generators and a Kontorovich--Lebedev transform in the radial direction, as described by Jones [Acoustic and Electromagnetic Waves, Oxford (1986)], both the exterior and interior fields can be obtained as a sum of impulsive terms, some of which are due to edge diffraction. If the wedge angle is a rational fraction of (pi), then the residue series can be summed and, after careful consideration of when and where impulsive disturbances can occur, the fields can be written in remarkably simple closed forms. This solution provides the zero-order field for a relatively small difference in wave speeds.