At infinite frequency, a medium only influences the waveform of a received signal emitted from a source along one or more infinitesimally thin rays. For transient signals at finite frequencies, an exact method is developed to compute the regions in a medium that significantly influence the received signal for any specified window of signal travel time. Results are (1) in homogeneous media, a significant portion of the signal originates from waves that traverse paths that are approximately integer and half-integer numbers of cycles greater than the straight path between source and receiver. Called ‘‘constructive and destructive paths of influence,’’ they do not obey Fermat’s principle of path stationarity. (2) For reflection from a flat interface in a homogeneous medium, the received signal is significantly influenced by constructive and destructive paths of influence whose angles of incidence and reflection differ. (3) Near 100 Hz in an oceanic waveguide, the region of influence markedly departs from a ray. Near upper turning points, the influential region is flat for O(10) km instead of O(1) km for a ray. (4) The first Fresnel zone is an inappropriate scale to characterize the region of influence for transient signals near a steep ray in inhomogeneous media.