Regions where transient signals are influenced between a source and receiver

At infinite frequency, the only place that a medium can influence the waveform of a received signal that is emitted from a source is along one or more infinitesimally thin ray paths. For any transient signal 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. This window is sometimes chosen to surround a peak. Results at finite frequencies differ from those at infinite frequency because of diffraction. The method has its foundation in the integral theorem of Helmholtz and Kirchhoff. Part of the method involves a filter that yields an imperfect but apparently useful picture of influential regions in the presence of interfering waves. The method is useful for quantifying differences between the region of influence and a ray, and for identifying regions in which medium fluctuations significantly influence signal aberrations at a receiver. Four principal results are found at low frequencies. They are: 1) For propagation in homogeneous media, a significant portion of the received signal is influenced by waves that traverse paths that are approximately integer and half-integer numbers of cycles greater than the straight path between source and receiver. Such paths are called 'constructive and destructive paths of influence', respectively. They correspond to edge-diffracted rays for the geometrical theory of diffraction. 2) For reflection from a flat interface in an otherwise homogeneous medium, the received signal is significantly influenced by constructive and destructive paths of influence whose angles of incidence and reflection differ (non-specular reflection). 3) For acoustic propagation centered at 100 Hz in an oceanic acoustic waveguide, the region of influence markedly departs from a ray path, particularly near the reflective ocean surface. 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, as assumed by at least one scattering theory. Modification of that theory may yield a better fit with data.