  Locating where transient signals travel between a source and receiver
requires a final step that is needed after using a theory of diffraction such
as the integral theorem of Helmholtz and Kirchhoff. Introduced here, the final
step accounts for interference between adjacent apertures on a phase screen by
adaptively adjusting their phase and amplitude, yielding a hierarchy of energy
contributions to any desired window of signal travel time at the receiver. The
method allows one to check errors in ray theory at finite wavelengths. Acoustic
propagation at long distance in the oceanic waveguide (50-100 Hz, 0.05 s
resolution) has significant deviations from ray theory. The boundary condition
of zero pressure at the surface of the ocean appears to cause sound to travel
in a nearly horizontal trajectory for a much greater distance near the surface
than predicted by rays. The first Fresnel zone is an inappropriate scale to
characterize where transient sounds travel near a ray path as assumed by a
standard scattering theory. Instead, the Fresnel zone is too large by an order
of magnitude for cases investigated here. Regions where sounds travel can have
complicated structures defying a simple length scale. These results are
applicable to the physics of underwater sound, optics, radio communication,
radar, geophysics, and theories of wave-scattering.
