Weakly dispersive spectral theory of transients, part II: Evaluation of the spectral integral

In the spectral theory of transients formulated in Part I of this paper, the transient response for weakly dispersive wave processes has been expressed in terms of canonical integrals in the complex spatial wavenumber domain. The real and complex singularities in the integrands, which dominate the behavior of the spectral integrals, have been categorized and associated with generic physical wave processes. The integrals are now evaluated by contour deformation around the singularities. This yields general expressions for the transient Green's function that are applicable to a broad class of propagation and diffraction problems. The generic results, which can be grouped into contributions from real or complex singularities, express the transient field in terms of compact (and therefore physically incisive) wave spectra, in contrast to alternative procedures that always constrain the spectra to be real. These aspects, together with simplifying explicit wavefront approximations, are explored in the present paper, with the application to specific problems relegated to Part III.