Modified modal theory of transient response in layered media

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Abstract
A modified modal theory is developed for the transient response due to a line source placed in a two‐layer slab medium. The double integral is evaluated by deformation on the complex wave number plane followed by a second deformation from the real frequency axis to the steepest descent path on the complex frequency plane. We first show that causality is preserved such that before the arrival time of the direct wave or the head wave, whichever takes the least time, the transient response vanishes. With the preservation of causality, we proceed to find the complete transient response at all times. The results obtained from the modal theory are then checked with those generated from direct numerical calculations with computers. Comparisons are also made with the explicit inversion technique which is applicable to the present case but appears to have severe restrictions that the modal theory does not have in generalizing to the solution of other transient problems.