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Transient solutions for a class of diffraction problems

L. B. Felsen

1965
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Quarterly of Applied Mathematics
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The study of the fields excited by impulsive sources in layered media has been facilitated by a technique employed originally by Cagniard and Pekeris, and simplified subsequently by de Hoop. The procedure involves a reformulation of the time-harmonic solution so as to permit the explicit recovery of the transient result by inspection. In the present paper, it is shown that this method may be applied conveniently to the inversion of a certain Sommerfeld-type integral which occurs frequently in
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... urs frequently in diffraction theory, thereby unifying the analysis of a class of pulse diffraction problems. Illustrative examples include the transient response to a line source in the presence of a dielectric half space, a perfectly absorbing and perfectly reflecting wedge, and a unidirectionally conducting infinite and semi-infinite screen. The latter applications illuminate the role of surface waves in the impulsive solution. It is found, in contrast to the time-harmonic case, that a different behavior characterizes the surface waves excited on a unidirectionally conducting half plane by the incident field and by the edge discontinuity, respectively. 1. Introduction. A standard procedure in the determination of the response to non-harmonic excitation is to apply the Fourier or Laplace inversion to the time-harmonic solution.! In diffraction problems involving unbounded regions, the steady-state response is generally given in the form of a single or double integral to which another integration is added for the recovery of the transient result. To facilitate the evaluation of the disturbance due to an impulsive source in the presence of an elastic half space, Cagniard5 proposed a method wherein the time-harmonic solution is transformed into a Laplace integral which may be inverted by inspection. The rather involved original treatment of Cagniard (see also Pekeris6 for a similar analysis) was rephrased and simplified by de Hoop7 who applied the procedure to certain elastodynamic diffraction problems and also to the determination of the fields radiated by an impulsive line or dipole source in the presence of a (non-dispersive) dielectric half-space, de Hoop's investigation, like the preceding ones, proceeds in the complex wavenumber plane (see also van der Pol and Levelt8), and the transformation of his time-harmonic solutions into the desired form necessitates the deformation of the original integration path away from the real axis into a hyperbolic contour in the complex plane. It is one of the purposes of the present paper to point out that the analysis is simplified further when the complex angle, rather than the complex wavenumber, plane is chosen for the representation of the steady-state results. Thus, the original integration contour is the well-known Sommerfeld path which arises in a variety of scattering problems, and the transformed contour is found to be a *

doi:10.1090/qam/184554
fatcat:jyhfxwqiircp3kdxzth2blekbm