Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X



Transient solutions for a class of diffraction problems

Author: L. B. Felsen
Journal: Quart. Appl. Math. 23 (1965), 151-169
MSC: Primary 78.35
DOI: https://doi.org/10.1090/qam/184554
MathSciNet review: 184554
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: 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 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.

References [Enhancements On Off] (What's this?)

  • [1] Joseph B. Keller and Albert Blank, Diffraction and reflection of pulses by wedges and corners, Comm. Pure Appl. Math. 4 (1951), 75–94. MR 0043714, https://doi.org/10.1002/cpa.3160040109
  • [2] Hendricus Bremmer, The pulse solution connected with the Sommerfeld problem for a dipole in the interface between two dielectrics, Electromagnetic waves, Univ. of Wisconsin Press, Madison, Wis., 1962, pp. 39–64. MR 0129751
  • [3] M. Papadopoulos, The refraction of a spherical pulse at a plane interface, Report No. 279, Dec. 1961; Diffraction of pulses by a half plane. I, Report No. 293, Feb. 1962. U. S. Army Mathematics Research Center, University of Wisconsin, Madison, Wis.
  • [4] F. G. Friedlander, Sound pulses, Cambridge University Press, New York, 1958. MR 0097233
  • [5] L. Cagniard, Reflection and refraction of progressive seismic waves, translated by E. Flinn and C. H. Dix, McGraw-Hill, New York, 1962
  • [6a] C. L. Pekeris, Solution of an integral equation occurring in impulsive wave propagation problems, Proc. Nat. Acad. Sci. U. S. A. 42 (1956), 439–443. MR 0078575
  • 1. Chaim L. Pekeris and Hanna Lifson, Motion of the surface of a uniform elastic half-space produced by a buried pulse, J. Acoust. Soc. Amer. 29 (1957), 1233–1238. MR 0090262, https://doi.org/10.1121/1.1908753
  • 2. c. C. L. Pekeris and Z. Alterman, Radiation resulting from an impulsive current in a vertical antenna placed on a dielectric ground, J. App. Physics 28 (1957) 1317
  • [7a] A. T. de Hoop, A modification of Cagniard's method for solving seismic pulse problems, App. Sci. Res., Sec. B, 8 (1960) 349
  • 3. b. A. T. de Hoop and H. J. Frankena, Radiation of pulses generated by a vertical electric dipole above a plane, non-conducting earth, App. Sci. Res., Sec. B, 8 (1960) 369
  • 4. c. A. T. de Hoop, Representation theorems for the displacement in an elastic solid and their application to elastodynamic diffraction theory, Dissertation, University of Delft, Netherlands, 1958
  • [8] Balth. van der Pol and A. H. M. Levelt, On the propagation of a discontinuous electromagnetic wave, Nederl. Akad. Wetensch. Proc. Ser. A 63 = Indag. Math. 22 (1960), 254–265. MR 0122335
  • [9] Murray F. Gardner and John L. Barnes, Transients in Linear Systems, John Wiley and Sons, Inc., New York, 1942. MR 0007508
  • [10] cf. L. B. Felsen and N. Marcuvitz, Modal analysis and synthesis of electromagnetic fields, Microwave Research Institute, Polytechnic Institute of Brooklyn, Report PIBMRI-841-60, Ch. 5, Sec. C3
  • [11] A. Sommerfeld, Mathematische Theorie der Diffraction, Math. Ann. 47 (1896), no. 2-3, 317–374 (German). MR 1510907, https://doi.org/10.1007/BF01447273
  • [12] reference 10, Chapter 6, Sec. D
  • [13] F. Oberhettinger, On the diffraction and reflection of waves and pulses by wedges and corners, J. Res. Nat. Bur. Standards 61 (1958), 343–365. MR 0098579, https://doi.org/10.6028/jres.061.030
  • [14] S. R. Seshadri, Excitation of surface waves on a unidirectionally conducting screen, IRE Transactions Microwave Theory and Techniques, MTT-10 (1962 ) 279
  • [15] Philip M. Morse and Herman Feshbach, Methods of theoretical physics. 2 volumes, McGraw-Hill Book Co., Inc., New York-Toronto-London, 1953. MR 0059774

Similar Articles

Retrieve articles in Quarterly of Applied Mathematics with MSC: 78.35

Retrieve articles in all journals with MSC: 78.35

Additional Information

DOI: https://doi.org/10.1090/qam/184554
Article copyright: © Copyright 1965 American Mathematical Society

American Mathematical Society