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
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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.
- Joseph B. Keller and Albert Blank, Diffraction and reflection of pulses by wedges and corners, Comm. Pure Appl. Math. 4 (1951), 75–94. MR 43714, DOI https://doi.org/10.1002/cpa.3160040109
- 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
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.
- F. G. Friedlander, Sound pulses, Cambridge University Press, New York, 1958. MR 0097233
L. Cagniard, Reflection and refraction of progressive seismic waves, translated by E. Flinn and C. H. Dix, McGraw-Hill, New York, 1962
- 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 78575, DOI https://doi.org/10.1073/pnas.42.7.439
- 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 90262, DOI https://doi.org/10.1121/1.1908753
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
A. T. de Hoop, A modification of Cagniard’s method for solving seismic pulse problems, App. Sci. Res., Sec. B, 8 (1960) 349
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
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
- 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
- Murray F. Gardner and John L. Barnes, Transients in Linear Systems, John Wiley and Sons, Inc., New York, 1942. MR 0007508
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
- A. Sommerfeld, Mathematische Theorie der Diffraction, Math. Ann. 47 (1896), no. 2-3, 317–374 (German). MR 1510907, DOI https://doi.org/10.1007/BF01447273
reference 10, Chapter 6, Sec. D
- 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, DOI https://doi.org/10.6028/jres.061.030
S. R. Seshadri, Excitation of surface waves on a unidirectionally conducting screen, IRE Transactions Microwave Theory and Techniques, MTT-10 (1962 ) 279
- Philip M. Morse and Herman Feshbach, Methods of theoretical physics. 2 volumes, McGraw-Hill Book Co., Inc., New York-Toronto-London, 1953. MR 0059774
J. B. Keller and A. A. Blank, Diffraction and reflection of pulses by wedges and corners, Comm. Pure and App. Math. 4 (1951) 75
H. Bremmer, The pulse solution connected with the Sommerfeld problem for a dipole in the interface between two dielectrics, Proceedings of a Symposium on Electromagnetic Waves, University of Wisconsin Press, Madison, 1962, p. 39
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.
F. G. Friedlander, Sound pulses, Cambridge University Press, 1958
L. Cagniard, Reflection and refraction of progressive seismic waves, translated by E. Flinn and C. H. Dix, McGraw-Hill, New York, 1962
C. L. Pekeris, Solution of an integral equation occurring in impulsive wave propagation problems, Proc. Nat. Acad. Sci. 42 (1956), 439 (references to earlier articles by the same author are also given here)
b. C. L. Pekeris and H. Lifson, Motion of the surface of a uniform elastic half space produced by a buried pulse, J. Acoust. Soc. Amer. 29 (1957) 1233
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
A. T. de Hoop, A modification of Cagniard’s method for solving seismic pulse problems, App. Sci. Res., Sec. B, 8 (1960) 349
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
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
B. van der Pol and A. H. M. Levelt, On the propagation of a discontinuous electromagnetic wave, Koninkl. Nederl. Akad. v. Wetenschappen, Amsterdam, 63 (1960) 254
cf. M. Gardner and J. Barnes, Transients in linear systems, John Wiley and Sons, 1942. Vol. I, Ch. 5
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
A. Sommerfeld, Mathematische Theorie der Diffraktion, Math. Ann. 47 (1896) 317
reference 10, Chapter 6, Sec. D
F. Oberhettinger, On the diffraction and reflection of waves and pulses by wedges and corners, J. Research (NBS) 61 (1958) 343 (an extensive list of references is contained in this paper)
S. R. Seshadri, Excitation of surface waves on a unidirectionally conducting screen, IRE Transactions Microwave Theory and Techniques, MTT-10 (1962 ) 279
cf. P. M. Morse and H. Feshbach, Methods of theoretical physics, McGraw-Hill Book Co., New York, 1953, p. 813–814
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© Copyright 1965
American Mathematical Society
