Abstract

Starting with the most elementary multiple scattering problem, a symmetrically excited pair of monopoles, we use both “successive-scattering” and “self-consistent” procedures to obtain the closed-form multiply scattered amplitude in terms of the singly scattered value. Then we represent the field for an arbitrary configuration of arbitrary scatterers in terms of the individual, multiply scattered amplitudes, and specify these in terms of singly scattered amplitudes by a system of self-consistent integral equations. Applications are made to fixed configurations (two scatterers, and the diffraction grating), and to random distributions (models for rough surfaces, and for random media). Simple explicit results in terms of isolated scatterer functions are derived, and some of their essential features are given in graphs: the coupling effects for two dielectric rods (intensity and phase) are graphed versus spacing for both polarizations; for the diffraction grating, when a surface wave is “near grazing,” the spectral orders show essential features of the anomalies measured by Wood, Strong, and Palmer; for a lossless rough surface, the graphs for both polarizations for the coherently reflected intensity and phase are analogous to those obtained for reflection from a lossy interface (the corresponding results for the incoherent scattering accounting for the loss from the “main beam”); and for a slab region containing a random distribution of nearly transparent large scatterers, we graph the macroscopic parameters, and the coherent and incoherent, forward-scattered intensities versus the fractional volume.

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  1. Additional introductory material on scattering problems and reviews of recent developments will be found in the writer's "Electromagnetic waves," Physics Today 13, (7) 30 (1960), and "On multiple scattering of waves," J. Research Natl. Bureau Standards 64D, 715 (1960).
  2. All numerical computations (three significant figures, rounded off) for the graphs of this paper were done by S. E. Bergstrom with a desk calculator.
  3. V. Twersky, J. Appl. Phys. 23, 407 (1952).
  4. Such series are discussed in some detail by the writer in J. Acoust. Soc. Am. 24, 42 (1952).
  5. F. Zaviska, Ann. Physik 40, 1023 (1913).
  6. K. A. Breuckner, Phys. Rev. 89, 834 (1953); 97, 1353 (1955).
  7. K. M. Watson, Phys. Rev. 87, 575 (1953).
  8. R. F. Millar, Can. J. Phys. 38, 272 (1960).
  9. See literature survey by V. Twersky, in J. Research Natl. Bur. Standards 64D, 715 (1960); part of the Report of the United States of America National Committee to the XIIIth General Assembly of the International Scientific Radio Union, London, England.
  10. The two-dimensional case is treated by V. Twersky, in Report EDL-E60, Sylvania Electronic Defense Laboratories, 1961; to be published in "Proceedings of Symposium on lectromagnetic Waves," University of Wisconsin (April, 1961). The three-dimensional case is treated by V. Twersky, in Report EDL-E61, Sylvania Electronic Electronic Defense Laboratories, 1961. [J. Math. Phys. (to be published)].
  11. S. N. Karp and N. Zitron, Research Report EM-126, Institute of Mathematical Sciences, New York University (1959).
  12. S. N. Karp, Proceedings of tle Symposium on Microwave Optics, 1953, p. 198 EElectronic Research Directorate, AFCRC (1959)].
  13. V. Twersky, IRE Trans. AP-4, 330 (1956), Proceedings of the URSI-Michigan International Symposium on Electromagnetic Theory; Report EDL-M105, Sylvania Electronic Defense Laboratories (1957); Report EDL-E28, Sylvania Electronic Defense Laboratories (1958).
  14. Lord Rayleigh, Proc. Roy. Soc. London A79, 399 (1907); see also Rayleigh's earlier paper in Phil. Mag. 14, 60 (1907).
  15. W. v. Ignatowsky, Ann. Physik 44, 369 (1914).
  16. F. M. Schwerd, Die Beugungsercheinungen aus den Fundamentalgesetzen der Undulations Theorie Analytisch Entwickelt (Schwan und Goetz'schen Hosbuchhandlung, Mannheim, Germany, 1835).
  17. R. W. Wood, Phys. Rev. 48, 928 (1935); Phil. Mag. 4, 396 (1902); ibid. 23, 310 (1912).
  18. L. R. Ingersoll, J. Astrophys. 51, 129 (1920); Phys. Rev. 17, 493 (1921).
  19. J. Strong, Phys. Rev. 49, 291 (1936).
  20. H. C. Palmer, Jr., J. Opt. Soc. Am. 42, 269 (1952); ibid. 46, 50 (1956).
  21. U. Fano, Ann. Physik 32, 393 (1938).
  22. K. Artmann, Z. Physik 119, 529 (1942).
  23. V. Twersky, J. Appl. Phys. 25, 859 (1954).
  24. J. E. Burke and V. Twersky, Report EDL-M266, Sylvania Electronic Defense Laboratories (1960).
  25. J. E. Burke and V. Twersky, Report EDL-E44, Sylvania Electronic Defense Laboratories (1960).
  26. S. N. Karp and J. Radlow, IRE Trans AP-4, 654 (1956).
  27. B. A. Lippmann and A. Oppenheim, "Towards a Theory of Wood's Anomalies," Technical Research Group, New York (1954).
  28. R. F. Millar, Can. J. PIlys. 3, 81 (1961).
  29. A. Hessel, "Guiding and scattering by sinusoidally-modulated reactance surfaces," Rept. PIBMRI-825-60, Polytechnic Institute of Brooklyn, 1960; A. Hessel and A. A. Oliner, "On the theory of Wood's anomalies" in Rept. R-452: 19–61, Polytechnic Institute of Brooklyn, 1961.
  30. V . Twersky, J. Appl. Phys. 23, 1099 (1952).
  31. L. L. Foldy, Phys. Rev. 67, 107 (1945).
  32. M. Lax, Revs. Modern Phys. 23 (1951); Phys. Rev. 80, 621 (1952).
  33. V. Twersky, J. Acoust. Soc. Am. 29, 209 (1957); IRE Trans. AP-7, S5307(1959), Proceedings of the URSI-Toronto International Symposium on Electromagnetics; IRE Trans. AP-5, 81 (1957).
  34. Lord Rayleigh, Phil. Mag. 14, 350 (1907).
  35. V. Twersky, J. Acoust. Soc. Am. 22, 539 (1950); ibid. 23, 336 (1951); J. Appl. Phys. 22, 825 (1951), Report NDA 18–3, Nuclear Development Associates; J. Appl. Phys. 24, 659 (1953).
  36. V. Twersky, Research Report EM-58, Institute of Mathematical Sciences, New York University (1953).
  37. M. A. Biot, J. Acoust. Soc. Am. 29, 1193 (1957); J. Appl. Phys. 28, 1455 (1957); 29, 998 (1958).
  38. J. R. Wait, IRE Trans. AP-7, Part I, S154; Part II, S163 (1959).
  39. V. Twersky, Report EDL-E26, Sylvania Electronic Defense Laboratories, 1958.
  40. V. Twersky, Report EDL-E36, Sylvania Electronic Defense Laboratories, 1959.
  41. V. Twersky, Research Report EM-59, Institute of Mathematical Sciences, New York University (1953).
  42. R. J. Urick and W. S. Ament, J. Acoust. Soc. Am. 21, 115 (1949).
  43. F. Reiche, Ann. Physik 50, 1, 121 (1916).
  44. Lord Rayleigh, Phil. Mag. 47, 375 (1899).
  45. C. I. Beard and V. Twersky, Report EDL-E46; Report EDL-E47, Sylvania Electronic Defense Laboratories, 1960.

Ament, W. S.

R. J. Urick and W. S. Ament, J. Acoust. Soc. Am. 21, 115 (1949).

Artmann, K.

K. Artmann, Z. Physik 119, 529 (1942).

Beard, C. I.

C. I. Beard and V. Twersky, Report EDL-E46; Report EDL-E47, Sylvania Electronic Defense Laboratories, 1960.

Biot, M. A.

M. A. Biot, J. Acoust. Soc. Am. 29, 1193 (1957); J. Appl. Phys. 28, 1455 (1957); 29, 998 (1958).

Breuckner, K. A.

K. A. Breuckner, Phys. Rev. 89, 834 (1953); 97, 1353 (1955).

Burke, J. E.

J. E. Burke and V. Twersky, Report EDL-M266, Sylvania Electronic Defense Laboratories (1960).

J. E. Burke and V. Twersky, Report EDL-E44, Sylvania Electronic Defense Laboratories (1960).

Fano, U.

U. Fano, Ann. Physik 32, 393 (1938).

Foldy, L. L.

L. L. Foldy, Phys. Rev. 67, 107 (1945).

Hessel, A.

A. Hessel, "Guiding and scattering by sinusoidally-modulated reactance surfaces," Rept. PIBMRI-825-60, Polytechnic Institute of Brooklyn, 1960; A. Hessel and A. A. Oliner, "On the theory of Wood's anomalies" in Rept. R-452: 19–61, Polytechnic Institute of Brooklyn, 1961.

Ignatowsky, W. v.

W. v. Ignatowsky, Ann. Physik 44, 369 (1914).

Ingersoll, L. R.

L. R. Ingersoll, J. Astrophys. 51, 129 (1920); Phys. Rev. 17, 493 (1921).

Karp, S. N.

S. N. Karp, Proceedings of tle Symposium on Microwave Optics, 1953, p. 198 EElectronic Research Directorate, AFCRC (1959)].

S. N. Karp and N. Zitron, Research Report EM-126, Institute of Mathematical Sciences, New York University (1959).

S. N. Karp and J. Radlow, IRE Trans AP-4, 654 (1956).

Lax, M.

M. Lax, Revs. Modern Phys. 23 (1951); Phys. Rev. 80, 621 (1952).

Lippmann, B. A.

B. A. Lippmann and A. Oppenheim, "Towards a Theory of Wood's Anomalies," Technical Research Group, New York (1954).

Millar, R. F.

R. F. Millar, Can. J. PIlys. 3, 81 (1961).

R. F. Millar, Can. J. Phys. 38, 272 (1960).

Oppenheim, A.

B. A. Lippmann and A. Oppenheim, "Towards a Theory of Wood's Anomalies," Technical Research Group, New York (1954).

Palmer, Jr., H. C.

H. C. Palmer, Jr., J. Opt. Soc. Am. 42, 269 (1952); ibid. 46, 50 (1956).

Radlow, J.

S. N. Karp and J. Radlow, IRE Trans AP-4, 654 (1956).

Rayleigh, Lord

Lord Rayleigh, Proc. Roy. Soc. London A79, 399 (1907); see also Rayleigh's earlier paper in Phil. Mag. 14, 60 (1907).

Lord Rayleigh, Phil. Mag. 14, 350 (1907).

Lord Rayleigh, Phil. Mag. 47, 375 (1899).

Reiche, F.

F. Reiche, Ann. Physik 50, 1, 121 (1916).

Schwerd, F. M.

F. M. Schwerd, Die Beugungsercheinungen aus den Fundamentalgesetzen der Undulations Theorie Analytisch Entwickelt (Schwan und Goetz'schen Hosbuchhandlung, Mannheim, Germany, 1835).

Strong, J.

J. Strong, Phys. Rev. 49, 291 (1936).

Twersky, V .

V . Twersky, J. Appl. Phys. 23, 1099 (1952).

Twersky, V.

V. Twersky, J. Acoust. Soc. Am. 29, 209 (1957); IRE Trans. AP-7, S5307(1959), Proceedings of the URSI-Toronto International Symposium on Electromagnetics; IRE Trans. AP-5, 81 (1957).

J. E. Burke and V. Twersky, Report EDL-E44, Sylvania Electronic Defense Laboratories (1960).

J. E. Burke and V. Twersky, Report EDL-M266, Sylvania Electronic Defense Laboratories (1960).

V. Twersky, J. Appl. Phys. 25, 859 (1954).

V. Twersky, IRE Trans. AP-4, 330 (1956), Proceedings of the URSI-Michigan International Symposium on Electromagnetic Theory; Report EDL-M105, Sylvania Electronic Defense Laboratories (1957); Report EDL-E28, Sylvania Electronic Defense Laboratories (1958).

See literature survey by V. Twersky, in J. Research Natl. Bur. Standards 64D, 715 (1960); part of the Report of the United States of America National Committee to the XIIIth General Assembly of the International Scientific Radio Union, London, England.

The two-dimensional case is treated by V. Twersky, in Report EDL-E60, Sylvania Electronic Defense Laboratories, 1961; to be published in "Proceedings of Symposium on lectromagnetic Waves," University of Wisconsin (April, 1961). The three-dimensional case is treated by V. Twersky, in Report EDL-E61, Sylvania Electronic Electronic Defense Laboratories, 1961. [J. Math. Phys. (to be published)].

V. Twersky, J. Appl. Phys. 23, 407 (1952).

C. I. Beard and V. Twersky, Report EDL-E46; Report EDL-E47, Sylvania Electronic Defense Laboratories, 1960.

V. Twersky, J. Acoust. Soc. Am. 22, 539 (1950); ibid. 23, 336 (1951); J. Appl. Phys. 22, 825 (1951), Report NDA 18–3, Nuclear Development Associates; J. Appl. Phys. 24, 659 (1953).

V. Twersky, Research Report EM-58, Institute of Mathematical Sciences, New York University (1953).

V. Twersky, Report EDL-E26, Sylvania Electronic Defense Laboratories, 1958.

V. Twersky, Report EDL-E36, Sylvania Electronic Defense Laboratories, 1959.

V. Twersky, Research Report EM-59, Institute of Mathematical Sciences, New York University (1953).

Urick, R. J.

R. J. Urick and W. S. Ament, J. Acoust. Soc. Am. 21, 115 (1949).

Wait, J. R.

J. R. Wait, IRE Trans. AP-7, Part I, S154; Part II, S163 (1959).

Watson, K. M.

K. M. Watson, Phys. Rev. 87, 575 (1953).

Wood, R. W.

R. W. Wood, Phys. Rev. 48, 928 (1935); Phil. Mag. 4, 396 (1902); ibid. 23, 310 (1912).

Zaviska, F.

F. Zaviska, Ann. Physik 40, 1023 (1913).

Zitron, N.

S. N. Karp and N. Zitron, Research Report EM-126, Institute of Mathematical Sciences, New York University (1959).

Other (45)

Additional introductory material on scattering problems and reviews of recent developments will be found in the writer's "Electromagnetic waves," Physics Today 13, (7) 30 (1960), and "On multiple scattering of waves," J. Research Natl. Bureau Standards 64D, 715 (1960).

All numerical computations (three significant figures, rounded off) for the graphs of this paper were done by S. E. Bergstrom with a desk calculator.

V. Twersky, J. Appl. Phys. 23, 407 (1952).

Such series are discussed in some detail by the writer in J. Acoust. Soc. Am. 24, 42 (1952).

F. Zaviska, Ann. Physik 40, 1023 (1913).

K. A. Breuckner, Phys. Rev. 89, 834 (1953); 97, 1353 (1955).

K. M. Watson, Phys. Rev. 87, 575 (1953).

R. F. Millar, Can. J. Phys. 38, 272 (1960).

See literature survey by V. Twersky, in J. Research Natl. Bur. Standards 64D, 715 (1960); part of the Report of the United States of America National Committee to the XIIIth General Assembly of the International Scientific Radio Union, London, England.

The two-dimensional case is treated by V. Twersky, in Report EDL-E60, Sylvania Electronic Defense Laboratories, 1961; to be published in "Proceedings of Symposium on lectromagnetic Waves," University of Wisconsin (April, 1961). The three-dimensional case is treated by V. Twersky, in Report EDL-E61, Sylvania Electronic Electronic Defense Laboratories, 1961. [J. Math. Phys. (to be published)].

S. N. Karp and N. Zitron, Research Report EM-126, Institute of Mathematical Sciences, New York University (1959).

S. N. Karp, Proceedings of tle Symposium on Microwave Optics, 1953, p. 198 EElectronic Research Directorate, AFCRC (1959)].

V. Twersky, IRE Trans. AP-4, 330 (1956), Proceedings of the URSI-Michigan International Symposium on Electromagnetic Theory; Report EDL-M105, Sylvania Electronic Defense Laboratories (1957); Report EDL-E28, Sylvania Electronic Defense Laboratories (1958).

Lord Rayleigh, Proc. Roy. Soc. London A79, 399 (1907); see also Rayleigh's earlier paper in Phil. Mag. 14, 60 (1907).

W. v. Ignatowsky, Ann. Physik 44, 369 (1914).

F. M. Schwerd, Die Beugungsercheinungen aus den Fundamentalgesetzen der Undulations Theorie Analytisch Entwickelt (Schwan und Goetz'schen Hosbuchhandlung, Mannheim, Germany, 1835).

R. W. Wood, Phys. Rev. 48, 928 (1935); Phil. Mag. 4, 396 (1902); ibid. 23, 310 (1912).

L. R. Ingersoll, J. Astrophys. 51, 129 (1920); Phys. Rev. 17, 493 (1921).

J. Strong, Phys. Rev. 49, 291 (1936).

H. C. Palmer, Jr., J. Opt. Soc. Am. 42, 269 (1952); ibid. 46, 50 (1956).

U. Fano, Ann. Physik 32, 393 (1938).

K. Artmann, Z. Physik 119, 529 (1942).

V. Twersky, J. Appl. Phys. 25, 859 (1954).

J. E. Burke and V. Twersky, Report EDL-M266, Sylvania Electronic Defense Laboratories (1960).

J. E. Burke and V. Twersky, Report EDL-E44, Sylvania Electronic Defense Laboratories (1960).

S. N. Karp and J. Radlow, IRE Trans AP-4, 654 (1956).

B. A. Lippmann and A. Oppenheim, "Towards a Theory of Wood's Anomalies," Technical Research Group, New York (1954).

R. F. Millar, Can. J. PIlys. 3, 81 (1961).

A. Hessel, "Guiding and scattering by sinusoidally-modulated reactance surfaces," Rept. PIBMRI-825-60, Polytechnic Institute of Brooklyn, 1960; A. Hessel and A. A. Oliner, "On the theory of Wood's anomalies" in Rept. R-452: 19–61, Polytechnic Institute of Brooklyn, 1961.

V . Twersky, J. Appl. Phys. 23, 1099 (1952).

L. L. Foldy, Phys. Rev. 67, 107 (1945).

M. Lax, Revs. Modern Phys. 23 (1951); Phys. Rev. 80, 621 (1952).

V. Twersky, J. Acoust. Soc. Am. 29, 209 (1957); IRE Trans. AP-7, S5307(1959), Proceedings of the URSI-Toronto International Symposium on Electromagnetics; IRE Trans. AP-5, 81 (1957).

Lord Rayleigh, Phil. Mag. 14, 350 (1907).

V. Twersky, J. Acoust. Soc. Am. 22, 539 (1950); ibid. 23, 336 (1951); J. Appl. Phys. 22, 825 (1951), Report NDA 18–3, Nuclear Development Associates; J. Appl. Phys. 24, 659 (1953).

V. Twersky, Research Report EM-58, Institute of Mathematical Sciences, New York University (1953).

M. A. Biot, J. Acoust. Soc. Am. 29, 1193 (1957); J. Appl. Phys. 28, 1455 (1957); 29, 998 (1958).

J. R. Wait, IRE Trans. AP-7, Part I, S154; Part II, S163 (1959).

V. Twersky, Report EDL-E26, Sylvania Electronic Defense Laboratories, 1958.

V. Twersky, Report EDL-E36, Sylvania Electronic Defense Laboratories, 1959.

V. Twersky, Research Report EM-59, Institute of Mathematical Sciences, New York University (1953).

R. J. Urick and W. S. Ament, J. Acoust. Soc. Am. 21, 115 (1949).

F. Reiche, Ann. Physik 50, 1, 121 (1916).

Lord Rayleigh, Phil. Mag. 47, 375 (1899).

C. I. Beard and V. Twersky, Report EDL-E46; Report EDL-E47, Sylvania Electronic Defense Laboratories, 1960.

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