The Atkinson-Wilcox expansion theorem for elastic waves
Author:
George Dassios
Journal:
Quart. Appl. Math. 46 (1988), 285-299
MSC:
Primary 73D25
DOI:
https://doi.org/10.1090/qam/950603
MathSciNet review:
950603
Full-text PDF Free Access
Abstract |
References |
Similar Articles |
Additional Information
Abstract: Consider the problem of scattering of an elastic wave by a three-dimensional bounded and smooth body. In the region exterior to a sphere that includes the scatterer, any solution of Navier’s equation that satisfies the Kupradze’s radiation condition has a uniformly and absolutely convergent expansion in inverse powers of the radial distance from the center of the sphere. Moreover, the coefficients of the expansion can recurrently be evaluated from the knowledge of the leading coefficient, known as radiation pattern. Therefore, a one-to-one correspondence between the scattered fields and the corresponding radiation patterns is established. The acoustic and electromagnetic cases are recovered as special cases.
- F. V. Atkinson, On Sommerfeld’s “radiation condition.”, Philos. Mag. (7) 40 (1949), 645–651. MR 30103
P. J. Barratt, W. D. Collins, The Scattering Cross-Section of an Obstacle in an Elastic Solid for Plane Harmonic Waves, Proc. Camb. Philos. Soc. 61, 969–981, (1965)
- David L. Colton and Rainer Kress, Integral equation methods in scattering theory, Pure and Applied Mathematics (New York), John Wiley & Sons, Inc., New York, 1983. A Wiley-Interscience Publication. MR 700400
- George Dassios and Kiriakie Kiriaki, The low-frequency theory of elastic wave scattering, Quart. Appl. Math. 42 (1984), no. 2, 225–248. MR 745101, DOI https://doi.org/10.1090/S0033-569X-1984-0745101-9
N. Dunford, J. T. Schwartz, Linear Operators I, Interscience, New York, 1958
- Fritz John, Partial differential equations, 4th ed., Applied Mathematical Sciences, vol. 1, Springer-Verlag, New York, 1982. MR 831655
- S. N. Karp, A convergent ’farfield’ expansion for two-dimensional radiation functions, Comm. Pure Appl. Math. 14 (1961), 427–434. MR 135451, DOI https://doi.org/10.1002/cpa.3160140318
V. D. Kupradze, Dynamical Problems in Elasticity, in Progress in Solid Mechanics, North-Holland, Amsterdam, 1963
- V. D. Kupradze, T. G. Gegelia, M. O. Basheleĭshvili, and T. V. Burchuladze, Three-dimensional problems of the mathematical theory of elasticity and thermoelasticity, Translated from the second Russian edition, North-Holland Series in Applied Mathematics and Mechanics, vol. 25, North-Holland Publishing Co., Amsterdam-New York, 1979. Edited by V. D. Kupradze. MR 530377
- Peter D. Lax and Ralph S. Phillips, Scattering theory, Pure and Applied Mathematics, Vol. 26, Academic Press, New York-London, 1967. MR 0217440
- L. A. Liusternik and V. J. Sobolev, Elements of functional analysis, Russian Monographs and Texts on Advanced Mathematics and Physics, Vol. 5, Hindustan Publishing Corp., Delhi; Gordon and Breach Publishers, Inc., New York, 1961. MR 0141967
- James Clerk Maxwell, A treatise on electricity and magnetism, Dover Publications, Inc., New York, 1954. 3d ed; Two volumes bound as one. MR 0063293
C. Müller, Die Grundzüge einer Mathematischen Theorie Elektromagnetischer Schwingungen, Archiv. der Math. 1, 296–302 (1948–49)
- Claus Müller, Radiation patterns and radiation fields, J. Rational Mech. Anal. 4 (1955), 235–246. MR 69026, DOI https://doi.org/10.1512/iumj.1955.4.54008
J. W. S. Rayleigh, On the incidence of aerial and electrical waves upon small obstacles in the form of ellipsoids, of elliptic cylinders, and on the passage of electric waves through a circular aperture in a conducting screen, Phil. Mag. 44, 28–52 (1897)
S. Silver, Microwave Antenna Theory and Design, M. I. T. Rad. Lab., McGraw-Hill, New York, 1949
A. Sommerfeld, Die Greensche Funktion der Schwingungsgleichung, Jahr. Der Deut. Math. Ver. 21, 309–353 (1912)
- J. J. Stoker, Some remarks on radiation conditions, Proceedings of Symposia in Applied Mathematics, Vol. V, Wave motion and vibration theory, McGraw-Hill Book Company, Inc., New York-Toronto-London, 1954, pp. 97–102. MR 0062562
- J. J. Stoker, On radiation conditions, Comm. Pure Appl. Math. 9 (1956), 577–595. MR 82295, DOI https://doi.org/10.1002/cpa.3160090327
J. A. Stratton, L. J. Chu, Diffraction Theory of Electromagnetic Waves, Phys. Rev. 56, 99–107 (1939)
C. Truesdell, Mechanics of Solids II, Encyclopedia of Physics, Springer-Verlag, Berlin, 1972
- Victor Twersky, Multiple scattering by arbitrary configurations in three dimensions, J. Mathematical Phys. 3 (1962), 83–91. MR 142299, DOI https://doi.org/10.1063/1.1703791
V. Twersky, Rayleigh Scattering, Appl. Opt. 3, 1150–1162 (1964)
- Calvin H. Wilcox, A generalization of theorems of Rellich and Atkinson, Proc. Amer. Math. Soc. 7 (1956), 271–276. MR 78912, DOI https://doi.org/10.1090/S0002-9939-1956-0078912-4
- C. H. Wilcox, An expansion theorem for electromagnetic fields, Comm. Pure Appl. Math. 9 (1956), 115–134. MR 89000, DOI https://doi.org/10.1002/cpa.3160090202
- Calvin H. Wilcox, Spherical means and radiation conditions, Arch. Rational Mech. Anal. 3 (1959), 133–148. MR 103705, DOI https://doi.org/10.1007/BF00284171
F. V. Atkinson, On Sommerfeld’s Radiation Condition, Phil. Mag. Series 7, 40, 645–651 (1949)
P. J. Barratt, W. D. Collins, The Scattering Cross-Section of an Obstacle in an Elastic Solid for Plane Harmonic Waves, Proc. Camb. Philos. Soc. 61, 969–981, (1965)
D. Colton, R. Kress, Integral Equation Methods in Scattering Theory, John Wiley, New York, 1983
G. Dassios, K. Kiriaki, The Low-Frequency Theory of Elastic Wave Scattering, Quart. Appl. Math. 42, 225–248 (1984)
N. Dunford, J. T. Schwartz, Linear Operators I, Interscience, New York, 1958
F. John, Partial Differential Equations, fourth edition, Springer-Verlag, New York, 1983
S. N. Karp, A Convergent Farfield Expansion for Two-Dimensional Radiation Functions, C. P. A. M. 14, 427–434 (1961)
V. D. Kupradze, Dynamical Problems in Elasticity, in Progress in Solid Mechanics, North-Holland, Amsterdam, 1963
V. D. Kupradze, Three-dimensional Problems of the Mathematical Theory of Elasticity and Thermoelasticity, North-Holland, Amsterdam, 1979
P. D. Lax, R. S. Phillips, Scattering Theory, Academic Press, New York, 1967
L. A. Liusternik, V. J. Sobolev, Elements of Functional Analysis, Ungar, New York, 1961
J. C. Maxwell, Treatise on Electricity and Magnetism, 2 vols., third edition, Dover, New York, 1954
C. Müller, Die Grundzüge einer Mathematischen Theorie Elektromagnetischer Schwingungen, Archiv. der Math. 1, 296–302 (1948–49)
C. Müller, Radiation Patterns and Radiation Fields, J. Rat. Mech. Anal. 4, 235–246 (1955)
J. W. S. Rayleigh, On the incidence of aerial and electrical waves upon small obstacles in the form of ellipsoids, of elliptic cylinders, and on the passage of electric waves through a circular aperture in a conducting screen, Phil. Mag. 44, 28–52 (1897)
S. Silver, Microwave Antenna Theory and Design, M. I. T. Rad. Lab., McGraw-Hill, New York, 1949
A. Sommerfeld, Die Greensche Funktion der Schwingungsgleichung, Jahr. Der Deut. Math. Ver. 21, 309–353 (1912)
J. J. Stoker, Some Remarks on Radiation Conditions, Proc. Symp. Appl. Math. A. M. S. 5, 97–102 (1954)
J. J. Stoker, On Radiation Conditions, C. P. A. M. 9, 577–595 (1956)
J. A. Stratton, L. J. Chu, Diffraction Theory of Electromagnetic Waves, Phys. Rev. 56, 99–107 (1939)
C. Truesdell, Mechanics of Solids II, Encyclopedia of Physics, Springer-Verlag, Berlin, 1972
V. Twersky, Multiple scattering by arbitrary configurations in three dimensions, J. M. P. 3, 83–91 (1962)
V. Twersky, Rayleigh Scattering, Appl. Opt. 3, 1150–1162 (1964)
C. H. Wilcox, A Generalization of Theorems of Rellich and Atkinson, Proc. Amer. Math. Soc. 7, 271–276 (1956)
C. H. Wilcox, An Expansion Theorem for Electromagnetic Fields, C. P. A. M. 9, 115–134 (1956)
C. H. Wilcox, Spherical Means and Radiation Conditions, A. R. M. A. 3, 133–148 (1959)
Similar Articles
Retrieve articles in Quarterly of Applied Mathematics
with MSC:
73D25
Retrieve articles in all journals
with MSC:
73D25
Additional Information
Article copyright:
© Copyright 1988
American Mathematical Society
