Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

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



The low-frequency theory of elastic wave scattering

Authors: George Dassios and Kiriakie Kiriaki
Journal: Quart. Appl. Math. 42 (1984), 225-248
MSC: Primary 73D25; Secondary 35L20
DOI: https://doi.org/10.1090/qam/745101
MathSciNet review: 745101
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Abstract: An incident longitudinal, or transverse, plane wave is scattered by a bounded region immersed in an infinite isotropic and homogeneous elastic medium. The region could be either a rigid scatterer or a cavity. Integral representations for the total displacement field, as well as for the introduced spherical scattering amplitudes are given explicitely in a compact form. Representations for the scattering cross-section whenever the incident wave is a longitudinal or a transverse wave are also provided. Using Papkovich potentials and low-frequency techniques the scattering problems are reduced to an iterative sequence of potential problems which can be solved successively in terms of expansions in appropriate harmonic functions. In each one of the four cases (longitudinal and transverse incidence on rigid scatterer and cavity) the corresponding exterior boundary value problems that specify the approximations as well as the analytic expressions for the scattering amplitudes and the scattering cross-section are given explicitly. The leading low-frequency term of the scattering cross-section for a rigid scatterer is independent of the wave number while for the case of a cavity it is proportional to the fourth power of the wave number. The low-frequency limit of the displacement field which corresponds to the static problem when the scatterer is a cavity, does not depend on the geometrical characteristics of the scatterer and it is always a constant.

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  • [1] R. P. Banaugh, Application of integral representations of displacement potentials in elasto-dynamics, Bull. Seism. Soc. Amer. 54, 1073 (1964)
  • [2] P. J. Barratt and W. D. Collins, The scattering cross-section of an obstacle in an elastic solid for plane harmonic waves, Proc. Camb. Philos. Soc. 61, 969 (1965)
  • [3] G. Dassios, Convergent low-frequency expansions for penetrable scatterers, j. Math. Phys. 18, 126 (1977)
  • [4] George Dassios, Second order low-frequency scattering by the soft ellipsoid, SIAM J. Appl. Math. 38 (1980), no. 3, 373–381. MR 579425, https://doi.org/10.1137/0138031
  • [5] -, Scattering of acoustic waves by a coated pressure--release ellipsoid, J. Acoust. Soc. Amer. 70, 176 (1981)
  • [6] George Dassios, Low frequency scattering theory for a penetrable body with an impenetrable core, SIAM J. Appl. Math. 42 (1982), no. 2, 272–280. MR 650223, https://doi.org/10.1137/0142021
  • [7] G. Dassios and K. Kiriaki, The rigid ellipsoid in the presence of a low-frequency elastic wave (to be published)
  • [8] -, The ellipsoidal cavity in the presence of a low-frequency elastic wave (to be published)
  • [9] Norman G. Einspruch, E. J. Witterholt, and Rohn Truell, Scattering of a plane transverse wave by a spherical obstacle in an elastic medium, J. Appl. Phys. 31 (1960), 806–818. MR 0112391
  • [10] V. D. Kupradze, Progress in solid mechanics. III, Dynamical problems in elasticity, North-Holland (1963)
  • [11] E. G. Lawrence, Diffraction of elastic waves by a rigid inclusion, Quart. J. Mech. Appl. Math. XXIII, 389 (1970)
  • [12] -, Diffraction of elastic waves by a rigid ellipsoid, Quart. J. Mech. Appl. Math. XXV, 161 (1972)
  • [13] Y. H. Pao and Varatharajulu, Huygens' principle, radiation conditions, and integral formulas for the scattering of elastic waves, J. Acoust. Soc. Amer. 59, 1361 (1976)
  • [14] V. Twersky, Certain transmission and reflection theorems, J. Appl. Phys. 25 (1954), 859–862. MR 0063930
  • [15] -, Ravleigh scattering, Appl. Opt. 3, 1150 (1964)
  • [16] P. C. Waterman, Matrix theory of elastic wave scattering, J. Acoust. Soc. Amer. 60 (1976), no. 3, 567–580. MR 0431874, https://doi.org/10.1121/1.381130
  • [17] P. C. Waterman, Matrix theory of elastic wave scattering. II. A new conservation law, J. Acoust. Soc. Amer. 63 (1978), no. 5, 1320–1325. MR 0478919, https://doi.org/10.1121/1.381884
  • [18] Lewis T. Wheeler and Eli Sternberg, Some theorems in classical elastodynamics, Arch. Rational Mech. Anal. 31 (1968), no. 1, 51–90. MR 1553519, https://doi.org/10.1007/BF00251514
  • [19] C. F. Ying and Rohn Truell, Scattering of a plane longitudinal wave by a spherical obstacle in an isotropically elastic solid, J. Appl. Phys. 27 (1956), 1086–1097. MR 0086519

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DOI: https://doi.org/10.1090/qam/745101
Article copyright: © Copyright 1984 American Mathematical Society

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