Interaction of a plane compressional elastic wave with a rigid spheroidal inclusion

Author:
Subhendu K. Datta

Journal:
Quart. Appl. Math. **31** (1973), 217-235

DOI:
https://doi.org/10.1090/qam/99704

MathSciNet review:
QAM99704

Full-text PDF Free Access

Abstract | References | Additional Information

Abstract: In this paper we have considered the problem of diffraction of a plane compressional harmonic elastic wave by a rigid spheroidal inclusion embedded in a homogeneous isotropic medium. For simplicity we have confined our attention to the axisymmetric case when the incident wave propagates along the axis of symmetry of the spheroid. The inclusion is assumed to be movable. Since the exact solution to this problem is not obtainable analytically we have used a boundary perturbation technique that is applicable at any finite frequency. We have derived exact analytical expression for the amplitude of oscillation of the inclusion correct to first order in a shape correction factor . It is shown that the low-frequency expansion of the amplitude agrees with the expansion derived by other means correct to first order in frequency. We have also given a high-frequency expansion of the amplitude. Furthermore, we have derived the asymptotic expansion of the field in the illuminated zone and have shown that these are compatible with those obtained by an application of Keller's ray theory.

**[1]**Yih-Hsing Pao and C. C. Mow,*Scattering of plane compressional waves by a spherical obstacle*, J. Appl. Phys.**34**(1963), 493–499. MR**0149745****[2]**C. C. Mow,*Transient response of a rigid spherical inclusion in an elastic medium*, J. Appl. Mech., Trans. ASME, Ser. E**32**, 637 (1965)**[3]**C. C. Mow,*On the transient motion of a rigid spherical inclusion in an elastic medium and its inverse problem*, J. Appl. Mech., Trans. ASME, Ser. E**33**, 807 (1966)**[4]**Leon Knopoff,*Scattering of shear waves by spherical obstacles*, Geophysics**24**(1959), 209–219. MR**0108090****[5]**Leon Knopoff,*Scattering of shear waves by spherical obstacles*, Geophysics**24**(1959), 209–219. MR**0108090****[6]**Horace Lamb,*Problems relating to the Impact of Waves on a Spherical Obstacle in an Elastic Medium*, Proc. London Math. Soc.**S1-32**, no. 1, 120. MR**1576215**, https://doi.org/10.1112/plms/s1-32.1.120**[7]**Horace Lamb,*On a Peculiarity of the Wave-System due to the Free Vibrations of a Nucleus in an Extended Medium*, Proc. London Math. Soc.**S1-32**, no. 1, 208. MR**1576221**, https://doi.org/10.1112/plms/s1-32.1.208**[8]**A. E. H. Love,*Some Illustrations of Modes of Decay of Vibratory Motions*, Proc. London Math. Soc.**S2-2**, no. 1, 88. MR**1577293**, https://doi.org/10.1112/plms/s2-2.1.88**[9]**P. Chadwick and E. A. Trowbridge,*Elastic wave fields generated by scalar wave functions*, Proc. Cambridge Philos. Soc.**63**(1967), 1177–1187. MR**0218047****[10]**A. K. Mal, D. D. Ang and L. Knopoff,*Diffraction of elastic waves by a rigid circular disc*, Proc. Camb. Philos. Soc.**64**, 237 (1968)**[11]**S. K. Datta,*The diffraction of a plane compressional elastic wave by a rigid circular disc*, Q. Appl. Math.**28**, 1 (1970)**[12]**A. K. Mal,*Motion of a rigid disc in an elastic solid*, Bull. Seism. Soc. Am.**61**, 1717 (1971)**[13]**Victor A. Erma,*An exact solution for the scattering of electromagnetic waves from conductors of arbitrary shape. I. Case of cylindrical symmetry*, Phys. Rev. (2)**173**(1968), 1243–1257. MR**0246572****[14]**S. K. Datta,*Rectilinear oscillations of a rigid inclusion in an infinite elastic medium*, Int. J. Engr. Sci.**9**, 947 (1971)**[15]**S. K. Datta,*Torsional oscillations of a rigid inclusion in an infinite elastic medium*, in*Proc*. 3*rd Canadian Congr. Appl. Mech*. (ed. P. G. Glockner), The University of Calgary, Calgary, Canada, 1971**[16]**D. S. Ahluwalia, J. B. Keller, and E. Resende,*Reflection of elastic waves from cylindrical surfaces*, J. Math. Mech.**19**(1969/1970), 93–105. MR**0250534****[17]**E. G. Lawrence,*Diffraction of elastic waves by a rigid inclusion*, Quart. J. Mech. Appl. Math.**23**, 389 (1970)**[18]**S. A. Thau and Y. H. Pao,*A perturbation method for boundary value problems in dynamic elasticity*, Q. Appl. Math.**25**, 243 (1967)**[19]**W. E. Williams,*A note on slow vibrations in an elastic medium*, Quart. J. Mech. Appl. Math.**19**, 413 (1966)**[20]**J. B. Keller, R. M. Lewis, and B. D. Seckler,*Asymptotic solution of some diffraction problems*, Comm. Pure Appl. Math.**9**(1956), 207–265. MR**0079182**, https://doi.org/10.1002/cpa.3160090205**[21]***Progress in Solid Mechanics. Vol.III*, Dynamical problems in elasticity by V. D. Kupradze. Edited by I. N. Sneddon and R. Hill, North-Holland Publishing Co., Amsterdam; Interscience Publishers John Wil ey & Sons, Inc. New York, 1963. MR**0164488****[22]**Masahumi Nagase,*Diffraction of elastic waves by a spherical surface*, J. Phys. Soc. Japan**11**(1956), 279–301. MR**0078858**, https://doi.org/10.1143/JPSJ.11.279**[23]**F. R. Norwood and J. Miklowitz,*Diffraction of transient elastic waves by a spherical cavity*, J. Appl. Mech., Trans. ASME, Ser. E**34**, 735 (1967)**[24]**H. M. Nussenzveig,*High-frequency scattering by an impenetrable sphere*, Ann. Physics**34**(1965), 23–95. MR**0189455**, https://doi.org/10.1016/0003-4916(65)90041-2**[25]**W. Maurice Ewing, Wenceslas S. Jardetzky, and Frank Press,*Elastic waves in layered media*, Lamont Geological Observatory Contribution No. 189. McGraw-Hill Series in the Geological Sciences, McGraw-Hill Book Co., Inc., New York-Toronto-London, 1957. MR**0094967****[26]**Freeman Gilbert,*Scattering of impulsive elastic waves by a smooth convex cylinder*, J. Acoust. Soc. Amer.**32**(1960), 841–857. MR**0120921**, https://doi.org/10.1121/1.1908238

Additional Information

DOI:
https://doi.org/10.1090/qam/99704

Article copyright:
© Copyright 1973
American Mathematical Society