The ellipsoidal cavity in the presence of a low-frequency elastic wave

Dassios, George; Kiriaki, Kiriakie

doi:10.1090/qam/872823

Authors: George Dassios and Kiriakie Kiriaki
Journal: Quart. Appl. Math. 44 (1987), 709-735
MSC: Primary 73D25
DOI: https://doi.org/10.1090/qam/872823
MathSciNet review: 872823
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We consider the problem of scattering of a longitudinal or a transverse plane elastic wave by a general ellipsoidal cavity in the low-frequency region. Explicit closed-form solutions for the zeroth- and first-order approximations are provided in terms of the physical and geometric characteristics of the scatterer, as well as the direction cosines of the incidence and observation points. This was made possible with the introduction of an analytical technique based on the Papkovich representations and their interdependence. The leading low-frequency term for the normalized spherical scattering amplitudes and the scattering cross section are also given explicitly. Degenerate ellipsoids corresponding to the prolate and oblate spheroids, the sphere, the needle, and the disc are considered as special cases.

P. J. Barratt and W. D. Collins, The scattering cross section of an obstacle in an elastic solid for plane harmonic waves, Proc. Cambridge Philos. Soc. 61, 969 (1965)

George Dassios, CONVERGENT LOW-FREQUENCY EXPANSIONS FOR PENETRABLE SCATTERERS, ProQuest LLC, Ann Arbor, MI, 1975. Thesis (Ph.D.)–University of Illinois at Chicago. MR 2625259

G. Dassios, Convergent low-frequency expansions for penetrable scatterers, J. Math. Phys. 18, 126 (1977)

George Dassios, Second order low-frequency scattering by the soft ellipsoid, SIAM J. Appl. Math. 38 (1980), no. 3, 373–381. MR 579425, DOI https://doi.org/10.1137/0138031
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, DOI https://doi.org/10.1137/0142021

G. Dassios, Scattering of acoustic waves by a coated pressure release ellipsoid, J. Acoust. Soc. Amer. 70, 176 (1981)

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
George Dassios and Kiriakie Kiriaki, The rigid ellipsoid in the presence of a low frequency elastic wave, Quart. Appl. Math. 43 (1986), no. 4, 435–456. MR 846156, DOI https://doi.org/10.1090/S0033-569X-1986-0846156-7
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 112391
E. W. Hobson, The theory of spherical and ellipsoidal harmonics, Chelsea Publishing Company, New York, 1955. MR 0064922

V. D. Kupradze, Dynamical problems in elasticity, in Progress in solid mechanics III. North-Holland, Amsterdam (1963)

E. G. Lawrence, Diffraction of elastic waves by a rigid ellipsoid, Quart. J. Mech. Appl. Math. XXV, 161 (1972)

L. Solomon, Elasticité Linéaire, Masson(1968)

V. Twersky, Certain transmission and reflection theorems, J. Appl. Phys. 25 (1954), 859–862. MR 63930

C. Truesdell, Mechanics of Solids II, in Encyclopedia of Physics, Vol. VI a/2, Springer-Verlag (1972)

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 86519