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IssuesArchive of Issues2002-3pp.89-100

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S. V. Novotny, "Generalization of the Pochhammer-Chree problem to the case of an inertially compliant boundary," Mech. Solids. 37 (3), 89-100 (2002)
Year 2002 Volume 37 Number 3 Pages 89-100
Title Generalization of the Pochhammer-Chree problem to the case of an inertially compliant boundary
Author(s) S. V. Novotny (Moscow)
Abstract A solution of the internal regular problem of propagation of stationary axially symmetric plane waves in an infinite isotropic circular cylinder (the problem of Pochhammer-Chree [1], [2]) is considered under boundary conditions of inertial type, i. e., the stresses on the boundary are proportional to the accelerations.

We obtain the corresponding frequency relation which determines the dispersion law in the presence a stiffened boundary. This relation is examined for real and imaginary propagation parameters and is compared with solutions satisfying the classical boundary conditions.

Qualitative distinctions are detected in the structure of the frequency spectrum as compared with the case of free boundary. We also describe how and to which extent the principal spectral characteristics of the harmonic wave process depend on the stiffening parameter and Poisson's ratio.
References
1.  L. Pochhammer, "Über die Fortpflanzungsgeschwindigkeiten kleiner Schwingungen in einem unbegrenzten isotropen Kreiszylinder," J. Reine und Angew. Bd. 81, S. 324-336, 1876.
2.  C. Chree, "The equations of an isotropic elastic solid in polar and cylindrical coordinates, their solutions and applications," Trans. Cambridge Phil. Soc., Vol. 14, pp. 250-369, 1889.
3.  A. Love, A Treatise on the Mathematical Theory of Elasticity [Russian translation], ONTI, Leningrad, 1935.
4.  M. Onoe, H. D. McNiven, and R. D. Mindlin, "Dispersion of axially symmetric waves in elastic rods," Trans. ASME. Appl. Mech., Vol. 62, No. 4, pp. 139-145, 1962.
5.  T. R. Meeker and A. H. Meitzler, "Wave propagation in rods and plates," in W. P. Mason (Editor), Physical Acoustics. Volume 1. Methods and Devices. Part A, pp. 140-203, New York, London, 1964.
6.  J. Miklowitz, The Theory of Elastic Waves and Waveguides, North-Holland, Amsterdam, 1978.
7.  V. T. Grinchenko and V. V. Meleshko, Harmonic Vibrations and Waves in Elastic Bodies [in Russian], Naukova Dumka, Kiev, 1981.
8.  S. V. Novotny, "Stationary torsional waves in a cylinder whose boundary has limited compliance," Vestnik MGU [Bulletin of Moscow State University], Ser. 1, Mat., Mekh., No. 4, pp. 77-85, 1996.
9.  J. Zemanek, "An experimental and theoretical investigation of elastic wave propagation in a cylinder," J. Acoust. Soc.Amer., Vol. 51, No. 1, Pt. 2, pp. 265-283, 1972.
10.  D. Bancroft, "The velocity of longitudinal waves in cylindrical bars," Phys. Rev., Vol. 59, pp. 588-593, 1941.
11.  Y. H. Pao and R. D. Mindlin, "Dispersion of flexural waves in an elastic circular cylinder," Trans. ASME, Ser. E., J. Appl. Mech., Vol. 27, No. 3, pp. 513-520, 1960.
12.  R. Sinclair, "Velocity dispersion of Rayleigh waves propagating along rough surfaces. Part 2," J. Acoust. Soc. Amer., Vol. 50, No. 3, pp. 841-845, 1971.
Received 22 September 2000
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