Abstract
Continuous media through which acoustic or elastic waves propagate often exhibit inhomogeneities of various types which are difficult to describe, either due to paucity of detailed physical measurements or to the vast complexity in both space and time of these inhomogeneities. The introduction of stochasticity in the description of a continuous medium offers an attractive alternative, due to the fact that randomness is able to reproduce the wave scattering phenomena associated with a naturally occuring medium. In this work, the phenomenon of acoustic or elastic wave propagation under time harmonic conditions is used as the vehicle through which the assumption of randomness in an otherwise homogeneous medium is validated against a deterministic, inhomogeneous medium whose properties vary with depth. The range of applicability of the former model is identified through a series of parametric studies and the results are followed by a discussion on the appropriateness of the various correlation functions that can be used for representing the medium randomness. The numerical methodology employed for both deterministic and random models is a Green's function approach for waves propagating from a point source, while techniques to account for the presence of boundaries are also discussed.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Adomian, G. (1983): Stochastic systems. New York: Academic Press
Askar, A.; Cakmak, A. S. (1988): Seismic waves in random media. Prob. Eng. Mech. 3, 124–129
Barucha-Reid, A. T. (1972): Random integral equations. New York: Academic Press
Bagtzoglou, A. C. (1990): Particle-grid methods with application to reacting flows and reliable solute source identification. Ph.D. dissertation, University of California, Irvine, California
Benaroya, H.; Rehak, M. (1987): The Neuman series/Born approximation applied to parametrically excited stochastic systems. Prob. Eng. Mech. 2, 74–81
Burczynski, T. (1985): The boundary element method for stochastic potential problems. Appl. Math. Mod. 9, 189–194
Chernov, L. A. (1962). Wave propagation in a random medium. New York: Dover
Chu, L.; Askar, A.; Cakmak, A. S. (1981): Earthquake waves in a, random medium. Int. J. Num. Anal. Meth. Geomech. 5, 79–96
Cruse, T. A.; Burnside, O. H.; Wu, Y. T.; Polch, E. Z.; Dias, I. B. (1988): Probabilistic structural analysis methods for select space propulsion system structural components (PSAM). Comput. Struct. 29, 891–901
Dagan, G. (1989). Flow and transport in, porous formations, Berlin, Heidelberg, New York: Springer
Dias, J. B.; Nagtegaal, J. C. (1986): Efficient algorithms for use in probabilistic finite element analysis. In Burnside, O. H.; Parr, C. H. (eds) Advances in Aerospace Structural Analysis, Vol. AD-09, pp. 37–49, New York: ASME Publication
Drewniak, J. (1985): Research communication: boundary elements for random heat conduction problems. Eng. Anal. 2, 168–169
Hoflord, R. L. (1981): Elementary source-type solutions of the reduced wave equation. J. Acoust. Soc. Am. 70, 1427–1436
Hryniewicz, Z. (1991): Mean response to distributed dynamic loads across the random layer for anti-plane shear motion. Acta Mech. 90, 81–89
Ishimaru, A. (1978): Wave propagation, and scattering in random media, Vols. 1 and 2. New York: Academic Press
Karal, F. C.; Keller, J. B. (1964): Elastic, electromagnetic and other waves in a random medium. J. Math. Phys. 5, 537–549
Kohler, W.; Papanikolaou, G.; White, B. (1991): Reflection of waves generated by a point source over a randomly layered medium. Wave Motion 13, 53–87
Kotulski, Z. (1990): Elastic waves in randomly stratified medium. Analytical results. Acta Mech. 83, 61–75
Lafe, O. E.; Cheng, A. H. D.: (1987) A perturbation boundary element code for steady-state groundwater flow in heterogeneous aquifers. Water Resour. Res. 23, 1079–1084
Li, Y. L.; Liu, C. H.; Franke, S. J. (1990): Three-dimensional Green's function for wave propagation in a linearly inhomogeneous medium—the exact analytic solution. J. Acoust. Soc. Am. 87, 2285–2291
Liu, K. C. (1991): Wave scattering in discrete random media by the discontinuous stochastic field method I: Basic method and general theory. J. Sound Vibr. 147, 301–311
Liu, W. K.; Belytschko, T.; Mani, A. (1986): Random field finite elements. Int. J. Num. Meth. Eng. 23, 1831–1845
Luco, J. E.; Wong, H. L. (1986): Response of a rigid foundation to a spatially random ground motion. Earthquake Eng. Struct. Dyn. 14, 891–908
Manolis, G. D.; Beskos, D. E. (1988): Boundary element methods in elastodynamics. London: Chapman and Hall
Manolis, G. D.; Shaw, R. P. (1990): Random wave propagation using boundary elements. In: Tanaka, M.; Brebbia, C. A.; Shaw, R. P. (eds). Advances in boundary element methods in Japan and USA, Topics in Engineering, Vol. 7. Southampton: Computational Mechanics Publications
Manolis, G. D.; Shaw, R. P. (1992): Wave motion in a random hydroacoustic medium using boundary integral/element methods. Eng. Anal. Bound. Elem 9, 61–70
McLachlan, N. W. (1954). Bessel functions for engineers. Oxford: Clarendon Press
Mindlin, R. D. (1936): Force at a point in the interior of a semi-infinite solid. J. Phys. 7, 195–202
Pai, D. M. (1991): Wave propagation in inhomogeneous media: a planewave layer interaction method. Wave Motion 13, 205–209
Pais, A. L.; Kausel, E. (1990): Stochastic response of rigid foundations. Earthquake Eng. Struct. Dyn. 19, 611–622
Pekeris, C. L. (1946): Theory of propagation of sound in a half-space of variable sound velocity under conditions of formation of a sound zone. J. Acoust Soc. Am. 18, 295–315
Shaw, R. P. (1991): Boundary integral equations for nonlinear problems by the Kirchhoff transformation. In: Brebbia, C. A.; Gipson, G. S. (eds): Boundary Elements XIII, pp. 59–69. London: Elsevier
Shaw, R. P.; Makris, N. (1991) Green's functions for Helmholtz and Laplace equations in heterogeneous media. In: Brebbia, C. A.; Gipson, G. S. (eds): Boundary Elements XIII, pp. 59–69. London: Elsevier
Shinozuka, M. (1972) Digital simulation of random processes and its applications. J. Sound Vibr 25, 111–128
Spanos, P. D.; Ghanem, R. (1991): Boundary element formulation for random vibration problems. J. Eng. Mech. ASCE 117, 409–423
Sobczyk, K. (1976): Elastic wave propagation in a discrete random medium. Acta Mech. 25, 13–28
Varadan, V. K.; Ma, Y.; Varadan, V. V. 1985): Multiple scattering theory for elastic wave propagation in discrete random media. J. Acous. Soc. Am. 77, 375–389
Vanmarke, E.; Shinozuka, M.; Nakagiri, S.; Schueller, G. I.; Grigoriu, M. (1986): Random fields and stochastic finite elements. Struct. Safety 3, 143–166
Vrettos, C. (1990a): Dispersive SH-surface waves in soil deposits of variable shear modulus. Soil Dyn. Earthquake Eng. 9, 255–264
Vrettos, C. (1990b): In-plane vibrations of soil deposits with variable shear modulus: I. Surface waves, and II. Line load. Int. J. Num. Anal. Meth. Geomech. 14, 209–222 and 649–662
Brettos, C. (1991a). Forced anti-plane vibrations at the surface of an inhomogeneous half-space. Soil Dyn. Earthquake Eng. 10, 230–235
Vrettos, C. (1991b): Time-harmonic Boussinesq problem for a continuously non-homogeneous soil. Earthquake Eng. Struct. Dyn. 20, 961–977
Author information
Authors and Affiliations
Additional information
Communicated by D. Beskos, May 10, 1992
Rights and permissions
About this article
Cite this article
Manolis, G.D., Bagtzoglou, A.C. A numerical comparative study of wave propagation in inhomogeneous and random media. Computational Mechanics 10, 397–413 (1992). https://doi.org/10.1007/BF00363995
Issue Date:
DOI: https://doi.org/10.1007/BF00363995