Mathematical model of acoustic waves in a bounded domain with “white noise”

E. V. Bychkov ; N. N. Solovyova ; G. A. Sviridyuk

Vestnik Ûžno-Uralʹskogo gosudarstvennogo universiteta. Seriâ, Matematika, mehanika, fizika, Tome 11 (2019) no. 3, pp. 12-19

Résumé

The article presents a fresh approach at the classical problem of the acoustic waves propagation in a bounded region with a constant phase velocity. The classical statement of the problem is formulated in deterministic spaces, but in that work this problem will be studied in spaces of $K$-“noise”. An initial-boundary value problem for an inhomogeneous stochastic hyperbolic equation is investigated. The initial data are a random $K$-variables, and the inhomogeneity function is a random $K$-process in the general case. The inhomogeneous function is defined as “white noise”, in the application. In this paper the term “white noise” refers to the first derivative in the sense of Nelson–Gliklich Wiener $K$-process. This problem can be considered as generalization of the classical, since the Nelson–Gliklich derivative of the deterministic function coincides with the classical derivative. In the article, the results are obtained for an abstract deterministic hyperbolic equation are shifted to the stochastic case. Abstract results are applied to the mathematical model of the acoustic waves propagation with additive “white noise” in a bounded region from $R^n$ with a smooth boundary.

Détail

Keywords: acoustic waves, Cauchy–Dirichlet problem, “white noise”, Wiener $K$-process, propagators.

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