Proceedings of the Second ISAAC Congress pp 1319-1333 | Cite as

# Wave Propagation in the Ice-Covered Ocean Wave Guide and Operator Polynomials

## Abstract

Wave propagation in a 2D ocean wave guide covered by a thin ice layer is considered. Various models of ice cover are used (pack ice, solid ice, ice with surface tension, etc.), and the eigenfunctions of the transverse cross—section are analyzed. For each model of ice, an operator polynomial equation for eigenfunctions is derived. It is proved that, for non—critical frequencies, the eigenfunctions form a basis in a suitable Hilbert space. At the critical frequencies, a chain of associated functions might appear. The method is based on some (known) results from the abstract operator theory and some orthogonality relations for eigenfunctions.

### Keywords

Propa Nite Acoustics Boris## Preview

Unable to display preview. Download preview PDF.

### References

- [1]I.V. Andronov, B.P. Belinskiy: Acoustic scattering by a narrow crack in an elastic plate.
*Wave Motion*24 (1996), 101–115.MATHCrossRefGoogle Scholar - [2]B.P. Belinskiy: Boundary-contact problem of acoustics in an infinite domain.
*Soviet Phys. Doklady*30 (1985), 560–562.Google Scholar - [3]B.P. Belinskiy: Ice-covered ocean wave guides and operator polynomials.
*Abstracts of Papers Presented to the**936th Meeting of the AMS*,*Wake Forest Univ.*,*Winston-Salem*,*North Carolina*,*October**9–10*, 462, 1998.Google Scholar - [4]B.P. Belinskiy. The bases of the system of eigenfunctions of the transverse cross-section of wave guides with flexible walls.
*USSR Comput. Mathem. and Mathem. Phys*. (1989) 6, 51–60.CrossRefGoogle Scholar - [5]B.P. Belinskiy, J.P. Dauer: Mathematical aspects of wave phenomena in a wave guide with elastic walls and operator polynomials.
*Festschrift Volumes for***H.***Uberall*,*World Scientific*,*Singapore*,*to appear.*Google Scholar - [6]B.P. Belinskiy, J.P. Dauer: Wave phenomena in a wave guide with elastic walls: abstract differential equation formulation.
*Proceedings of the Symposium on Volterra Equations*,to appear.Google Scholar - [7]B.P. Belinskiy, J.P. Dauer, Y. Xu: Inverse scattering of acoustic waves in an ocean with ice cover.
*Applicable Analysis*61 (1996), 255–283.MathSciNetCrossRefGoogle Scholar - [8]B.P. Belinskiy, J.P. Dauer: On a regular Sturm - -Liouville problem on a finite interval with the eigenvalue parameter appearing linearly in the boundary conditions.
*Spectral theory and computational methods of Sturm — —Liouville problem*. Eds. D. Hinton and P.W. Schaefer, 1997.Google Scholar - [9]B.P. Belinskiy, J.P. Dauer: Eigenoscillations of mechanical systems with boundary conditions containing the frequency.
*Quarterly of Applied Math*. 56 (1998), 521–541.MathSciNetMATHGoogle Scholar - [10]A. Erdélyi:
*Asymptotic expansions*. Dover Publications, Inc., New York, 1956.MATHGoogle Scholar - [11]Cg. Greenhild: Scattering on the thin ice.
*The Lond. Edinb. and Dubl. Phil. Mag. and J. of Sci.*, 31, 1916.Google Scholar - [12]I.C. Gohberg, M.G. Krein:
*Introduction to the theory of linear nonselfadjoint operators*Trans. Math. Monographs 18, AMS, Providence, Rhode Island, 1969.MATHGoogle Scholar - [13]M.C. Junger, D. Feit:
*Sound structures*,*and their interaction*. The MIT Press, Cambridge, 1972.MATHGoogle Scholar - [14]J.B. Keller, E. Goldstein: Water wave reflection due to surface tension and floating ice.
*Trans. Am. Geophys. Union*34 (1953), 43–48.MathSciNetGoogle Scholar - [15]J.B. Keller, M. Weitz: Reflection and transmission coefficients for waves entering or leaving an ice field.
*Comm. Pure Appl. Math.*415–417, 1953.Google Scholar - [16]D.P. Kousov: Diffraction of a plane hydro-acoustic wave on a crack in an elastic plate.
*J. Appl. Math. Mech.*27 (1963), 1037–1043.MathSciNetGoogle Scholar - [17]D.P. Kousov: On resonance phenomenon for diffraction of a plane hydro-acoustic wave on a system of cracks in an elastic plate.
*J. Appl. Math. Mech.*28 (1964), 409–417.MathSciNetGoogle Scholar - [18]O.A. Ladyzhenskaia:
*The boundary value problems of mathematical physics*. Springer-Verlag, New York, 1985.Google Scholar - [19]M. Reed, B. Simon:
*Methods of modern math. physics 3: Scattering theory*. Academic Press, New York, 1979.Google Scholar - [20]M. Weitz, J.B. Keller: Reflection of water waves from floating ice in water of finite depth.
*Comm. Pure Appl. Math.*3 (1950) 305–318.MathSciNetMATHCrossRefGoogle Scholar