Abstract
The paper is concerned with the consideration of some problems of wave propagation on the surface of fluid with gradually varying depth. The asymptotics have been obtained for the following problems: wave diffraction by a bottom obstacle, evolution of an initial disturbance for one-and two-layer fluids, for gravity - capillary waves and for fluid layer on an elastic base. All the mentioned asymptotics take into account the focusing effects. Their construction is reduced to solution of certain Hamiltonian systems. The final result is obtained with the help of Maslov’s canonical operator. For the simplest cases, this result is in agreement with that obtained by the standard ray method [1].
For the Cauchy-Poisson problem, the asymptotics is obtained with the help of the Pearcey functions. The calculation algorithm consists of two stages: first, all the rays coming to the observation point are found, then, second, the free surface elevation is calculated by means of explicit formulas. As compared to the ray method, the algorithm doesn’t require additional data. A new method is proposed for calculating the Pearcey functions. The calculations were carried out for the case of nonstationary wave focusing on a single bottom hill.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Shen M.C., Keller J.B. Uniform ray theory of surface, internal and acoustic wave propagation in a rotating ocean or atmosphere. SIAM J. Appl. Math, 1975, vol. 28, N 4, pp.857–875.
Dobrokhotov S.Yu. Maslov’s methods in the linearized theory of gravity waves on the liquid suface. Dokl. Akad. Nayk, 1983, vol. 269 N 1, pp.76–80 (in Russian).
Dobrokhotov S.Yu., Zhevandrov P.N. Nonstandard characteristics and Maslov operator method in linear problems of unsteady waters waves. Funktsional. Anal. i Prilozhen, 1985, vol. 19, N 4, pp. 43–54, Englisn transl. in Functional Anal. Applic., 1985, vol. 19.
Garipov R.M. On the linear theory of gravity waves: the theorem of existence and uniqueness. Arch. Rat. Mech. Anal, 1967, vol. 24, N 5, pp. 352–362.
Zhevandrov P.N., Isakov R.V. Cauchy-Poisson problem for a two-layered liquid of variable-depth. Mat. zametki, 1990, vol. 47, N 6, pp. 31–44, Englisn transl. in Math Notes, 1990, vol. 47.
Dobrokhotov S.Yu., Tolstova O.L. Application of conservation laws to asymptotic problems for equations with an operator - valued symbol. Mat. zametki, 1990, vol. 47, N 5, pp.148–151(in Russian).
Dobrokhotov S.Yu., Zhevandrov P.N. Surface waves on a liquid of variable depth generated by a moving submerged source. Oscillations and waves in fluids, Gorky, 1988, pp. 32–41 (in Russian).
Korobkin A.A., Sturova I.V. Three-dimensional Cauchy-Poisson problem for a basin with gradually varying depth. Dinamika sploshnoi sredy, 1990, vyp.97, pp.37–47 (in Russian).
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1992 Springer Basel AG
About this chapter
Cite this chapter
Dobrokhotov, S.Y., Zhevandrov, P.N., Korobkin, A.A., Sturova, I.V. (1992). Asymptotic Theory of Propagation of Nonstationary Surface and Internal Waves over Uneven Bottom. In: Antontsev, S.N., Khludnev, A.M., Hoffmann, KH. (eds) Free Boundary Problems in Continuum Mechanics. International Series of Numerical Mathematics / Internationale Schriftenreihe zur Numerischen Mathematik / Série Internationale d’Analyse Numérique, vol 106. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-8627-7_12
Download citation
DOI: https://doi.org/10.1007/978-3-0348-8627-7_12
Publisher Name: Birkhäuser, Basel
Print ISBN: 978-3-0348-9705-1
Online ISBN: 978-3-0348-8627-7
eBook Packages: Springer Book Archive