ARTICLE
Long gravity waves in a canal with a corrugated bottom in the asymptotic description
 
More details
Hide details
1
Institute of Geophysics, Polish Academy of Sciences, Warsaw, Poland
2
Institute of Fundamental Technological Research, PAS, Warsaw, Poland
CORRESPONDING AUTHOR
Włodzimierz Bielski   

Theoretical Geophysics, Institute of Geophysics, Księcia Janusza 64, 01-452, Warszawa, Poland
Submission date: 2020-09-25
Final revision date: 2021-05-22
Acceptance date: 2021-06-11
Online publication date: 2021-06-27
Publication date: 2021-07-25
 
Journal of Theoretical and Applied Mechanics 2021;59(3):443–454
 
KEYWORDS
TOPICS
ABSTRACT
We consider the classic Lagrange long gravitational wave of a homogeneous incompressible fluid in a shallow canal with a corrugated bottom. We use the asymptotic expansion method to find the effective depth of a one-dimensional canal and, hence, the effective wave velocity. A flow in a two-dimensional tank with a corrugated bottom is also studied by this method.
 
REFERENCES (21)
1.
Adler P.M., Malevich A.E., Mityushev V.V., 2013, Nonlinear correction to Darcy’s law for channels with wavy walls, Acta Mechanica, 224, 8, 1823-1848.
 
2.
Andrianov I.V., Awrejcewicz J., Danishevskyy V.V., 2018, Asymptotical Mechanics of Composites: Modelling Composites without FEM, Springer International Publishing AG, Cham.
 
3.
Bakhvalov N.S., Panasenko G.P., 1989, Homogenisation: Averaging Processes in Periodic Media: Mathematical Problems in the Mechanics of Composite Materials, Kluwer, Dordrecht-Boston-London.
 
4.
Bensoussan A., Lions J.-L., Papanicolaou G., 1980, Asymptotic Analysis of Periodic Structures, North Holland Publishing Company, Amsterdam-New York-Oxford.
 
5.
Dingemans M.W., 1997, Water Wave Propagation Over Uneven Bottoms, Part 1 – Basic Equations, Part 2 – Wave propagation formulation, Advanced Series on Ocean Engineering: Vol. 13, World Scientific, Delft.
 
6.
Evangelos K., 2012, The impact of vegetation on the characteristics of the flow in an inclined open channel using the piv method, Water Resources and Ocean Science, 1, 1, 1-6.
 
7.
Gill A.E., 1982, Gravity wave, Atmosphere Ocean Dynamics, Academic Press.
 
8.
Karaeva D.A., Karaev A.D., Nazaikinskii V.E., 2018, Homogenization method in the problem of long wave propagation from a localized source in a basin over an uneven bottom, Differential Equations, 54, 8, 1057-1072.
 
9.
Kubrak E., Kubrak J., Rowiński P.M., 2013, Application of one-dimensional model to calculate water velocity distributions over elastic elements simulating Canadian waterweed plants (Elodea canadensis), Acta Geophysica, 61, 1, 194-210.
 
10.
Lagrange J.-L., 1781, Mémoire sur la théorie du mouvement des fluides, Nouveaux mémoires de l’Académie royale des sciences et belles-lettres de Berlin, année 1781, also: Oéuvres complétes, tome 4, 695-748.
 
11.
Lamb H., 1916, Hydrodynamics, 4th ed., University Press, Cambridge.
 
12.
Landau L.D., Lifshitz E.M., 1987, Fluid Mechanics, transl. by J.B. Sykes and W.H. Reid, 2nd ed., Pergamon Press, Oxford-New York.
 
13.
Lighthill J., 2001, Waves in Fluids, Cambridge University Press.
 
14.
Malevich A.E., Mityushev V.V., Adler P.M., 2006, Stokes flow through a channel with wavy walls, Acta Mechanica, 182, 3-4, 151-182.
 
15.
Mei C.C., Stiassnie M., Yue D.K.-P., 2005, Theory and Applications of Ocean Surface Waves, Advanced Series on Ocean Engineering, World Scientific, Singapore, 1136 pp.
 
16.
Mei C.C., Vernescu B., 2010, Homogenization Methods for Multiscale Mechanics, World Scientific Publishers, New Jersey-London-Singapore-Beijing-Shanghai-Hong Kong-Taipei-Chennai.
 
17.
Nappo C.J., 2012, An Introduction to Atmospheric Gravity Waves, 2nd ed. Waltham, Elsevier Academic Press, Massachusetts.
 
18.
Popescu I., 2014, Computational Hydraulics – Numerical Methods and Modelling, IWA Publishing Alliance House, London.
 
19.
Sanchez-Palencia E., 1980, Non-homogeneous Media and Vibration Theory, Springer Verlag, Berlin, Heidelberg, New York.
 
20.
Telega J.J., Bielski W., 2003, Flow in random porous media: effective models, Computers and Geotechnics, 30, 4, 271-288.
 
21.
Wojnar R., Bielski W., 2014, Flow in the canal with plants on the bottom, [In:] Complex Analysis and Potential Theory with Applications, T. Aliev Azerogly, A. Golberg, S.V. Rogosin (Edit.), Cambridge Scientific Publishers, 167-183.
 
eISSN:2543-6309
ISSN:1429-2955