TY - JOUR

T1 - On frequency–amplitude dependences for surface and internal standing waves

AU - Lizárraga-Celaya, Carlos

AU - Shingareva, Inna

N1 - Funding Information:
This work was partly supported by SEP-CONACYT, México (the Grant No. 41421-F). We would also like to thank the referees for their helpful comments and referring us to important works.

PY - 2007/3/15

Y1 - 2007/3/15

N2 - The analytic approach proposed by Sekerzh-Zenkovich [On the theory of standing waves of finite amplitude, Dokl. Akad. Nauk USSR 58 (1947) 551–554] is developed in the present study of standing waves. Generalizing the solution method, a set of standing wave problems are solved, namely, the infinite- and finite-depth surface standing waves and the infinite- and finite-depth internal standing waves. Two-dimensional wave motion of an irrotational incompressible fluid in a rectangular domain is considered to study weakly nonlinear surface and internal standing waves. The Lagrangian formulation of the problems is used and the fifth-order perturbation solutions are determined. Since most of the approximate analytic solutions to these problems were obtained using the Eulerian formulation, the comparison of the results, as an example the analytic frequency–amplitude dependences, obtained in Lagrangian variables with the corresponding ones known in Eulerian variables has been carried out in the paper. The analytic frequency–amplitude dependences are in complete agreement with previous results known in the literature. Computer algebra procedures were written for the construction of asymptotic solutions. The application of the model constructed in Lagrangian formulation to a set of different problems shows the ability to correctly reproduce and predict a wide range of situations with different characteristics and some advantages of Lagrangian particle models (for example, the bigger radius of convergence of an expansion parameter than in Eulerian variables, simplification of the boundary conditions, parametrization of a free boundary).

AB - The analytic approach proposed by Sekerzh-Zenkovich [On the theory of standing waves of finite amplitude, Dokl. Akad. Nauk USSR 58 (1947) 551–554] is developed in the present study of standing waves. Generalizing the solution method, a set of standing wave problems are solved, namely, the infinite- and finite-depth surface standing waves and the infinite- and finite-depth internal standing waves. Two-dimensional wave motion of an irrotational incompressible fluid in a rectangular domain is considered to study weakly nonlinear surface and internal standing waves. The Lagrangian formulation of the problems is used and the fifth-order perturbation solutions are determined. Since most of the approximate analytic solutions to these problems were obtained using the Eulerian formulation, the comparison of the results, as an example the analytic frequency–amplitude dependences, obtained in Lagrangian variables with the corresponding ones known in Eulerian variables has been carried out in the paper. The analytic frequency–amplitude dependences are in complete agreement with previous results known in the literature. Computer algebra procedures were written for the construction of asymptotic solutions. The application of the model constructed in Lagrangian formulation to a set of different problems shows the ability to correctly reproduce and predict a wide range of situations with different characteristics and some advantages of Lagrangian particle models (for example, the bigger radius of convergence of an expansion parameter than in Eulerian variables, simplification of the boundary conditions, parametrization of a free boundary).

KW - Nonlinear waves

KW - Surface standing waves

KW - Internal standing waves

KW - Perturbation methods

KW - Computer algebra methods

UR - http://www.scopus.com/inward/record.url?scp=33846074410&partnerID=8YFLogxK

U2 - https://doi.org/10.1016/j.cam.2006.01.002

DO - https://doi.org/10.1016/j.cam.2006.01.002

M3 - Article

SN - 0377-0427

VL - 200

SP - 459

EP - 470

JO - Journal of Computational and Applied Mathematics

JF - Journal of Computational and Applied Mathematics

IS - 2

ER -