Linear waves in shallow water over an uneven bottom, slowing down near the shore

Cover Page

Cite item

Full Text

Open Access Open Access
Restricted Access Access granted
Restricted Access Subscription Access

Abstract

The exact solutions of the system of equations of the linear theory of shallow water are discussed, representing traveling waves with specific properties for the time propagation, which is infinite when approaching the shore and finite when leaving for deep water. These solutions are obtained by reducing one-dimensional shallow water equations to the Euler–Poisson–Darboux equation with a negative integer coefficient before the lower derivative. The analysis of the wave field dynamics is carried out. It is shown that the shape of a wave approaching the shore will be differentiated a certain number of times, which is illustrated by a number of examples. When a wave moves away from the shore, its profile is integrated. The solutions obtained in the framework of linear theory are valid only for a finite interval of depth variation.

Full Text

Restricted Access

About the authors

I. E. Melnikov

Национальный исследовательский университет “Высшая школа экономики”; Институт прикладной физики им. А. В. Гапонова-Грехова РАН

Email: melnicovioann@gmail.com
Russian Federation, Нижний Новгород; Нижний Новгород

E. N. Pelinovsky

Национальный исследовательский университет “Высшая школа экономики”; Институт прикладной физики им. А. В. Гапонова-Грехова РАН

Author for correspondence.
Email: pelinovsky@appl.sci-nnov.ru
Russian Federation, Нижний Новгород; Нижний Новгород

References

  1. Mei C.C. The Applied Dynamics of Ocean Surface Waves. W.S.: Singapore, 1989. 740 p.
  2. Brekhovskikh L.M. Waves in Layered Media. Cambridge Univ. Press, USA, 1976. 520 p.
  3. Dingemans M.W. Water Wave Propagation over Uneven Bottom. W.S.: Singapore, 1997. 700 p.
  4. Kravtsov Y.A., Orlov Y.I. Geometrical Optics of Inhomogeneous Media. Spring.: N.Y., 1990. 325 p.
  5. Babich V.M., Buldyrev V.S. Asymptotic Methods In Short-Wavelength Diffraction Theory. Alpha Sci., 2009. 495 p.
  6. Капцов О.В., Капцов Д.О. Решения некоторых волновых моделей механики // ПММ. 2023. T. 87. № 2. С. 176–185.
  7. Полянин А.Д., Зайцев В.Ф. Справочник по нелинейным уравнениям математической физики: точные решения. М.: Физматлит, 2002. 432 с.
  8. Zaitsev V.F., Polyanin A.D. Exact solutions and transformations of nonlinear heat and wave equations // Dokl. Math. 2001. V. 64. № 3. P. 416–420.
  9. Didenkulova I.I., Pelinovsky E.N., Soomere T. Long surface wave dynamics along a convex bottom // J. Geophys. Res. 2008. V. 114. № C7. 14 p.
  10. Didenkulova I.I., Pelinovsky E.N. Travelling water waves along a quartic bottom profile // Proc. Estonian Acad. Sci. 2010. V. 59. № 2. Р. 166–171. doi: 10.3176/proc.2010.2.16.
  11. Didenkulova I.I., Pelinovsky D.E., Tyugin D.Y., Giniyatullin A.R., Pelinovsky E.N. Travelling long waves in water rectangular channels of variable cross section // Geogr. Environ. and Liv. Syst. 2012. № 5. P. 89–93.
  12. Пелиновский Е.Н., Диденкулова И.И., Шургалина Е.Г. // Динамика волн в каналах переменного сечения. Морской гидр. жур. 2017. № 3. С. 22–31.
  13. Pelinovsky E.N., Kaptsov O.V. Traveling Waves in Shallow Seas of Variable Depths // Symm. 2022. V. 14. № 7. Р. 1448.
  14. Melnikov I.E., Pelinovsky E.N. Euler-Darboux-Poisson Equation in Context of the Traveling Waves in a Strongly Inhomogeneous Media // Math. 2023. V. 11. № 15. Р. 3309. DOI: https://doi.org/10.3390/math11153309.
  15. Эйлер Л. Интегральное исчисление. T. 3. M: ГИФМЛ. 1958. 447 с.
  16. Kaptsov O.V. Equivalence of linear partial differential equations and Euler-Darboux transformations // Comput. Technol. 2007. V. 12. № 4. P. 59–72.
  17. Copson E.T. Partial differential equations. Cambridge Univ. Press, 1975. 292 p.
  18. Didenkulova I.I. New Trends in the Analytical Theory of Long Sea Wave Runup // In Appl. Wave Math. Spring. 2009. P. 265–296.

Supplementary files

Supplementary Files
Action
1. JATS XML
Download

Copyright (c) 2024 Russian Academy of Sciences