Boundary value problem of calculating ray characteristics of ocean waves reflected from coastline

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Abstract

A direct variational method for solving the problem of reflection of ocean waves ray paths from the coastline with given positions of the source and the point of registration is considered. It is shown that the original boundary value problem can be reduced to the direct search of stationary points of the functional. Information about the objective function in the area of solutions of the ray problem allows us to construct a systematic procedure for finding minima and saddle points. A feature of the proposed approach is the optimization of the ray reflection point along a given coastline.

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About the authors

I. A. Nosikov

Demidov Yaroslavl State University

Author for correspondence.
Email: ianosikov@wdizmiran.ru
Russian Federation, ul. Sovetskaya, 14, Yaroslavl, 150003

A. A. Tolchennikov

Insitute of Applied Mechanics, Russian Academy of Sciences

Email: tolchennikovaa@gmail.com
Russian Federation, 101 Vernadskogo Ave., Moscow, 117526

M. V. Klimenko

West Department of the Pushkov Institute of Terrestrial Magnetism, Ionosphere and Radio Wave Propagation; St. Petersburg State University

Email: mvklimenko@wdizmiran.ru
Russian Federation, Pionerskaya str., 61, Kaliningrad, 236035; University Embankment, 7-9, St. Petersburg, 1199034

References

  1. Dobrokhotov S.Y., Sekerzh-Zenkovich S.Y., Tirozzi B., Tudorovskii T.Y. Description of tsunami propagation based on the Maslov canonical operator // Dokl. Math. 2006. V. 74. No 1. P. 592–596.
  2. Dobrokhotov S.Y., Shafarevich A.I., Tirozzi B. Localized wave and vortical solutions to linear hyperbolic systems and their application to linear shallow water equations // Russ. J. Math. Phys. 2008. V. 15. No 2. P. 192–221.
  3. Dobrokhotov S.Y., Nazaikinskii V.E. Punctured Lagrangian manifolds and asymptotic solutions of the linear water wave equations with localized initial conditions // Math. Notes. 2017. V. 101. No 5. P. 1053–1060.
  4. Dobrokhotov S.Y., Minenkov D.S., Nazaikinskii V. E. Asymptotic solutions of the Cauchy problem for the nonlinear shallow water equations in a basin with a gently sloping beach // Russ. J. Math. Phys. 2022. V. 29. No 1. P. 28–36.
  5. Казанцев А.Н., Лукин Д.С., Спиридонов Ю.Г. Метод исследования распространения радиоволн в неоднородной магнитоактивной ионосфере // Косм. исследования. 1967. Т. 5. №. 4. С. 593–600.
  6. Пелиновский Е.Н. Гидродинамика волн цунами. Н. Новгород: ИПФ РАН, 1996. 276 с.
  7. Марчук А.Г., Чубаров Л.Б., Шокин Ю.И. Численное моделирование волн цунами. М.: Наука, 1983. 175 c.
  8. Доброхотов С.Ю., Клименко М.В., Носиков И.А., Толченников А.А. Вариационный метод расчета лучевых траекторий и фронтов волн цунами, порожденных локализованным источником // Ж. вычисл. матем. и матем.физ. 2020. Т. 60. №. 8. С. 1439–1448.
  9. Koketsu K., Sekine S. Pseudo-bending method for three-dimensional seismic ray tracing in a spherical earth with discontinuities // Geophys. J. Int. 1998. V. 132. No 2. С. 339–346.
  10. Rawlinson N., Hauser J., Sambridge M. Seismic ray tracing and wavefront tracking in laterally heterogeneous media // Adv. geophys. 2008. V. 49. P. 203–273.
  11. Nosikov I.A., Klimenko M.V., Zhbankov G.A., Podlesnyi A.V., Ivanova V.A., Bessarab P.F. Generalized force approach to point-to-point ionospheric ray tracing and systematic identification of high and low rays // IEEE Trans. Antennas Propag. 2019. V. 68. No 1. P. 455–467.
  12. Jónsson H., Mills G., Jacobsen K.W. Nudged elastic band method for finding minimum energy paths of transitions. В сб. Classical and Quantum Dynamics in Condensed Phase Simulations. World Scientific, 1998. P. 385–404.
  13. Носиков И.А., Клименко М.В., Бессараб П.Ф. Применение метода поперечных смещений для расчета коротковолновых радиотрасс. Постановка задачи и предварительные результаты // Известия вузов. Радиофиз. 2016. Т. 59. №. 1. С. 1–14.
  14. Доброхотов С.Ю., Назайкинский В.Е. Характеристики с особенностями и граничные значения асимптотического решения задачи Коши для вырождающегося волнового уравнения // Матем. заметки. 2016. Т. 100. №. 5. С. 710–731.
  15. Носиков И.А., Клименко М.В. Исследование функционала верхних и нижних лучей в задаче расчета радиотрасс в модельной ионосфере // Хим. физика. 2017. Т. 36. № 12. С. 61–65.
  16. Dobrokhotov S.Y., Nazaikinskii V.E., Tirozzi B. Two-dimensional wave equation with degeneration on the curvilinear boundary of the domain and asymptotic solutions with localized initial data // Russ. J. Math. Phys. 2013. V. 20. P. 389–401.
  17. Назайкинский В.Е. Канонический оператор Маслова на лагранжевых многообразиях в фазовом пространстве, соответствующем вырождающемуся на границе волновому уравнению // Матем. заметки. 2014. Т. 96. №. 2. С. 261–276.
  18. Dobrokhotov S.Y., Nazaikinskii V.E., Tolchennikov A.A. Uniform asymptotics of the boundary values of the solution in a linear problem on the run-up of waves on a shallow beach // Math. Notes. 2017. V. 101. P. 802–814.
  19. Аникин, А.Ю., Доброхотов, С.Ю., Назайкинский, В.Е., Руло, М. Лагранжевы многообразия и конструкция асимптотик для (псевдо) дифференциальных уравнений с локализованными правыми частями // Теор. и матем. физ. 2023. Т. 214. №. 1. С. 3–29.

Supplementary files

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2. Fig. 1. a – Family of ray trajectories (green lines) – solutions of the Hamiltonian system (1), reflected from the coastline; b – ocean wave fronts obtained on the basis of ray trajectories for moments of time –  with a step . The source is located at the point with coordinates . The initial radiation angles are specified in the range  with a step of π/10. The wave propagation speed is , where . The coast is shown as a horizontal gray line.

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3. Fig. 2.a Comparison of rays 1, 2 and 3 obtained from analytical expressions [14] (solid lines) and calculated by the variational method (circles). The set of virtual rays used in the express analysis procedure is represented by thin grey lines. The coast is represented as a horizontal grey line; b dependence of the propagation time on the position of the reflection point of the virtual rays. Local minima and maxima are designated by numbers 1, 2 and 3.

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4. Fig. 3.a Rays with one, two, three reflections from the shore. Solutions obtained by the variational method are shown as circles. Analytical solutions are shown as solid lines.

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5. Fig. 4. Map of the dependence of the propagation time along virtual rays on the positions of two reflection points  and . Stationary points correspond to rays 1 – 7 from Fig. 3a, b.

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6. Fig. 5. Results of calculations of ray trajectories between points  and  with one (a) and two (b) reflection points in a round pool with a parabolic bottom. Solutions obtained by the variational method are shown by circles. Ray trajectories calculated by the bicharacteristic method are shown by solid lines. (c) Wave fronts reconstructed on the basis of ray calculations by the bicharacteristic method for moments of time —  with a step . The color scale corresponds to the distribution of the reservoir depth function . The shore, where the condition is met, is shown by a solid black curve.

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