Russian Journal of Mathematical Physics

, Volume 26, Issue 1, pp 32–39 | Cite as

Asymptotics of Linear Surface Waves Generated by a Localized Source Moving Along the Bottom of the Basin. I. One-Dimensional Case

  • S. Yu. DobrokhotovEmail author
  • P. N. PetrovEmail author


In the paper, water waves generated by a source moving along the bottom of the basin are considered. The complete system of equations for water waves is very complicated and does not have an exact solution at present. Assuming that the change of the bottom function is slow, the authors of the paper [S. Yu. Dobrokhotov and V. E. Nazaikinskii, RJMP 25 (1), 1–16 (2018)] showed that the linearized system of hydrodynamic equations is reduced to a pseudodifferential equation on the surface of the liquid, and the unknown function in this equation is the elevation of the free surface. Assume that the source of the waves is localized in the vicinity of a point moving along the bottom of the basin at a speed less than the speed of the long waves. Assume also that the time of motion of the source and the horizontal sizes of the source are sufficiently small, and, during the motion, the source changes its shape and also changes its velocity. We show that, under some conditions, the last assumption leads to the occurrence of waves on the surface of the liquid with a wavelength approximately equal to the product of the typical speed of long waves in the vicinity of the place where the source moves and the time of its motion, which exceeds the size of the source. In this way, one of the possible mechanisms for generating the so-called landslide tsunami waves can be described. In this paper, we consider the situation with a single horizontal spatial variable.


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  1. 1.
    S. Tinti, E. Bortolucci, and C. Chiavettieri, “Tsunami Excitation by Submarine Slides in Shallow–Water Approximation,” Pure Appl. Geophysics 158 (4), 759–797 (2001).ADSCrossRefGoogle Scholar
  2. 2.
    E. Pelinovsky and A. Poplavsky, “Simplified Model of Tsunami Generation by Submarine Landslides,” Physics and Chemistry of the Earth 21 (1–2), 13–17 (1996).ADSCrossRefGoogle Scholar
  3. 3.
    J. P. Bardet et al., “Landslide Tsunamis: Recent Findings and Research Directions,” Landslide Tsunamis: Recent Findings and Research Directions, Birkhauser, Basel, 1793–1809 (2003).CrossRefGoogle Scholar
  4. 4.
    I. Didenkulova et al., “Tsunami Waves Generated by Submarine Landslides of Variable Volume: Analytical Solutions for a Basin of Variable Depth,” Natural Hazards and Earth System Sciences 10 (11), 2407–2419 (2010).ADSCrossRefGoogle Scholar
  5. 5.
    I. Didenkulova and E. Pelinovsky, “Analytical Solutions for Tsunami Waves Generated by Submarine Landslides in Narrow Bays and Channels,” Pure Appl. Geophysics 170 (9–10), 1661–1671 (2013).ADSCrossRefGoogle Scholar
  6. 6.
    G. A. Papadopoulos et al., “Numerical Modeling of Sediment Mass Sliding and Tsunami Generation: The Case of February 7, 1963, in Corinth Gulf, Greece,” Marine Geodesy 30 (4), 315–331 (2007).CrossRefGoogle Scholar
  7. 7.
    S. Yu. Dobrokhotov and V. E. Nazaikinskii, “Waves on the Free Surface Described by Linearized Equations of Hydrodynamics with Localized Right–Hand Sides,” Russ. J. Math. Phys. 25 (1), 1–16 (2018).MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    V. P. Maslov, Operator Methods (Nauka, Moscow, 1973); English transl.: Operational Methods (Mir, Moscow, 1976).zbMATHGoogle Scholar
  9. 9.
    V. V. Belov, S. Yu. Dobrokhotov, and T. Ya. Tudorovskiy, “Operator Separation of Variables for Adiabatic Problems in Quantum and Wave Mechanics,” J. Engrg. Math. 55 (1–4), 183–237 (2006).Google Scholar
  10. 10.
    V. P. Maslov, Asymptotic Methods and Perturbation Theory (Fizmatlit, Moscow, 1988) [in Russian].zbMATHGoogle Scholar
  11. 11.
    S. Yu. Dobrokhotov, S. O. Sinitsyn, and B. Tirozzi, “Asymptotics of Localized Solutions of the One–Dimensional Wave Equation with Variable Velocity. I. The Cauchy Problem,” Russ. J. Math. Phys. 14 (1), 28–56 (2007).MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    A. I. Allilueva and A. I. Shafarevich, “Localized Asymptotic Solutions of the Wave Equation with Variable Velocity on the Simplest Graphs,” Russ. J. Math. Phys. 24 (3), 279–289 (2017).MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    S. A. Sergeev, “Asymptotic Solutions of One–Dimensional Linear Evolution Equations for Surface Waves with Account for Surface Tension,” Math. Notes 103 (3–4), 499–504 (2018).MathSciNetCrossRefzbMATHGoogle Scholar

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© Pleiades Publishing, Ltd. 2019

Authors and Affiliations

  1. 1.Ishlinsky Institute for Problems in Mechanics of the Russian Academy of SciencesMoscowRussia
  2. 2.Moscow Institute of Physics and Technology (State University), DolgoprudnyMoscow regionRussia

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