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Multiple Scattering of Water Waves by N Floating Bodies Using Boundary Integral Equation Methods

  • Lahcene Bencheikh

Abstract

The use of boundary integral equation methods for solving exterior problems of scattering of time harmonic waves leads to a problem of nonuniqueness of the solutions of the relevant integral equations. This difficulty is inherent to the methods used. Various methods have been devised to overcome this problem. One of these methods consists in modifying the fundamental solution by adding to it a series of multipoles. With a mild condition on the coefficients of these multipoles, the problem of nonuniqueness is removed. In the context of water waves, Ursell [1,2] was the first to propose the modification of the fundamental solution for problems involving one floating cylindrical body. He proved uniqueness for the high frequencies. Sayer [3] did the same for the low frequencies. Later, Ursell [4] generalized the result for all frequencies. Martin [5] considered the case of one three-dimensional floating body. Later, Martin [6] treated the problem for two infinitely long floating cylinders. More recently, we extended Martin’s work [6] to the case of two three-dimensional floating bodies [7]. Here, we generalize the case to N three-dimensional floating bodies.We found out that the conditions on some of the coefficients of the multipoles depend on the number, N, of the scatterers. It is worth noting that Martin [8] treated the case of N scattering bodies for three-dimensional acoustic waves. He found out that the conditions on all the coefficients depended on N.

Keywords

Integral Equation Fundamental Solution Multiple Scattering Water Wave Boundary Integral Equation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    F. Ursell, Short surface waves due to an oscillating immersed body, Proc. Roy. Soc. London A 220 (1953), 90–103.MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    F. Ursell, The transmission of surface waves under surface obstacles, Proc. Cam. Phil. Soc. 57 (1961), 638–668.MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    P. Sayer, An integral equation method for determining the fluid motion due to a cylinder heaving on water of finite depth, Proc. Roy. Soc. London A 372 (1980), 93–110.MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    F. Ursell, Irregular frequencies and the motion of floating bodies, J.Fluid Mech. 105 (1981), 143–156.MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    P.A. Martin, On the null-field equations for water wave radiation problems, J. Fluid Mech. 113 (1981), 315–332.MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    P.A. Martin, Integral equation methods for multiple scattering problems. II. Water waves, Q. Jl. Mech. Appl. Math. 38 Pt.1, (1985), 119–133.CrossRefMATHGoogle Scholar
  7. 7.
    M. Sidi and L. Bencheikh, Uniqueness for a problem of multiple scattering of water waves, Proc. Roy. Soc. London A (submitted for publication).Google Scholar
  8. 8.
    P.A. Martin, Multiple scattering and modified Green’s functions, J. Math. Anal. Appl. (in press).Google Scholar
  9. 9.
    F. John, On the motion of floating bodies II, Comm. Pure Appl. Math. 3 (1950), 45–101.MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2004

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  • Lahcene Bencheikh

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