Skip to main content
SpringerLink
Log in

Scattering of a surface wave by a submerged sphere

  • Published:
Journal of Engineering Mathematics Aims and scope Submit manuscript

Summary

The problem of the scattering of a surface wave in a nonviscous, incompressible fluid of infinite depth by a fully submerged, rigid, stationary sphere has been reduced to the solution of an infinite set of linear algebraic equations for the expansion coefficients in spherical harmonics of the velocity potential. These equations are easily solved numerically, so long as the sphere is not too close to the surface. The approach has been to formulate the problem as an integral equation, expand the Green's function, the velocity potential of the incident wave, and the total velocity potential in spherical harmonics, impose the boundary condition at the surface of the sphere, and carry out the integrations. The scattering cross section has been evaluated numerically and is shown to peak for values of the product of radius and wave number somewhat less than unity. Also, the Born approximation to the cross section is obtained in closed form.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Sir W. Thomson, On stationary waves in flowing water, Phil. Mag. (5) 22 (1886) 353–357, 445–452, 517–530 and 23 (1887) 52–58.

    Google Scholar 

  2. Sir Horace Lamb, On some cases of wave-motion in deep water, Annali di Matematica (3) 21 (1913) 237–250.

    Google Scholar 

  3. N. E. Kochin, The theory of waves generated by oscillations of a body under the free surface of a heavy, incompressible fluid, Uchenye Zapiski Moskov. Gos. Univ., 46 (1940) 85–106. Transl. in Soc. Nav. Arch. Marine Eng., Tech. Res. Bull. No. 1–10 (1952).

    Google Scholar 

  4. W. R. Dean, On the reflection of surface waves by a submerged plane barrier, Proc. Camb. Phil. Soc. 41 (1945) 231–238.

    Google Scholar 

  5. W. R. Dean, On the reflexion of a surface wave by a submerged cylindrical barrier, Proc. Camb. Phil. Soc. 44 (1948) 483–491.

    Google Scholar 

  6. Fritz John, On the motion of floating bodies II. Simple harmonic motion, Comm. Pure App. Math., 3 (1950) 45–101.

    Google Scholar 

  7. Sir Thomas Havelock, Waves due to a sphere making periodic heaving oscillations, Proc. Roy. Soc. London, Ser. A 231 (1955) 1–7.

    Google Scholar 

  8. A. J. Hermans, A perturbation method for the radiation of surface waves, J. Eng. Math., 6 (1972) 323–330.

    Google Scholar 

  9. A. J. Hermans, A perturbation method for the radiation of short surface waves in three dimensions, J. Eng. Math., 7 (1973) 75–84.

    Google Scholar 

  10. F. G. Leppington, On the radiation and scattering of short surface waves, I–III, J. Fluid Mech., 56 (1972) 101–119; 59 (1973) 129–146, 147–157.

    Google Scholar 

  11. G. Alker, On the radiation of short surface waves by a heaving, circular cylinder, J. Eng. Math., 9 (1975) 197–205.

    Google Scholar 

  12. C. A. N. Morris, A variational approach to an unsymmetric water-wave scattering problem, J. Eng. Math., 9 (1975) 291–300.

    Google Scholar 

  13. John V. Wehausen and Edmund V. Laitone, Surface waves, in: Handbuch der Physik, Vol. 9, Pt. 3, Fluid Dynamics, pp. 445–778, Ed. S. Flugge, (1960).

  14. See for example L. S. Rodberg and R. M. Thaler, The Quantum Theory of Scattering, Academic Press (New York and London), 1967. The theorem is so called because of the analogy with light passing through a medium. The imaginary part of the index of refraction is related to the absorption coefficient, i.e., the total absorption cross section. See L. Rosenfeld, Theory of Electrons, North-Holland Publishing Co. (Amsterdam), 1951.

    Google Scholar 

  15. See for example H. C. van der Hulst, Light Scattering by Small Particles, Wiley (New York) 1957.

    Google Scholar 

  16. Philip M. Morse and Herman Feshbach, Methods of Theoretical Physics, McGraw-Hill, New York (1953) p. 887.

    Google Scholar 

  17. Philip M. Morse and Herman Feshbach, Methods of Theoretical Physics, McGraw-Hill, New York (1953), p. 1466.

    Google Scholar 

  18. A. Erdélyi, Higher Transcendental Functions, McGraw-Hill, New York, Vol. 1 (1953) p. 148.

    Google Scholar 

  19. A. Erdélyi, Higher Transcendental Functions, McGraw-Hill, New York, Vol. 1 (1953), p. 164.

    Google Scholar 

  20. Philip M. Morse and Herman Feshbach, Methods of Theoretical Physics, McGraw-Hill, New York (1953), p. 1573.

    Google Scholar 

  21. E. O. Steinborn and K. Ruedenberg, Rotation and translation of regular and irregular solid spherical harmonics, in: Advances in Quantum Chemistry, Academic Press, New York and London, Vol. 7 (1973) 1–81.

    Google Scholar 

  22. Philip M. Morse and Herman Feshbach, Methods of Theoretical Physics, McGraw-Hill, New York (1953), p. 1467; also A. Erdélyi, loc. cit., Vol. 2, p. 57.

    Google Scholar 

  23. Eldon R. Hansen, A Table of Series and Products, p. 449 (74.2.24), Prentice-Hall (Englewood, N. J.) 1975; also G. N. Watson, Theory of Bessel Functions, Second ed., p. 22, § 2.22(3), Cambridge University Press (London) 1962.

    Google Scholar 

  24. G. N. Watson, loc. cit., p. 79, § 3.71 (9).

  25. Ibid., p. 373, § 12.11 (1).

  26. A. M. J. Davis, Short surface waves due to an oscillating half-immersed sphere, Mathematika, 18 (1971) 20–39.

    Google Scholar 

  27. A. M. J. Davis, Short surface waves in the presence of a submerged sphere, SIAM J. Appl. Math., 27 (1974) 464–478.

    Google Scholar 

  28. A. M. J. Davis, Short surface waves in the presence of a submerged circular cylinder, SIAM J. Appl. Math., 27 (1974) 479–491.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Applied Physics Laboratory, The Johns Hopkins University, 20810, Laurel, Maryland, U.S.A.

    E. P. Gray

Authors
  1. E. P. Gray
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Supported by the Department of the Navy, Naval Sea Systems Command under Contract No. N00017-72-C-4401.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Gray, E.P. Scattering of a surface wave by a submerged sphere. J Eng Math 12, 15–41 (1978). https://doi.org/10.1007/BF00042802

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF00042802

Keywords

Search

Navigation