Skip to main content
SpringerLink
Log in

Wave scattering by circular are shaped plates

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

Abstract

This work investigates the reflection and transmission properties of a circular are plate which is submerged in deep water. The purpose is to compare the reflective properties of a circular arc plate with those for a submerged, circular cylinder in order to assess the suitability of using circular arc plates when constructing a water wave lens. Linear theory is assumed and two separate techniques are used to determine the wave field. The first involves expanding the potential as a series of multipole potentials outside a circular region and a series of nonsingular solutions of Laplace's equation within the region and matching the expansions on the boundary. The second technique is based on a variational procedure and is used to derive an explicit, approximate expression for the reflection coefficient, under the assumption that the plate is short compared with the other length scales in the problem. Results are presented which compare the approximate solution with the full numerical method for a variety of plates. Finally, the full numerical calculations of the reflection and transmission coefficients for a plate are compared with those for a submerged, circular cylinder.

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. E. Mehlum and J. J. Stamnes, On the focusing of ocean swells and its significance in Power Production,Central Institute for Industrial Research, Blindern, Oslo, SI Rep 78 04 08-3 (1978).

  2. S. Murashige and T. Kinoshita, An ideal ocean wave focusing lens and its shape.Applied Ocean Research, Vol. 14, No. 5, (1992) pp 275–290.

    Google Scholar 

  3. E. Mehlum, A circular cylinder in water waves,Applied Ocean Research, Vol. 2, No. 4, (1980) pp. 171–177.

    Google Scholar 

  4. W. R. Dean, On the reflexion of surface waves by a submerged circular cylinder,Proc. Camb. Phil. Soc., Vol. 44, (1948) pp. 483–491.

    Google Scholar 

  5. M. McIver, Diffraction of water waves by a moored, horizontal flat plate,J. Engineering Maths, Vol. 19, (1985) pp 297–319.

    Google Scholar 

  6. F. Ursell, The effect of a fixed, vertical barrier on surface waves in deep water,Proc. Camb. Phil. Soc., Vol. 43, (1947) pp. 374–382.

    Google Scholar 

  7. F. John, Waves in the presence of an inclined barrier,Comm. Pure. Appl. Maths., Vol. 1, (1948) pp. 149–200

    Google Scholar 

  8. D. V. Evans, Diffraction of water waves by a submerged vertical plate.J. Fluid Mechanics, Vol. 40, (1970) pp 433–451.

    Google Scholar 

  9. D. V. Evans and C. A. N. Morris, The effect of a fixed vertical barrier on obliquely incident surface waves in deep water,J. Inst. Maths Applics., Vol. 9, (1972) pp. 198–204.

    Google Scholar 

  10. D. Porter, The radiation and scattering of surface waves by vertical barriers,J. Fluid Mechanics, Vol. 63, Part 4, (1974) pp. 625–634.

    Google Scholar 

  11. M. Patarapanich, Maximum and zero reflection from a submerged plate,J. Waterway, Port, Coastal and Ocean Engineering, Vol. 110, No. 2, (1984) pp. 171–181.

    Google Scholar 

  12. J. E. Burke, Scattering of surface waves on an infinitely deep fluid,J. Mathematical Physics, Vol. 5, (1964) pp 805–819.

    Google Scholar 

  13. A. E. Heins, Water waves over a channel of finite depth with a submerged plane barrier,Canadian J. Maths, Vol. 2, (1950) pp 210–222

    Google Scholar 

  14. N. F. Parsons and P.A. Martin, Scattering of water waves by submerged plates using hypersingular integral equations,Applied Ocean Research, Vol. 14, (1992) pp. 313–321.

    Google Scholar 

  15. N. F. Parsons and P. A. Martin, Scattering of water waves by submerged curved plates and by surface-piercing flat plates,Applied Ocean Research, Vol 16, (1994) pp. 129–139

    Google Scholar 

  16. F. Ursell, Surface waves on deep water in the presence of a submerged cylinder. Part I,Proc. Camb. Phil. Soc., Vol. 46, (1950) pp. 141–152.

    Google Scholar 

  17. C. C. Mei and J. L. Black, Scattering of surface waves by rectangular obstacles in water of finite depth,J. Fluid Mechanics, Vol. 38, Part 3, (1969) pp. 499–511.

    Google Scholar 

  18. A. Norris and Y. Yang, Dynamic stress on a partially bonded fiber,J. Appl. Mech., Vol. 58, (1991) pp. 404–409.

    Google Scholar 

  19. J. W. Miles, Surface-wave scattering matrix for a shelf,J. Fluid Mechanics, Vol. 28, Part 4, (1967) pp. 755–767.

    Google Scholar 

  20. C. C. Mei and R. P. Petroni, Waves in a harbor with protruding breakwaters.J. Waterways, Harbors, Coastal Eng. Proc. ASCE, Vol. 99, (1973) pp 209–229.

    Google Scholar 

  21. I. S. Gradshteyn and I. M. Ryzhik,Tables of Integrals, Series and Products, Academic Press, New York (1980).

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematical Sciences, Loughborough University of Technology, LE11 3TU, Loughborough, Leicestershire, UK

    M. Mciver & U. Urka

Authors
  1. M. Mciver
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. U. Urka
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Mciver, M., Urka, U. Wave scattering by circular are shaped plates. J Eng Math 29, 575–589 (1995). https://doi.org/10.1007/BF00044123

Download citation

  • Received:

  • Accepted:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF00044123

Keywords

Search

Navigation