Skip to main content
Log in

Asymptotic reflection of linear water waves by submerged horizontal porous plates

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

Abstract

On the basis of linear water-wave theory, an explicit expression is presented for the reflection coefficient R when a plane wave is obliquely incident upon a semi-infinite porous plate in water of finite depth. The expression, which correctly models the singularity in velocity at the edge of the plate, does not rely on knowledge of any of the complex-valued eigenvalues or corresponding vertical eigenfunctions in the region occupied by the plate. The solution R is the asymptotic limit of the reflection coefficient R as a → ∞, for a plate of finite length a bounded by a rigid vertical wall, and forms the basis of a rapidly convergent expansion for R over a wide range of values of a. The special case of normal incidence is relevant to the design of submerged wave absorbers in a narrow wave tank. Modifications necessary to account for a finite submerged porous plate in a fluid extending to infinity in both horizontal directions are discussed.

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

Access this article

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

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

Instant access to the full article PDF.

Similar content being viewed by others

Explore related subjects

Discover the latest articles and news from researchers in related subjects, suggested using machine learning.

References

  1. Wu J, Wan Z, Fang Y (1998) Wave reflection by a vertical wall with a horizontal submerged porous plate. Ocean Eng 25(9): 767–779

    Article  Google Scholar 

  2. Chwang AT, Wu J (1994) Wave scattering by submerged porous disk. J Eng Mech 120(12): 2575–2587

    Article  Google Scholar 

  3. Hassan M, Meylan MH, Peter MA (2009) Water-wave scattering by submerged elastic plates. Q J Mech Appl Math 62(3): 321–344

    Article  MATH  Google Scholar 

  4. Porter R, Evans DV (1995) Complementary approximations to wave scattering by vertical barriers. J Fluid Mech 294: 155–180

    Article  MATH  MathSciNet  ADS  Google Scholar 

  5. Linton CM, McIver P (2001) Handbook of mathematical techniques for wave structure interactions. Chapman & Hall/CRC, Boca Raton

    Book  MATH  Google Scholar 

  6. McIver P (1998) The dispersion relation and eigenfunction expansions for water waves in a porous structure. J Eng Math 34: 319–334

    Article  MATH  MathSciNet  Google Scholar 

  7. Heins AE (1950) Water waves over a channel of finite depth with a submerged plane barrier. Can J Math 2: 210–222

    Article  MATH  MathSciNet  Google Scholar 

  8. Linton CM, Evans DV (1991) Trapped modes above a submerged horizontal plate. Q J Mech Appl Math 44(3): 487–506

    Article  MATH  MathSciNet  Google Scholar 

  9. Evans DV, Davies TV (1968) Wave–ice interaction. Technical report 1313, Davidson Laboratory, Stevens Institute of Technology, New Jersey

  10. Balmforth NJ (1999) Craster RV Ocean waves and ice sheets. J Fluid Mech 395: 89–124

    Article  MATH  MathSciNet  ADS  Google Scholar 

  11. Tkacheva LA (2004) The diffraction of surface waves by a floating elastic plate at oblique incidence. J Appl Math Mech 68(3): 425–436

    Article  MathSciNet  Google Scholar 

  12. Chung H, Fox C (2002) Calculation of wave–ice interaction using the Wiener–Hopf technique. N Z J Math 31: 1–18

    MathSciNet  ADS  Google Scholar 

  13. Linton CM, Chung H (2003) Reflection and transmission at the ocean/sea-ice boundary. Wave Motion 38(1): 43–52

    Article  MATH  MathSciNet  Google Scholar 

  14. Evans DV, Peter MA (2009) Reflection of water waves by a submerged horizontal porous plate. In: Korobkin A, Plotnikov P (eds), Proceedings of 24th international workshop on water waves and floating bodies, Zelenogorsk, Russia, pp 82–85

  15. Chwang AT (1983) A porous-wavemaker theory. J Fluid Mech 132: 395–406

    Article  MATH  ADS  Google Scholar 

  16. Gradshteyn, IS, Ryzhik, IM, Jeffrey, A, Zwillinger, D (eds) (2000) Table of integrals, series, and products, 6 edn. Academic Press, New York

    MATH  Google Scholar 

  17. Cho IH, Kim MH (2008) Development of wave absorbing system using an inclined porous plate. In: Choi HS, Kim Y (eds) Proceedings of 23rd international workshop on water waves and floating bodies, Jeju, Korea, pp 25–28

  18. Mittra R, Lee SW (1971) Analytical techniques in the theory of guided waves. Macmillan, New York

    MATH  Google Scholar 

  19. Evans DV (1992) Trapped acoustic modes. IMA J Appl Math 49: 45–60

    Article  MathSciNet  Google Scholar 

  20. Meylan MH, Gross L (2003) A parallel algorithm to find the zeros of a complex analytic function. ANZIAM J 44(E): E216–E234

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Malte A. Peter.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Evans, D.V., Peter, M.A. Asymptotic reflection of linear water waves by submerged horizontal porous plates. J Eng Math 69, 135–154 (2011). https://doi.org/10.1007/s10665-009-9355-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10665-009-9355-2

Keywords