Hostname: page-component-f554764f5-wjqwx Total loading time: 0 Render date: 2025-04-12T01:41:13.162Z Has data issue: false hasContentIssue false
  • English
  • Français

Trapping of water waves by submerged plates using hypersingular integral equations

Published online by Cambridge University Press:  26 April 2006

Neil F. Parsons  and
P. A. Martin
Neil F. Parsons
Affiliation:
Department of Mathematics, University of Manchester, Manchester M13 9PL, UK
P. A. Martin
Affiliation:
Department of Mathematics, University of Manchester, Manchester M13 9PL, UK
Get access Rights & Permissions [Opens in a new window]

Abstract

The trapping of surface water waves by a thin plate in deep water is reduced to finding non-trivial solutions of a homogeneous, hypersingular integral equation for the discontinuity in velocity potential across the plate. The integral equation is discretized using an expansion-collocation method, involving Chebyshev polynomials of the second kind. A non-trivial solution to the problem is given by the vanishing of the determinant inherent in such a method. Results are given for inclined flat plates, and for curved plates that are symmetric with respect to a line drawn vertically through their centre. Comparisons with published results for horizontal flat plates (in water of finite depth) and for circular cylinders are made.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 284 , 10 February 1995 , pp. 359 - 375
Copyright
© 1995 Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

Abramowitz, M. & Stegun, I. A. (EDS.) 1965 Handbook of Mathematical Functions. Dover.
Apelblat, A. & Kravitsky, N. 1985 Integral representations of derivatives and integrals with respect to order of the Bessel functions Jv(t), Iv(t) the Anger function Jv(t) and the integral Bessel function Jiv(t). IMA J. Appl. Maths 34, 187–210.Google Scholar
Bonnet-Ben Dhia, A.-S. & Joly, P. 1993 Mathematical analysis of guided water waves. SIAM J. Appl. Maths 53, 1507–1550.Google Scholar
Golberg, M. A. 1983 The convergence of several algorithms for solving integral equations with finite-part integrals. J. Integ. Equat. 5, 329–340.Google Scholar
Golberg, M. A. 1985 The convergence of several algorithms for solving integral equations with finite-part integrals. II. J. Integ. Equat. 9, 267–275.Google Scholar
Greene, T. R. & Heins, A. E. 1953 Water waves over a channel of infinite depth. Q. Appl. Maths 11, 201–214.Google Scholar
Jones, D. S. 1953 The eigenvalues of ∇2u + λu = 0 when the boundary conditions are given in semi-infinite domains. Proc. Camb. Phil. Soc. 49, 668–684.Google Scholar
LeBlond, P. H. & Mysak, L. A. 1978 Waves in the Ocean. Elsevier.
Lewin, L. 1958 Dilogarithms and Associated Functions. Macdonald.
Linton, C. M. & Evans, D. V. 1991 Trapped modes above a submerged horizontal plate. Q. J. Mech. Appl. Maths 44, 487–506.Google Scholar
McIver, P. & Evans, D. V. 1985 The trapping of surface waves above a submerged, horizontal cylinder. J. Fluid Mech. 151, 243–255.Google Scholar
Martin, P. A. 1989 On the computation and excitation of trapping modes. Proc. 4th Intl Workshop on Water Waves and Floating Bodies, Øystese, Norway (ed. J. Grue), pp. 145–147. University of Oslo.
Martin, P. A. 1994 Asymptotic approximations for functions defined by series, with some applications to the theory of guided waves. Preprint.
Martin, P. A. & Rizzo, F. J. 1989 On boundary integral equations for crack problems. Proc. R. Soc. Lond. A 421, 341–355.Google Scholar
Meyer, R. E. 1971 Resonance of unbounded water bodies. In Mathematical Problems in Geophysical Sciences, vol. 1 (ed. W. H. Reid), pp. 189–227. American Mathematical Society, Providence.
Parsons, N. F. 1994 The interaction of water waves with thin plates. PhD thesis. University of Manchester, UK.
Parsons, N. F. & Martin, P. A. 1992 Scattering of water waves by submerged plates using hypersingular integral equations. Appl. Ocean Res. 14, 313–321.Google Scholar
Parsons, N. F. & Martin, P. A. 1994 Scattering of water waves by submerged curved plates and by surface-piercing flat plates. Appl. Ocean Res. 16, 129–139.Google Scholar
Ursell, F. 1951 Trapping modes in the theory of surface waves. Proc. Camb. Phil. Soc. 47, 347–358.Google Scholar
Ursell, F. 1952 Edge waves on a sloping beach. Proc. R. Soc. Lond. A 214, 79–97.Google Scholar
Ursell, F. 1962 Slender oscillating ships at zero forward speed. J. Fluid Mech. 14, 496–516.Google Scholar
Ursell, F. 1968 The expansion of water-wave potentials at great distances. Proc. Camb. Phil. Soc. 64, 811–826.Google Scholar
Wimp, J. 1984 Computation with Recurrence Relations. Pitman.