Hostname: page-component-cd9895bd7-q99xh Total loading time: 0 Render date: 2024-12-19T16:04:08.007Z Has data issue: false hasContentIssue false
  • English
  • Français

On the normal incidence of linear waves over a plane incline partially covered by a rigid lid

Published online by Cambridge University Press:  06 March 2009

ULF T. EHRENMARK
ULF T. EHRENMARK*
Affiliation:
Department of Mathematics, University of Reading, Whiteknights, Reading RG6 6AX, UK
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

The effect is examined on infinitesimal standing waves over a plane beach when restricted by the arbitrary placing of a finite rigid (or permeable) lid of length ℓ on the undisturbed surface. A uniformly bounded solution for the potential function is obtained by a Green's function method. The Green's function is derived and manipulated, for subsequent computational expedience, from a previously known solution for the problem of an oscillating line source placed at an arbitrary location in the sector. Applications are made to both the case of plate anchored at the origin and the case of plate anchored some distance at sea (drifted plate problem). In both cases water column potentials and equipotentials are constructed from the numerical solution of a Fredholm equation of the second kind by finite difference discretization. Solutions are further extended to include the logarithmically singular standing wave, combination with which allows the construction of progressing waves. Computation of initially incoming progressing wave envelopes demonstrates the emergence of a partially standing wave pattern shoreward of the plate. There is no difficulty, in principle, to extend the theory to any number of plates, and this is verified by computation for the case of two plates. A new shoreline radiation condition is constructed to allow formulation, in the usual way, of the reflection/transmission problem for the plate, and results are in good qualitative agreement with a similar model on a horizontal plane bed. It is argued that the Green's function constructed here could be used in a number of diverse problems, of this linear nature, where all, or part, of the submerged boundary is that of a plane incline.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 623 , 25 March 2009 , pp. 209 - 240
Copyright
Copyright © Cambridge University Press 2009

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.)

References

REFERENCES

Blondeaux, P. & Vittori, G. 1995 The non-linear excitation of synchronous edge waves by a monochromatic wave normally approaching a plane beach. J. Fluid Mech. 301, 251–268.CrossRefGoogle Scholar
Booij, N. 1983 A note on the accuracy of the mild-slope equation. Coastal Engng 7, 191–203.CrossRefGoogle Scholar
Chung, H. & Linton, C. M. 2005 Reflection and transmission of waves across a gap between two semi-infinite elastic plates on water. Quart. J. Mech. Appl. Math. 58 (1), 1–15.CrossRefGoogle Scholar
Davies, A. G. & Heathershaw, A. D. 1983 Surface-wave propagation over sinusoidally varying topography: theory and observation. Inst. Oceonogr. Sci. Rep. No. 159, 181.Google Scholar
Davies, A. G. & Heathershaw, A. D. 1984 Surface-wave propagation over sinusoidally varying topography. J. Fluid Mech. 144, 419–443.CrossRefGoogle Scholar
Ehrenmark, U. T. 1987 Far-field asymptotics of the two-dimensional linearised sloping beach problem. SIAM J. Appl. Math. 47, 965–981.CrossRefGoogle Scholar
Ehrenmark, U. T. 1989 Overconvergence of the near-field expansion for linearised waves normally incident on a sloping beach. SIAM J. Appl. Math. 49, 109–134.CrossRefGoogle Scholar
Ehrenmark, U. T. 1992 A mixed problem for the bi-harmonic equation in a wedge with application to linear dissipating waves on a plane beach. Quart. J. Mech. Appl. Math. 45 (3), 407–433.CrossRefGoogle Scholar
Ehrenmark, U. T. 1996 Eulerian mean current and Stokes drift under non-breaking waves on a perfect fluid over a plane beach. Fluid Dyn. Res. 18, 117–150.CrossRefGoogle Scholar
Ehrenmark, U. T. 2003 Wave trapping above a plane beach by obstacles partially or totally submerged. J. Fluid Mech. 486, 261–285.CrossRefGoogle Scholar
Friedrichs, K. O. & Lewy, H. 1948 The dock problem. Comm. Pure Appl. Math. 1, 135–148.CrossRefGoogle Scholar
Gradshteyn, I. & Ryzhik, I. 1965 Tables of Integrals Series and Products. Academic.Google Scholar
Heathershaw, A. D. & Davies, A. G. 1985 Resonant wave reflection by transverse bedforms and its relation to beaches and offshore sandbars. Marine Geol. 62, 321–338.CrossRefGoogle Scholar
Heins, A. E. 1948 Water waves over a channel of finited depth with a dock. Am. J. Math. 70, 730–748.CrossRefGoogle Scholar
Hermans, A. J. 2003 Interaction of free-surface waves with a floating dock. J. EngNG Math. 45, 39–53.CrossRefGoogle Scholar
Hough, D. M. 1982 The accurate computation of certain integrals arising in integral equation methods for problems in potential theory. In Treatment of Integral Equations by Numerical Methods (ed. Baker, C. T. H. & Miller, G. F.), pp. 209–220. Academic.Google Scholar
Lamb, H. 1993 Hydrodynamics, 6th ed. Cambridge Mathematical Society.Google Scholar
Linton, C. M. 2001 The finite dock problem. ZAMP 52, 640–656.Google Scholar
Linton, C. M. & Chung, H. 2003 Reflection and transmission at the ocean/sea–ice boundary. Wave Motion 38, 43–52.CrossRefGoogle Scholar
Miles, J. 1990 Wave reflection from a gently sloping beach. J. Fluid Mech. 214, 59–66.CrossRefGoogle Scholar
Minzoni, A. A. & Whitham, G. B. 1977 On the excitation of edge waves on beaches. J. Fluid Mech. 79 (2), 273–287.CrossRefGoogle Scholar
Porter, D. & Porter, R. 2004 Approximations to wave scattering by an ice sheet of variable thickness over undulating topography. J. Fluid Mech. 509, 145–179.CrossRefGoogle Scholar
Porter, D. & Stirling, D. S. G. 1996 Integral Equations. Cambridge University Press.Google Scholar
Roseau, M. 1958 Short waves parallel to the shore over a sloping beach. Comm. Pure Appl. Math. 11 (4), 433–493.CrossRefGoogle Scholar
Roseau, M. 1976 Asymptotic Wave Theory. Elsevier/North Holland.Google Scholar
Sidi, A. 1988 A user-friendly extrapolation method for oscillatory infinite integrals. Math. Comp. 51, 249–266.CrossRefGoogle Scholar
Stoker, J. J. 1947 Surface waves in water of variable depth. Quart. J. Appl. Math. 5, 1–54.CrossRefGoogle Scholar
Stoker, J. J. 1957 Water Waves. Wiley Interscience.Google Scholar
Whittaker, E. T. & Watson, G. N. 1952 A Course of Modern Analysis, 4th edn. Cambridge University Press.Google Scholar