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Extensions of the mild-slope equation

  • D. Porter (a1) and D. J. Staziker (a1)
Abstract

The use of the mild-slope approximation, which is invoked to simplify the problem of linear water wave diffraction-refraction by bed undulations, is reassessed by using a variational method. It is found that smooth approximations to the free surface elevation obtained by using the long-standing mild-slope equation are not consistent with the continuity of mass flow at locations where the bed slope is discontinuous. The use of interfacial jump conditions at such locations significantly improves the accuracy of approximations generated by the mild-slope equation and by the recently derived modified mild-slope equation. The variational principle is also used to produce a generalization of these equations and of the associated jump condition. Numerical results are presented to illustrate the main points of the theory.

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References
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Bartholomeusz, E. F. 1958 The reflexion of long waves at a step. Proc. Camb. Phil. Soc. 54, 106118.
Berkhoff, J. C. W. 1973 Computation of combined refraction-diffraction. In Proc. 13th Intl Conf. on Coastal Engng, Vancouver, Canada, ASCE pp. 471490.
Booij, N. 1983 A note on the accuracy of the mild-slope equation. Coastal Engng 7, 191203.
Chamberlain, P. G. & Porter, D. 1995a The modified mild-slope equation. J. Fluid Mech. 291, 393407.
Chamberlain, P. G. & Porter, D. 1995b Decomposition methods for wave scattering by topography with application to ripple beds. Wave Motion (to appear).
Davies, A. G. & Heathershaw, A. D. 1984 Surface-wave propagation over sinusoidally varying topography. J. Fluid Mech. 144, 419443.
Fehlberg, E. 1970 Klassiche Runge-Kutta Formeln vierter und niedregerer Ordnung mit Schrittweiten-Kontrolle und ihre anwendung auf Warmeleitungsprobleme. Computing 6, 6171.
Forsythe, G. E., Malcolm, M. A. & Moler, C. B. 1977 Computer Methods for Mathematical Computations. Prentice-Hall.
Kirby, J. T. 1986 A general wave equation for waves over rippled beds. J. Fluid Mech. 162, 171186.
Lamb, H. 1932 Hydrodynamics, 6th edn. Cambridge University Press.
Massel, S. R. 1993 Extended refraction-diffraction equation for surface waves. Coastal Engng 19, 97126.
Rey, V. 1992 Propagation and local behaviour of normally incident gravity waves over varying topography. Eur. J. Mech. B/Fluids 11, 213232.
Smith, R. & Sprinks, T. 1975 Scattering of surface waves by a conical island. J. Fluid Mech. 72, 373384.
Staziker, D. J. 1995 Water wave scattering by undulating bed topography. PhD thesis, University of Reading.
Staziker, D. J., Porter, D. & Stirling, D. S. G. 1995 The scattering of surface waves by local bed elevations. (submitted).
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Journal of Fluid Mechanics
  • ISSN: 0022-1120
  • EISSN: 1469-7645
  • URL: /core/journals/journal-of-fluid-mechanics
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