Hostname: page-component-848d4c4894-m9kch Total loading time: 0 Render date: 2024-06-02T17:17:11.426Z Has data issue: false hasContentIssue false
  • English
  • Français

An angular spectrum model for propagation of Stokes waves

Published online by Cambridge University Press:  26 April 2006

Kyung Duck Suh ,
Robert A. Dalrymple  and
James T. Kirby
Kyung Duck Suh
Affiliation:
Center for Applied Coastal Research, Department of Civil Engineering, University of Delaware, Newark, DE 19716, USA Present address: Virginia Institute of Marine Science, Gloucester Point, VA 23062, USA.
Robert A. Dalrymple
Affiliation:
Center for Applied Coastal Research, Department of Civil Engineering, University of Delaware, Newark, DE 19716, USA
James T. Kirby
Affiliation:
Center for Applied Coastal Research, Department of Civil Engineering, University of Delaware, Newark, DE 19716, USA
Get access Rights & Permissions [Opens in a new window]

Abstract

An angular spectrum model for predicting the transformation of Stokes waves on a mildly varying topography is developed, including refraction, diffraction, shoaling and nonlinear wave interactions. The equations governing the water-wave motion are perturbed using the method of multiple scales and Stokes expansions for the velocity potential and free-surface displacement. The first-order solution is expressed as an angular spectrum, or directional modes, of the wave field propagating on a beach with straight iso-baths whose depth is given by laterally averaged depths. The equations for the evolution of the angular spectrum due to the effects of bottom variation and cubic resonant interaction are obtained from the higher-order problems. Comparison of the present model with existing models is made for some simple cases. Numerical examples of the time-independent version of the model are presented for laboratory experiments for wave diffraction behind a breakwater gap and wave focusing over submerged shoals: an elliptic shoal on a sloping beach and a circular shoal on a flat bottom.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 221 , December 1990 , pp. 205 - 232
Copyright
© 1990 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.)

References

Benney, D. J. & Newell, A. C., 1967 The propagation of nonlinear wave envelopes. J. Math. Phys. 46, 133–139.Google Scholar
Benney, D. J. & Roskes, G. J., 1969 Wave instabilities. Stud. Appl. Maths 48, 377–385.Google Scholar
Berkhoff, J. C. W.: 1972 Computation of combined refraction-diffraction. Proc. 13th Intl Conf. Coastal Engng, ASCE, Vancouver, pp. 471–490.Google Scholar
Berkhoff, J. C. W., Booij, N. & Radder, A. C., 1982 Verification of numerical wave propagation models for simple harmonic linear waves. Coastal Engng. 6, 255–279.Google Scholar
Booker, H. G. & Clemmow, P. C., 1950 The concept of an angular spectrum of plane waves and its relation to that of polar diagram and aperture distribution. Proc. Inst. Elect. Engrs, part 3, 97, 11–17.Google Scholar
Clemmow, P. C.: 1986 The Plane Wave Spectrum Representation of Electromagnetic Fields, Pergamon. 185 pp.
Dalrymple, R. A.: 1989 Water waves past abrupt channel transitions. Appl. Ocean Res. (In Press.)Google Scholar
Dalrymple, R. A. & Kirby, J. T., 1988 Models for very wide-angle water waves and wave diffraction. J. Fluid Mech. 192, 33–50.Google Scholar
Dalrymple, R. A., Suh, K. D., Kirby, J. T. & Chae, J. W., 1989 Models for very wide-angle water waves and wave diffraction. Part 2. Irregular bathymetry. J. Fluid Mech. 201, 299–322.Google Scholar
Davey, A. & Stewartson, K., 1974 On three-dimensional packets of surface waves. Proc. R. Soc. Lond. A 338, 101–110.Google Scholar
Djordjević, V. D. & Redekopp, L. G. 1978 On the development of packets of surface gravity wave moving over an uneven bottom. Z. angew. Math. Phys. 29, 950–962.Google Scholar
Gabor, D.: 1961 Light and information. In Progress in Optics (ed. E. Wolf), vol. 1. North-Holland.
Gottlieb, D., Lustman, L. & Orszag, S. A., 1981 Spectral calculations of one-dimensional inviscid compressible flows. SIAM J. Sci. Stat. Comput. 2, 296–310.Google Scholar
Ito, Y. & Tanimoto, K., 1972 A method of numerical analysis of wave propagation: Application to wave diffraction and refraction. Proc. 13th Intl Conf. Coastal Engng, ASCE, Vancouver, pp. 503–522.Google Scholar
Kirby, J. T.: 1986 Rational approximations in the parabolic equation method for water waves. Coastal Engng. 10, 355–378.Google Scholar
Kirby, J. T. & Dalrymple, R. A., 1983 A parabolic equation for the combined refraction-diffraction of Stokes waves by mildly varying topography. J. Fluid Mech. 136, 453–466.Google Scholar
Kirby, J. T. & Dalrymple, R. A., 1986 An approximate model for nonlinear dispersion in monochromatic wave propagation models. Coastal Engng 9, 545–561.Google Scholar
Longuet-Higgins, M. S.: 1962 Resonant interactions between two trains of gravity waves. J. Fluid Mech. 12, 321–332.Google Scholar
Majda, A., Mcdonough, J. & Osher, S., 1978 The Fourier method for nonsmooth initial data. Math. Comput. 32, 1041–1081.Google Scholar
Osher, S.: 1984 Smoothing for spectral methods. In Spectral Methods for Partial Differential Equations (ed. R. G. Voigt, D. Gottlieb & M. Y. Hussaini). SIAM.
Penney, W. G. & Price, A. T., 1952 The diffraction theory of sea waves and the shelter afforded by breakwaters. Phil. Trans. R. Soc. Lond. A 244, 236–253.Google Scholar
Phillips, O. M.: 1960 On the dynamics of unsteady gravity waves of finite amplitude. Part 1. The elementary interactions. J. Fluid Mech. 9, 193–217.Google Scholar
Pos, J. D. & Kilner, F. A., 1987 Breakwater gap wave diffraction: An experimental and numerical study. J. Waterway Port Coast. Ocean Engng 113, 1–21.Google Scholar
Ratcliffe, J. A.: 1956 Some aspects of diffraction theory and their application to the ionosphere. In Reports on Progress in Physics, vol. 19, (ed. A. C. Strickland). The Physical Society, London.
Roskes, G. J.: 1976a Some nonlinear multiphase interactions. Stud. Appl. Maths 55, 231–238.Google Scholar
Roskes, G. J., Roskes, G. J.: 1976b Nonlinear multiphase deep-water wavetrains. Phys. Fluids. 19, 1253–1254.Google Scholar
Sharma, J. N. & Dean, R. G., 1979 Development and evaluation of a procedure for simulating a random directional second order sea surface and associated wave forces. Ocean Engng Rep. 20. University of Delaware.
Stamnes, J. J.: 1986 Waves in Focal Regions. Bristol: Adam Hilger. 600 pp.
Stamnes, J. J., Løvhaugen, O., Spjelkavik, B., Mei, C. C., Lo, E. & Yue, D. K. P. 1983 Nonlinear focusing of surface waves by a lens - theory and experiment. J. Fluid Mech. 135, 71–94.Google Scholar
Sim, K. D.: 1989 Angular spectrum models for propagation of weakly nonlinear surface gravity waves in water of varying depth. PhD dissertation, University of Delaware.
Willmott, C. J.: 1981 On the validation of models. Phys. Geog. 2, 184–194.Google Scholar
Yue, D. K.-P. & Mei, C. C. 1980 Forward diffraction of Stokes waves by a thin wedge. J. Fluid Mech. 99, 33–52.Google Scholar