Hostname: page-component-5cf477f64f-tgq86 Total loading time: 0 Render date: 2025-04-02T07:41:32.408Z Has data issue: false hasContentIssue false
  • English
  • Français

On the parabolic equation method for water-wave propagation

Published online by Cambridge University Press:  19 April 2006

A. C. Radder
A. C. Radder
Affiliation:
Ministry of Transport and Public Works, Rijkswaterstaat, Data Processing Division, Rijswijk, The Netherlands
Get access Rights & Permissions [Opens in a new window]

Abstract

A parabolic approximation to the reduced wave equation is investigated for the propagation of periodic surface waves in shoaling water. The approximation is derived from splitting the wave field into transmitted and reflected components.

In the case of an area with straight and parallel bottom contour lines, the asymptotic form of the solution for high frequencies is compared with the geometrical optics approximation.

Two numerical solution techniques are applied to the propagation of an incident plane wave over a circular shoal.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 95 , Issue 1 , 14 November 1979 , pp. 159 - 176
Copyright
© 1979 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

Babich, V. M. 1970 Mathematical Problems in Wave Propagation Theory, parts I, II and III, translated from Russian. New York: Consultants Bureau.
Berkhoff, J. C. W. 1972 Computation of combined refraction-diffraction. Proc. 13th Coastal Engng Conf., Vancouver, A.S.C.E. 1 (1973), 471–490.Google Scholar
Berkhoff, J. C. W. 1976 Mathematical models for simple harmonic linear water waves. Report on Mathematical investigation, Delft Hydraulics Laboratory, Rep. W 154-IV.Google Scholar
Bettess, P. & Zienkiewicz, O. C. 1977 Diffraction and refraction of surface waves using finite and infinite elements. Int. J. Num. Meth. Engng 11, 1271–1290.Google Scholar
Biéasel, F. 1972 Réfraction de la houle avec diffraction modérée. Proc. 13th Coastal Engng Conf., Vancouver, A.S.C.E. 1 (1973), 491–501.Google Scholar
Booij, N. 1978 Refraction and diffraction of water waves in regions with variable depth and current. Delft University of Technology. Manuscript.
Candel, S. M. 1979 Numerical solution of wave scattering problems in the parabolic approximation. J. Fluid Mech. 90, 465–507.Google Scholar
Corones, J. 1975 Bremmer series that correct parabolic approximations. J. Math. Anal. Appl. 50, 361–372.Google Scholar
Flokstra, C. & Berkhoff, J. C. W. 1977 Propagation of short waves over a circular shoal. Delft Hydraulics Laboratory, Rep. W 154-V (in Dutch).Google Scholar
Fock, V. A. & Wainstein, L. A. 1965 On the transverse diffusion of short waves diffracted by a convex cylinder. In Electromagnetic Diffraction and Propagation Problems (ed. V. A. Fock), chap. 9. Pergamon Press.
Ito, Y. & Tanimoto, K. 1972 A method of numerical analysis of wave propagation - Application to wave diffraction and refraction. Proc. 13th Coastal Engng Conf., Vancouver, A.S.C.E. 1 (1973), 503–522.Google Scholar
Jonsson, I. G. & Brink-Kjaer, O. 1973 A comparison between two reduced wave equations for gradually varying depth. Inst. Hydrodyn. Hydraul. Engng, Tech. Univ. Denmark, Progr. Rep. 31, 13–18.Google Scholar
Kriegsmann, G. A. & Larsen, E. W. 1978 On the parabolic approximation to the reduced wave equation. SIAM J. Appl. Math. 34, 200–204.Google Scholar
Langer, R. E. 1937 On the connection formulas and the solutions of the wave equation. Phys. Rev. 51, 669–676.Google Scholar
Leontovich, M. A. & Fock, V. A. 1965 Solution of the problem of propagation of electromagnetic waves along the earth's surface by the method of parabolic equations. In Electromagnetic Diffraction and Propagation Problems (ed. V. A. Fock), chap. 11. Pergamon Press.
Lundgren, H. 1976 Physics and mathematics of waves in coastal zones. Proc. 15th Coastal Engng Conf., Hawaii, A.S.C.E., 880–885.Google Scholar
Malyuzhinets, G. D. 1959 Developments in our concepts of diffraction phenomena. Soviet Physics Uspekhi 69, 749–758.Google Scholar
Mcdaniel, S. T. 1975 Parabolic approximations for underwater sound propagation. J. Acoust. Soc. Am. 58, 1178–1185.Google Scholar
Messiah, A. 1969 Quantum Mechanics. Amsterdam: North Holland.
Radder, A. C. 1977 Propagation of short gravity waves in shoaling water, refraction and diffraction. Parts I and II. Rijkswaterstaat, Data Processing Division, Rep. no. 74347 and 77476.Google Scholar
Richtmyer, R. D. & Morton, K. W. 1967 Difference methods for Initial-Value Problems. New York: Interscience.
Schönfeld, J. CH. 1972 Propagation of two-dimensional short waves. Delft University of Technology. Manuscript (in Dutch).
Seckler, B. D. & Keller, J. B. 1959 Asymptotic theory of diffraction in inhomogeneous media. J. Acoust. Soc. Am. 31, 206–216.Google Scholar
Smith, R. & Sprinks, T. 1975 Scattering of surface waves by a conical island. J. Fluid Mech. 72, 373–384.Google Scholar
Svendsen, I. A. 1967 The wave equation for gravity waves in water of gradually varying depth. Coastal Engng Lab. and Hydraulic Lab. Tech. Univ. Denmark, Basic Res.-Progr. Rep., 15 2–7.Google Scholar
Tappert, F. D. 1977 The parabolic approximation method. In Wave Propagation and Underwater Acoustics (ed. J. B. Keller & J. S. Papadakis), chap. V. Springer.