Hostname: page-component-848d4c4894-m9kch Total loading time: 0 Render date: 2024-06-12T10:54:37.779Z Has data issue: false hasContentIssue false
  • English
  • Français

Scattering analysis and synthesis of wave trains

Published online by Cambridge University Press:  17 February 2009

R. J. Sobey  and
E. J. Colman
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

The Zakharov-Shabat scattering transform is an exact solution technique for the nonlinear Schrödinger equation, which describes the time evolution of weakly nonlinear wave trains. Envelope soliton and uniform wave train solutions of the nonlinear Schrödinger equation are separable in scattering transform space. The scattering transform is a potential analysis and synthesis technique for natural wave trains. Discrete versions of the direct and inverse scattering transform are presented, together with proven algorithms for their numerical computation from typical ocean wave records. The consequences of discrete resolution are considered.

Type
Research Article
Information
The ANZIAM Journal , Volume 25 , Issue 1 , July 1983 , pp. 44 - 63
Copyright
Copyright © Australian Mathematical Society 1983

References

[1]Ablowitz, M. J., Kaup, D. J., Newell, A. C. and Segur, H., “The inverse scattering transform — Fourier analysis for nonlinear problems”, Stud. Appl. Math. 53 (1974), 249–315.Google Scholar
[2]Bolt, B. A. and Brillinger, D. R., “Estimation of uncertainties in eigenspectral estimates from decaying geophysical time series”, Geophys. J. Roy. Astron. Soc. 59 (1979), 593–603.Google Scholar
[3]Brigham, E. O., The fast Fourier transform (Prentice-Hall, Englewood Cliffs, 1974).Google Scholar
[4]Churchill, R. V., Complex variables and applications (McGraw-Hill, New York, 2nd edition, 1960).Google Scholar
[5]Colman, E. J. and Sobey, R. J., “Analysis and synthesis of natural wave trains in deep water”, Dept. Civil and Systems Engrg., James Cook Univ. Res. Bull. CS 23 (1982).Google Scholar
[6]Davey, A. and Stewartson, K., “On three-dimensional packets of surface waves”, Proc. Roy. Soc. London Ser. A 338 (1974), 101–110.Google Scholar
[7]Deutsch, R., Nonlinear transformations of random processes (Prentice-Hall, Englewood Cliffs, 1962).Google Scholar
[8]Gardner, C. S., Greene, J. M., Kruskal, M. D. and Miura, R. M., “Method for solving the Korteweg-de Vries equation”, Phys. Rev. Lett. 19 (1967), 1095–1097.CrossRefGoogle Scholar
[9]Hasimoto, H. and Ono, H., “Nonlinear modulation of gravity waves”, J. Phys. Soc. Japan 33 (1972), 805–811.CrossRefGoogle Scholar
[10]Lake, B. M., Yuen, H. C., Rungaldier, H. and Ferguson, W. E., “Nonlinear deep-water waves: theory and experiment, Part 2. Evolution of a continuous wave train”, J. Fluid Mech. 83 (1977), 49–74.Google Scholar
[11]Lighthill, M. J., Introduction to Fourier analysis and generalised functions (Cambridge University Press, Cambridge, 1958).CrossRefGoogle Scholar
[12]Ma, Y.-C., “The perturbed plane-wave solution of the cubic Schrödinger equation”, Stud. Appl. Math. 60 (1979), 43–58.Google Scholar
[13]Resch, F. J. and Abel, R., “Spectral analysis using Fourier transform techniques”, Internal. J. Numer. Methods. Engrg. 9 (1975), 869–902.Google Scholar
[14]Saffman, P. G. and Yuen, H. C., “Stability of a plane soliton to infinitesimal two-dimensional perturbation”, Phys. Fluids 21 (1978), 1450–1451.CrossRefGoogle Scholar
[15]Sobey, R. J. and Colman, E. J., “Natural wave trains and scattering transform”, J. Waterway Port Coastal and Ocean Div. ASCE 108 (1982), 272–290.Google Scholar
[16]Tennekes, H. and Lumley, J. L., A first course in turbulence (MIT Press, Cambridge, Mass., 1972).Google Scholar
[17]Yuen, H. C. and Lake, B. M., “Nonlinear deep water waves: theory and experiment”, Phys. Fluids 18 (1975), 956–960.Google Scholar
[18]Zakharov, V. E., “Stability of periodic waves of finite amplitude on the surface of a deep fluid”, J. Appl. Mech. Tech. Phys. 9 (1968), 86–94.Google Scholar
[19]Zakharov, V. E. and Shabat, A. B., “Exact theory of two dimensional self focussing and one dimensional self modulation of waves in nonlmear media”, Soviet Phys. JETP 34 (1972), 62–69.Google Scholar