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Wave Propagation and Diffraction pp 25–75Cite as

Spectral Methods in the Theory of Wave Propagation

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Abstract

This chapter deals with spectral methods in the theory of wave propagation. The main focus is given to the Fourier methods in application to studying the Stokes (gravity) waves on the surface of an inviscid fluid. A spectral method for calculating the limiting Stokes wave with a corner at the crest is considered as well. We also briefly consider the evolution of narrow-band wave trains on the surface of an ideal finite-depth fluid. Finally, a two-parameter method for describing the non-linear evolution of narrow-band wave trains is described by the example of the Klein–Gordon equation with a cubic nonlinearity. The problem is reduced to a high-order nonlinear Schrödinger equation for the complex amplitude of wave envelope. This equation is integrated numerically using a split-step Fourier technique to describe the evolution of quasi-solitons.

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Authors and Affiliations

  1. Institute of Hydromechanics, National Academy of Sciences of Ukraine, Kyiv, Ukraine

    Igor T. Selezov

  2. Institute of Cybernetics, National Academy of Sciences of Ukraine, Kyiv, Ukraine

    Yuriy G. Kryvonos

  3. Institute of Physics, National Academy of Sciences of Ukraine, Kyiv, Ukraine

    Ivan S. Gandzha

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  1. Igor T. Selezov
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  2. Yuriy G. Kryvonos
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  3. Ivan S. Gandzha
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Selezov, I.T., Kryvonos, Y.G., Gandzha, I.S. (2018). Spectral Methods in the Theory of Wave Propagation. In: Wave Propagation and Diffraction. Foundations of Engineering Mechanics. Springer, Singapore. https://doi.org/10.1007/978-981-10-4923-1_2

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