Weakly nonlinear surface waves over a 
random seabed

Mei, Chiang C.; Hancock, Matthew

doi:10.1017/s002211200200280x

Cited by 36 publications

(74 citation statements)

References 36 publications

(51 reference statements)

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“…It thus appears that our rather surprising result for the convergence as n → ∞ is justified by the convergence of the discrete spectrum to the continuous power spectrum and the theory of Mei and Hancock (2003) applied to non-random bottoms (see Appendix). It shows that the far field scattered energy by small amplitude depth variations only depends on the power spectrum of the scatterers at the Bragg scale, and not on its localization in space, as long as the bottom amplitude remains small.…”

Section: First Test Case: Small Depth Changementioning

confidence: 77%

“…Mei (1985) developed a more accurate approximation that is valid at resonance using a multiple scale theory. This approach was further extended to random bottom topography in one dimension (Mei and Hancock, 2003). The Bragg resonance theory can be extended to any arbitrary topography in two dimensions, that is statistically uniform (Hasselmann 1966).…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Topographical Scattering of Waves: Spectral Approach

Magne

Ardhuin

Rey

et al. 2005

J. Waterway, Port, Coastal, Ocean Eng.

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The topographical scattering of gravity waves is investigated using a spectral energy balance equation that accounts for first order wave-bottom Bragg scattering. This model represents the bottom topography and surface waves with spectra, and evaluates a Bragg scattering source term that is theoretically valid for small bottom and surface slopes and slowly varying spectral properties. The robustness of the model is tested for a variety of topographies uniform along one horizontal dimension including nearly sinusoidal, linear ramp and step profiles. Results are compared with reflections computed using an accurate method that applies integral matching along vertical boundaries of a series of steps. For small bottom amplitudes, the source term representation yields accurate reflection estimates even for a localized scatterer. This result is proved for small bottom amplitudes h relative to the mean water depth H. Wave reflection by small amplitude bottom topography thus depends primarily on the bottom elevation variance at the Bragg resonance scales, and is insensitive to the detailed shape of the bottom profile. Relative errors in the energy reflection coefficient are found to be typically 2h/H.

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Section: First Test Case: Small Depth Changementioning

confidence: 77%

Section: Introductionmentioning

confidence: 99%

Topographical Scattering of Waves: Spectral Approach

Magne

Ardhuin

Rey

et al. 2005

J. Waterway, Port, Coastal, Ocean Eng.

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“…Using results by Mei and Hancock (2003), this scattering term was shown to be applicable to deterministic bottom topographies such as an isolated step or a ramp (Magne et al, 2005a), provided that the topography amplitude is small compared to the mean water depth. This result establishes that in this case the Bragg scattering mechanism fully explains the reflection: waves reflection only depends on the variance in the bottom elevation at the Bragg scale.…”

Section: 3wave Scattering and Reflectionmentioning

confidence: 99%

“…Reflection coefficients may thus be obtained from any bottom topography of small amplitude, including steps or ramps, as can be seen by the correspondence between the Green function method and Fourier transforms (Elter and Molyneux, 1972;Mei and Hancock, 2003;Magne et al, 2005b).…”

Section: Limitations Of Geometrical Optics: Diffraction Reflection Amentioning

confidence: 99%

Wave modelling – The state of the art

Cavaleri¹,

Alves²,

Ardhuin³

et al. 2007

Progress in Oceanography

451

177

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This paper is the product of the wave modelling community and it tries to make a picture of the present situation in this branch of science, exploring the previous and the most recent results and looking ahead towards the solution of the problems we presently face. Both theory and applications are considered.The many faces of the subject imply separate discussions. This is reflected into the single sections, seven of them, each dealing with a specific topic, the whole providing a broad and solid overview of the present state of the art. After an introduction framing the problem and the approach we followed, we deal in sequence with the following subjects: (Section) 2, generation by wind; 3, non-linear interactions in deep water; 4, white-capping dissipation; 5, non-linear interactions in shallow water; 6, dissipation at the sea bottom; 7, wave propagation; 8, numerics. The two final sections, 9 and 10, summarize the present situation from a general point of view and try to look at the future developments. Keywords

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“…This agreement was far superior than that obtained from any of the nondissipative models examined. Dissipation of this form in NLS has been examined by Miles (1967), Lake et al (1977) and Mei & Hancock (2003). It is important to note that this is one possible model of dissipation and that many other models exist including those of Trulsen & Dysthe (1990), Hara & Mei (1994), Shemer & Chamesse (1999), Joseph & Wang (2004) and Bridges & Dias (2004).…”

Section: Dissipative Bmnlsmentioning

confidence: 99%

Stability of plane waves on deep water with dissipation

Canney

Carter

2007

Mathematics and Computers in Simulation

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The Benjamin-Feir modulational instability effects the evolution of perturbed planewave solutions of the cubic nonlinear Schrödinger equation (NLS), the modified NLS, and the band-modified NLS. Recent work demonstrates that the BenjaminFeir instability in NLS is "stabilized" when a linear term representing dissipation is added. In this paper, we add a linear term representing dissipation to the modified NLS and band-modified NLS equations and establish that the plane-wave solutions of these equations are linearly stable. Although the plane-wave solutions are stable, some perturbations grow for a finite period of time. We analytically bound this growth and present approximate time-dependent regions of wave-number space that correspond to perturbations that have increasing amplitudes.

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Weakly nonlinear surface waves over a random seabed

Cited by 36 publications

References 36 publications

Topographical Scattering of Waves: Spectral Approach

Topographical Scattering of Waves: Spectral Approach

Wave modelling – The state of the art

Stability of plane waves on deep water with dissipation

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