On amphidromic points

1994 ◽  
Vol 444 (1920) ◽  
pp. 91-104 ◽  
Keyword(s):  
Differential Equations ◽  
Wave Field ◽  
Water Waves ◽  
Complex Potential ◽  
Wave Amplitude ◽  
Mild Slope Equation ◽  
Local Approximations ◽  
Isolated Points

Amphidromic points are isolated points at which the wave amplitude vanishes. We investigate the consequences of their existence in a wave field. For example, one method for solving the mild-slope equation (this models the propagation of water waves over a variable bathymetry) begins by writing the complex potential in terms of a real amplitude A and a real phase S , both of which are functions of position. We show that S is not continuous at amphidromic points, whereas its gradient is singular there. We also find local approximations for A and S . We discuss various differential equations governing A and S , with emphasis on their properties in the presence of amphidromic points, and find a new pair that is well behaved there. We discuss two simple examples for which the amphidromic points can be found explicitly. Finally, we show that our analysis can also be extended to Laplace’s tidal equations.

A pseudospectral approach for scattering of water waves

1994 ◽  
Vol 445 (1925) ◽  
pp. 619-636 ◽  
Keyword(s):  
Wave Field ◽  
Water Waves ◽  
Parabolic Model ◽  
Beach Profile ◽  
Flat Bottom ◽  
Propagation Models ◽  
Wave Refraction ◽  
Collocation Points ◽  
Mild Slope Equation ◽  
Backward Wave

Wave propagation models for scattering of water waves are developed based on the mild-slope equation. The pseudospectral Fourier approach is used to reduce the mild-slope equation to a set of ordinary differential equations for the mod­ified potential, ϕ√CC g , at collocation points in the alongshore direction. The wave field is then decoupled into a series of wave modes including all forward and backward propagating modes. Ignoring the backward wave field as a first approximation, a wide-angle parabolic model is derived. When the backward wave field is important, both forward and backward wave fields are obtained by construct­ing the Bremmer series solution. A small-angle parabolic model is also developed for comparison. Numerical results are presented for wave refraction over an equi­librium beach profile and wave focusing over a submerged circular shoal on a flat bottom. The importance of the backward scattering is illustrated by the latter example.

The Stochastic Parametric Mechanism for Growth of Wind-Driven Surface Water Waves

2008 ◽  
Vol 38 (4) ◽  
pp. 862-879 ◽  
Author(s):  
Brian F. Farrell ◽  
Petros J. Ioannou
Keyword(s):  
Surface Water ◽  
Growth Rates ◽  
Atmospheric Pressure ◽  
Water Waves ◽  
Wave Amplitude ◽  
Pressure Fluctuations ◽  
Theoretical Understanding ◽  
Wave Growth ◽  
Atmospheric Pressure Fluctuations ◽  
Surface Water Waves

Abstract Theoretical understanding of the growth of wind-driven surface water waves has been based on two distinct mechanisms: growth due to random atmospheric pressure fluctuations unrelated to wave amplitude and growth due to wave coherent atmospheric pressure fluctuations proportional to wave amplitude. Wave-independent random pressure forcing produces wave growth linear in time, while coherent forcing proportional to wave amplitude produces exponential growth. While observed wave development can be parameterized to fit these functional forms and despite broad agreement on the underlying physical process of momentum transfer from the atmospheric boundary layer shear flow to the water waves by atmospheric pressure fluctuations, quantitative agreement between theory and field observations of wave growth has proved elusive. Notably, wave growth rates are observed to exceed laminar instability predictions under gusty conditions. In this work, a mechanism is described that produces the observed enhancement of growth rates in gusty conditions while reducing to laminar instability growth rates as gustiness vanishes. This stochastic parametric instability mechanism is an example of the universal process of destabilization of nearly all time-dependent flows.

Energy Content Characterization of Water Waves Using Linear and Nonlinear Spectral Analysis

2021 ◽  
pp. 1-30
Author(s):  
Ali Mohtat ◽  
Solomon Yim ◽  
Alfred R. Osborne
Keyword(s):  
Wave Field ◽  
Nonlinear Wave ◽  
Water Waves ◽  
Ad Hoc ◽  
Energy Content ◽  
Energy Calculation ◽  
Linear And Nonlinear ◽  
Definition Of ◽  
Exact Energy ◽  
Wave Modes

Abstract The survivability, safe operation, and design of marine vehicles and wave energy converters are highly dependent on accurate characterization and estimation of the energy content of the ocean wave field. In this study, analytical solutions of the nonlinear Schrödinger equation (NLS) using periodic inverse scattering transformation (IST) and its associated Riemann spectrum are employed to obtain the nonlinear wave modes (eigen functions of the nonlinear equation consisting of multiple phase-locked harmonic components). These nonlinear wave modes are used in two approaches to develop a more accurate definition of the energy content. First, in an ad hoc approach, the amplitudes of the nonlinear wave modes are used with a linear energy calculation resulting in a semi-linear energy estimate. Next, a novel, mathematically exact definition of the energy content taking into account the nonlinear effects up to fifth order is introduced in combination with the nonlinear wave modes, the exact energy content of the wave field is computed. Experimental results and numerical simulations were used to compute and analyze the linear, ad hoc, and exact energy contents of the wave field, using both linear and nonlinear spectra. The ratio of the ad hoc and exact energy estimates to the linear energy content were computed to examine the effect of nonlinearity on the energy content. In general, an increasing energy ratio was observed for increasing nonlinearity of the wave field, with larger contributions from higher-order harmonic terms. It was confirmed that the significant increase in nonlinear energy content with respect to its linear counterpart is due to the increase in the number of nonlinear phase-locked (bound wave) modes.

Consistent nonlinear deterministic and stochastic wave evolution equations from deep water to the breaking region

2019 ◽  
Vol 877 ◽  
pp. 373-404
Author(s):  
T. Vrecica ◽  
Y. Toledo
Keyword(s):  
Wave Field ◽  
Large Scale ◽  
Evolution Equations ◽  
Deterministic Model ◽  
Two Dimensional ◽  
Spectral Evolution ◽  
Wave Interactions ◽  
Nonlinear Part ◽  
Mild Slope Equation ◽  
Wave Models

Modelling the evolution of the wave field in coastal waters is a complicated task, partly due to triad nonlinear wave interactions, which are one of the dominant mechanisms in this area. Stochastic formulations already implemented into large-scale operational wave models, whilst very efficient, are one-dimensional in nature and fail to account for the majority of the physical properties of the wave field evolution. This paper presents new two-dimensional (2-D) formulations for the triad interactions source term. A quasi-two-dimensional deterministic mild slope equation is improved by including dissipation and first-order spatial derivatives in the nonlinear part of equation, significantly enhancing the accuracy in the breaking zone. The newly defined deterministic model is used to derive an updated stochastic model consistent from deep waters to the breaking region. It is localized following the approach derived in Vrecica & Toledo (J. Fluid Mech., vol. 794, 2016, pp. 310–342), to which several improvements are also presented. The model is compared to measurements of breaking and non-breaking spectral evolution, showing good agreement in both cases. Finally, the model is used to analyse several interesting 2-D properties of the shoaling wave field including the evolution of directionally spread seas.

An Efficient Numerical Model of Hyperbolic Mild-Slope Equation

2007 ◽  
Author(s):  
Zhiyao Song ◽  
Honggui Zhang ◽  
Jun Kong ◽  
Ruijie Li ◽  
Wei Zhang
Keyword(s):  
Wave Propagation ◽  
Numerical Model ◽  
Water Waves ◽  
Computational Time ◽  
Transient Wave ◽  
Flat Bottom ◽  
The Real ◽  
Wave Elevation ◽  
Mild Slope Equation ◽  
Effective Wave

Introduction of an effective wave elevation function, the simplest time-dependent hyperbolic mild-slope equation has been presented and an effective numerical model for the water wave propagation has been established combined with different boundary conditions in this paper. Through computing the effective wave elevation and transforming into the real transient wave motion, then related wave heights are computed. Because the truncation errors of the presented model only induced by the dissipation terms, but those of Lin’s model (2004) contributed by the convection terms, dissipation terms and source terms, the error analysis shows that calculation stability of this model is enhanced obviously compared with Lin’s one. The tests show that this model succeeds to the merit in Lin’s one and the computer program simpler, computational time shorter because of calculation stability enhanced efficiently and computer memory decreased obviously. The presented model has the capability of simulating exactly the location of transient wave front by the speed of wave propagation in the first test, which is important for the real-time prediction of the arrival time of water waves generated in the deep sea. The model is validated against experimental data for combined wave refraction and diffraction over submerged circular shoal on a flat bottom in the second test. Good agreements are gained. The model can be applied to the theory research and engineering applications about the wave propagation in the coastal waters.

A second-order discretization for forward-backward SDEs using local approximations with Malliavin calculus

2019 ◽  
Vol 25 (4) ◽  
pp. 341-361
Author(s):  
Riu Naito ◽  
Toshihiro Yamada
Keyword(s):  
Monte Carlo ◽  
Monte Carlo Method ◽  
Differential Equations ◽  
Least Squares ◽  
Stochastic Differential Equations ◽  
Malliavin Calculus ◽  
Second Order ◽  
Discretization Method ◽  
Local Approximations ◽  
Backward Sdes

Abstract The paper proposes a new second-order discretization method for forward-backward stochastic differential equations. The method is given by an algorithm with polynomials of Brownian motions where the local approximations using Malliavin calculus play a role. For the implementation, we introduce a new least squares Monte Carlo method for the scheme. A numerical example is illustrated to check the effectiveness.

Nonlinear refraction–diffraction of water waves: the complementary mild-slope equations

2009 ◽  
Vol 641 ◽  
pp. 509-520 ◽  
Author(s):  
YARON TOLEDO ◽  
YEHUDA AGNON
Keyword(s):  
Water Waves ◽  
Numerical Models ◽  
Nonlinear Refraction ◽  
Wave Interaction ◽  
Domain Model ◽  
Nonlinear Frequency ◽  
Class Iii ◽  
Mild Slope Equation ◽  
Bottom Boundary Condition ◽  
Analytical Perturbation

A second-order nonlinear frequency-domain model extending the linear complementary mild-slope equation (CMSE) is presented. The nonlinear model uses the same streamfunction formulation as the CMSE. This allows the vertical profile assumption to accurately satisfy the kinematic bottom boundary condition in the case of nonlinear triad interactions as well as for the linear refraction–diffraction part. The result is a model with higher accuracy of wave–bottom interactions including wave–wave interaction. The model's validity is confirmed by comparison with accurate numerical models, laboratory experiments over submerged obstacles and analytical perturbation solutions for class III Bragg resonance.

Accurate refraction–diffraction equations for water waves on a variable-depth rough bottom

2001 ◽  
Vol 449 ◽  
pp. 301-311 ◽  
Author(s):  
YEHUDA AGNON ◽  
EFIM PELINOVSKY
Keyword(s):  
Water Waves ◽  
Mean Field ◽  
Variable Coefficients ◽  
Test Case ◽  
Random Roughness ◽  
Operator Calculus ◽  
Mild Slope Equation ◽  
The Mean ◽  
Model Equations ◽  
Systematic Derivation

The extended mild-slope equation and the modified mild-slope equation have been used successfully to study refraction–diffraction of linear water waves by steep bottom roughness. Their consistency has been questioned. A systematic derivation of these model equations exposes and illuminates their rationale. Their good performance stems from an accurate representation of (Class I) Bragg resonance. As a benchmark test case, we consider scattering by a sloping bottom with random roughness. The rates of scattering found for the mean field in both of the approximate models agree exactly with the full theory for scattering by small roughness. This greatly improves the limited agreement which was found for the mild-slope equation, and establishes the validity of the above model equations. The study involves operator calculus, a powerful method for simplifying problems with variable coefficients. The augmented mild-slope equation serves to consistently derive accurate model equations.

A scalar form of the complementary mild-slope equation

2010 ◽  
Vol 656 ◽  
pp. 407-416 ◽  
Author(s):  
YARON TOLEDO ◽  
YEHUDA AGNON
Keyword(s):  
Boundary Condition ◽  
Water Waves ◽  
Linear Time ◽  
Laboratory Data ◽  
Computational Effort ◽  
Scalar Equation ◽  
Scalar Form ◽  
Additional Boundary Condition ◽  
Mild Slope Equation ◽  
Pseudo Potential

Mild-slope (MS) type equations are depth-integrated models, which predict under appropriate conditions refraction and diffraction of linear time-harmonic water waves. Among these equations, the complementary mild-slope equation (CMSE) was shown to give better agreement with exact two-dimensional linear theory compared to other MS-type equations. Nevertheless, it has a disadvantage of being a vector equation, i.e. it requires solving a system of two coupled partial differential equations. In addition, for three-dimensional problems, there is a difficulty in constructing the additional boundary condition needed for the solution. In the present work, it is shown how the vector CMSE can be transformed into an equivalent scalar equation using a pseudo-potential formulation. The pseudo-potential mild-slope equation (PMSE) preserves the accuracy of the CMSE while avoiding the need of an additional boundary condition. Furthermore, the PMSE significantly reduces the computational effort relative to the CMSE, since it is a scalar equation. The accuracy of the new model was tested numerically by comparing it to laboratory data and analytical solutions.

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