Waves In Finite Depth Research Articles

Benjamin–Feir Instability of Stokes Waves in Finite Depth

Whitham and Benjamin predicted in 1967 that small-amplitude periodic traveling Stokes waves of the 2d-gravity water waves equations are linearly unstable with respect to long-wave perturbations, if the depth {mathtt h} is larger than a critical threshold texttt{h}_{scriptscriptstyle {textsc {WB}}}approx 1.363 . In this paper, we completely describe, for any finite value of mathtt h >0 , the four eigenvalues close to zero of the linearized equations at the Stokes wave, as the Floquet exponent mu is turned on. We prove, in particular, the existence of a unique depth texttt{h}_{scriptscriptstyle {textsc {WB}}}, which coincides with the one predicted by Whitham and Benjamin, such that, for any 0< mathtt h < texttt{h}_{scriptscriptstyle {textsc {WB}}}, the eigenvalues close to zero are purely imaginary and, for any mathtt h > texttt{h}_{scriptscriptstyle {textsc {WB}}}, a pair of non-purely imaginary eigenvalues depicts a closed figure “8”, parameterized by the Floquet exponent. As {mathtt h} rightarrow texttt{h}_{scriptscriptstyle {textsc {WB}}}^{, +} the “8” collapses to the origin of the complex plane. The complete bifurcation diagram of the spectrum is not deduced as in deep water, since the limits texttt{h}rightarrow +infty (deep water) and mu rightarrow 0 (long waves) do not commute. In finite depth, the four eigenvalues have all the same size mathcal {O}(mu ), unlike in deep water, and the analysis of their splitting is much more delicate, requiring, as a new ingredient, a non-perturbative step of block-diagonalization. Along the whole proof, the explicit dependence of the matrix entries with respect to the depth texttt{h} is carefully tracked.

Differential Galoisian approach to Jacobi integrability of general analytic dynamical systems and its application

The Morales-Ramis theory provides an effective and powerful non-integrability criterion for complex analytic Hamiltonian systems via the differential Galoisian obstruction. In this paper, we give a new Morales-Ramis type theorem on the meromorphic Jacobi non-integrability of general analytic dynamical systems. The key point is to show that the existence of Jacobian multipliers of a nonlinear system implies the existence of common Jacobian multipliers of Lie algebra associated with the identity component. In addition, we apply our results to the polynomial integrability of Karabut systems for stationary gravity waves in finite depth.

A high-order asymptotic analysis of the Benjamin–Feir instability spectrum in arbitrary depth

We investigate the Benjamin–Feir (or modulational) instability of Stokes waves, i.e. small-amplitude, one-dimensional periodic gravity waves of permanent form and constant velocity, in water of finite and infinite depth. We develop a perturbation method to describe to high-order accuracy the unstable spectral elements associated with this instability, obtained by linearizing Euler's equations about the small-amplitude Stokes waves. These unstable elements form a figure-eight curve centred at the origin of the complex spectral plane, which is parametrized by a Floquet exponent. Our asymptotic expansions of this figure-eight are in excellent agreement with numerical computations as well as recent rigorous results by Bertiet al.(Full description of Benjamin–Feir instability of Stokes waves in deep water, 2021, arXiv:2109.11852) and Bertiet al.(Benjamin–Feir instability of Stokes waves in finite depth, 2022, arXiv:2204.00809). From our expansions, we derive high-order estimates for the growth rates of the Benjamin–Feir instability and for the parametrization of the Benjamin–Feir figure-eight curve with respect to the Floquet exponent. We are also able to compare the Benjamin–Feir and high-frequency instability spectra analytically for the first time, revealing three different regimes of the Stokes waves, depending on the predominant instability.

A Fourier Galerkin method for ship waves

An efficient approximate solution to the problem of gravity–capillary ship-induced waves is derived using a Fourier Galerkin spectral approach. The proposed method solves linearised equations of finite-depth wave motion generated by a moving pressure distribution in a spectral framework. The method admits ship hulls of various shapes defined by their pressure signature and its Fourier transform in physical and spectral domains, respectively. For the present analysis, the ship hull of a given geometry is represented in a simplified form as a sum of two-dimensional Gaussian functions allowing analysis of interaction of wakes generated by bow-stern hull components or transverse multi-hull systems. An arbitrary movement of the pressure is realised through appropriate mathematical operations of translation and rotation of rigidly coupled component Gaussian forms directly in a spectral domain to preserve high model performance. Thus, the method is capable of giving fast predictions of wakes generated by an arbitrarily moving pressure distribution for different shapes of a simulated vessel.

Experimental study on the growth and conversion of duration- and fetch-limited wind waves in water of finite depth

This paper presents laboratory experiment results on the initial growth of wind waves in water of finite depth. Under the action of a steady wind flow, mixtures of duration- and fetch-limited waves were generated over a constant water depth. Variations of dimensionless growth rate data against finite-depth wave age and dimensionless depth for both scenarios are found to be highly depth-dependent; however, the growth rates of duration-limited waves are larger than those for fetch-limited waves. Our experiment showed that when wave age increases up to the so-called critical wave age, wind-induced wave growth reaches a limit due to the depth effect. The critical age values determined in our flume agree well with the theoretical ones. Dimensionless power-law predictions are presented for significant wave height and spectral peak period in each wave regime, based on which a set of time-space conversion relationships is obtained. However, our results suggest different conversion equations for wave height and wave period. Even if our data are obtained in a laboratory environment, the widely-used hyperbolic model fitted over our measurements could forecast available field data with a Scatter Index of 0.06 for wave height and 0.17 for wave period.

Higher-order gap resonance and heave response of two side-by-side barges under Stokes and cnoidal waves

Coupled piston-mode fluid response and the heave motion of two identical barges in side-by-side configuration is studied under finite-depth and shallow-water waves using a two-dimensional fully nonlinear numerical wave tank. To understand possible critical responses of the gap flow and the floating barge, regular-wave conditions which are able to excite up to 5th-order nonlinear gap resonance and also the resonant heave motion of the barge are considered. In shallow-water waves, high-frequency oscillations, featured by secondary peaks in the time histories, are observed for both wave elevation in the gap and the heave motion of the barge. The shallow-water wave-induced 4th- or 5th-order gap resonance can be equally crucial as the 1st- and 2nd-order resonances due to finite-depth waves. At higher-order gap resonance, the higher-harmonic heave motion of the barge is negligibly small, in contrast to the gap-flow response. Compared with fixed barges, the free-heave motion of an upstream barge tends to increase the wave elevation in the gap in most of the resonant conditions, except at 1st-order gap resonance where the gap response is greatly reduced. When the resonant heave motion of a floating barge, either located upstream or downstream, is excited, significant barge motion is observed. However, the relative motion between the gap flow and the floating barge is seen to be very small, ascribed by small phase difference between the two. The present study suggests that the effects of heave motion and water-depth should be carefully considered in the design of side-by-side marine operations, and hiding the small bunkering ships behind the large receiving ships is regarded as a preferred arrangement during the bunkering operations in offshore and coastal environments.

Growth of surface wind-waves in water of finite depth: A laboratory experiment

This paper reports on laboratory experiment results on wind-driven surface waves in finite depth and their comparison with theoretical predictions and experimental in-situ studies. We introduce the Miles theory’s extension to the case of finite depth, as well as the rules transforming theoretical expressions to formulae commonly used in experimental routines, in particular the important rules transforming theoretical growth rates to experimental ones. Wind waves depend strongly on boundary marine layer parameters, namely, the aerodynamic roughness length, the Charnock constant, as well as the drag coefficient. Consequently, this work gives detailed measurements of these parameters in finite depths. Experiments conducted in the IRPHÉ/Pythéas wind-wave tank (Marseille, France), reveal that for a given wind speed, these values are higher in finite depth than in deep water. In the limit case of fully developed surface, due to depth, theoretical and empirical formulas relating the critical values of wave age to the non-dimensional depth have been experimentally verified. Plots of non-dimensional peak frequency, non-dimensional energy, and the inverse of wave age, against non-dimensional depth are presented. The plots clearly show that these quantities admit a limit due to the depth influence. All data obey the range of empirical values established in field experiments. Experimental data, obtained in the facility, agree with the theoretical family of depth-dependent wave growth rate as a function of wave age in finite depth. The non-dimensional growth-rate data obtained in our laboratory, as a function of the inverse of wave age, are consistent with the theoretical predictions of Miles theory in finite depth (Montalvo et al., 2013a,b, Latifi et al.. 2017), as well as measurements from other laboratories. • Wind-wave tank experiment on wave growth by the wind in finite depth environment. • Low-water depth have effects on transfers, drag, roughness and Charnock parameters. • For certain conditions of water depth and other parameters, the wave growth rate tend to zero. • Good comparison between Montalvo et al 2013 theory and experimental data. • All our experimental data are within the Young 1997 asymptotic depth limits.

Higher-order gap resonance between two identical fixed barges: A study on the effect of water depth

Nonlinear piston-mode fluid resonance in the gap formed by two identical fixed barges in close proximity is investigated using a two-dimensional (2D) fully nonlinear numerical wave tank. To delve into the effect of water depth on higher-order resonances in the gap, consistent models are employed to describe the incident waves and wave-structure interactions for finite and shallow water depths. In contrast to previous studies, strong fourth- and fifth-order nonlinear gap resonances are observed under the action of shallow-water waves, resulting in significant higher-harmonic responses that are comparable to the corresponding first and second harmonics. For a given wave height, those waves are found to be more critical than the finite-depth waves, which are capable of evoking second- or third-order gap resonance. Highly oscillatory behavior is observed at the trough under the action of shallow-water waves. This is ascribed to the higher-harmonic diffraction effects, breaking the “perfect cancellation” between the incident-wave harmonics, which should, otherwise, produce smooth and flatter wave trough. Unlike what a linear diffraction theory will predict, the gap response does not behave completely in a quasi-static manner under long cnoidal waves, and the higher harmonics generally show larger phase differences to the corresponding incident-wave components until the resonance occurs. The present study suggests that the water-depth effect and higher harmonics should be consistently accounted for in the design and analysis of side-by-side marine operations in coastal environments involving piston-mode gap responses.

Effect of Water Vorticity on Wind-Generated Gravity Waves in Finite Depth

The generation of wind waves at the surface of an established underlying vertically sheared water flow, of constant vorticity, is considered. A particular attention is paid to the role of the vorticity in water on wind-wave generation in finite depth. The present theoretical results are compared with experimental data obtained by Young and Verhagen (Coast Eng 29:47–78, 1996), in the shallow Lake George (Australia), and the least squares fit of these data by Young (Coast Eng 32:181–195, 1997). It is shown that without vorticity in water, there is a deviation between theory and experimental data. However, a good agreement between the theory and the fit of experimental data is obtained when negative vorticity is taken into account. Furthermore, it is shown that the amplitude growth rate increases with vorticity and depth. A limit to the wave energy growth, corresponding to the vanishing of the growth rate, is obtained. The corresponding limiting wave age is derived in a closed form showing its explicit dependence on vorticity and depth. The limiting wave age is found to increase with both vorticity and depth.

Progressive flexural–gravity waves with constant vorticity

Abstract

Atmospheric infrasound generation by ocean waves in finite depth: unified theory and application to radiation patterns

SUMMARYBetween 0.1 and 0.5 Hz, infrasound signals recorded in the atmosphere are dominated by ocean-generated noise called microbaroms. Microbaroms propagate through the atmosphere over thousands of kilometres due to low absorption and efficient ducting between the ground and the stratopause. Different theoretical models have been developed to characterize the source of microbaroms, all based on the second-order nonlinear interaction of ocean waves. While early theories considered an infinite ocean depth and a source radiation depending on the acoustic wave elevation angle, other works have approximated the radiation pattern as a monopole, and found a considerable effect of the water depth. This paper reviews these models and extends the previous theories to the combined effects of both finite depth ocean and source directivity in both elevation and azimuth angles. It is found that the water depth has a negligible effect for the near-horizontally propagating acoustic waves that should dominate the measured microbarom records. Another important result is that the microbarom azimuthal variation can be highly directive locally, but it generally becomes isotropic when integrated over a realistic source region.

Two-dimensional modulation instability of wind waves

It is widely known that deep-water waves are modulationally unstable and that this can be modelled by a nonlinear Schrödinger equation. In this paper, we extend the previous studies of the effect of wind forcing on this instability to water waves in finite depth and in two horizontal space dimensions. The principal finding is that the instability is enhanced and becomes super-exponential and that the domain of instability in the modulation wavenumber space is enlarged. Since the outcome of modulation instability is expected to be the generation of rogue waves, represented within the framework of the nonlinear Schrödinger equation as a Peregrine breather, we also examine the effect of wind forcing on a Peregrine breather. We find that the breather amplitude will grow at twice the rate of a linear instability.

On the Wave Bottom Shear Stress in Shallow Depths: The Role of Wave Period and Bed Roughness

Lagoons and coastal semi-enclosed basins morphologically evolve depending on local waves, currents, and tidal conditions. In very shallow water depths, typical of tidal flats and mudflats, the bed shear stress due to the wind waves is a key factor governing sediment resuspension. A current line of research focuses on the distribution of wave shear stress with depth, this being a very important aspect related to the dynamic equilibrium of transitional areas. In this work a relevant contribution to this study is provided, by means of the comparison between experimental growth curves which predict the finite depth wave characteristics and the numerical results obtained by means a spectral model. In particular, the dominant role of the bottom friction dissipation is underlined, especially in the presence of irregular and heterogeneous sea beds. The effects of this energy loss on the wave field is investigated, highlighting that both the variability of the wave period and the relative bottom roughness can change the bed shear stress trend substantially.

KAM for gravity water waves in finite depth

We present the recent result in [3] concerning the existence of Cantor families of small amplitude, linearly stable, time quasi-periodic standing water wave solutions – i.e. periodic and even in the space variable x – of a bi-dimensional ocean with finite depth under the action of pure gravity. Such a result holds for all the values of the depth parameter in a Borel set of asymptotically full measure.

Green–Naghdi dynamics of surface wind waves in finite depth

The Miles’ quasi laminar theory of waves generation by wind in finite depth h is presented. In this context, the fully nonlinear Green–Naghdi model equation is derived for the first time. This model equation is obtained by the non perturbative Green–Naghdi approach, coupling a nonlinear evolution of water waves with the atmospheric dynamics which works as in the classic Miles’ theory. A depth-dependent and wind-dependent wave growth γ is drawn from the dispersion relation of the coupled Green–Naghdi model with the atmospheric dynamics. Different values of the dimensionless water depth parameter δ = gh/U1, with g the gravity and U1 a characteristic wind velocity, produce two families of growth rate γ in function of the dimensionless theoretical wave-age c0: a family of γ with h constant and U1 variable and another family of γ with U1 constant and h variable. The allowed minimum and maximum values of γ in this model are exhibited.

Existence and amplitude bounds for irrotational water waves in finite depth.

We prove the existence of solutions to the irrotational water-wave problem in finite depth and derive an explicit upper bound on the amplitude of the nonlinear solutions in terms of the wavenumber, the total hydraulic head, the wave speed and the relative mass flux. Our approach relies upon a reformulation of the water-wave problem as a one-dimensional pseudo-differential equation and the Newton-Kantorovich iteration for Banach spaces.This article is part of the theme issue 'Nonlinear water waves'.

Linear and Weakly Nonlinear Models of Wind Generated Surface Waves in Finite Depth

This work regards the extension of the Miles’ and Jeffreys’ theories of growth of wind-waves in water of finite depth. It is divided in two major sections. The first one corresponds to the surface water waves in a linear regimes and the second one to the surface water waver considered in a weak nonlinear, dispersive and anti-dissipative regime. In the linear regime, we extend the Miles’ theory of wind wave amplification to finite depth. The dispersion relation provides a wave growth rate depending to depth. A dimensionless water depth parameter depending to depth and a characteristic wind speed, induces a family of curves representing the wave growth as a function of the wave phase velocity and the wind speed. We obtain a good agreement between our theoretical results and the data from the Australian Shallow Water Experiment as well as the data from the Lake George experiment. In a weakly nonlinear regime the evolution of wind waves in finite depth is reduced to an anti-dissipative Korteweg-de Vries-Burgers equation and its solitary wave solution is exhibited. Anti-dissipation phenomenon accelerates the solitary wave and increases its amplitude which leads to its blow-up and breaking. Blow-up is a nonlinear, dispersive and anti-dissipative phenomenon which occurs in finite time. A consequence of anti-dissipation is that any solitary waves’ adjacent planes of constants phases acquire different velocities and accelerations and ends to breaking which occurs in finite space and in a finite time prior to the blow-up. It worth remarking that the theoretical amplitude growth breaking time are both testable in the usual experimental facilities. At the end, in the context of wind forced waves in finite depth, the nonlinear Schr odinger equation is derived and for weak wind inputs, the Akhmediev, Peregrine and Kuznetsov-Ma breather solutions are obtained

An exact Hamiltonian coupled-mode system with application to extreme design waves over variable bathymetry

A novel, exact, Hamiltonian system of two nonlinear evolution equations, coupled with a time-independent system of horizontal differential equations providing the Dirichlet-to-Neumann operator for any bathymetry, is applied to the study of the evolution of wave trains in finite depth, aiming at the identification of nonlinear high waves in finite depth, and over a sloping bottom. The vertical structure of the wave field is exactly represented up to the instantaneous free surface, by means of an appropriately constructed, rapidly convergent, local vertical series expansion of the wave potential. This Hamiltonian system is used for studying the fully nonlinear refocusing of transient wave groups, obtained by linear backpropagation of high-amplitude wave trains constructed by the theory of quasi-determinism. The results presented give a first quantification of the effects of sloping bottom and of spectral bandwidth on rogue-wave dynamics and kinematics, in finite depth.

Gravity-capillary waves in finite depth on flows of constant vorticity.

This paper considers two-dimensional periodic gravity-capillary waves propagating steadily in finite depth on a linear shear current (constant vorticity). A perturbation series solution for steady periodic waves, accurate up to the third order, is derived using a classical Stokes expansion procedure, which allows us to include surface tension effects in the analysis of wave-current interactions in the presence of constant vorticity. The analytical results are then compared with numerical computations with the full equations. The main results are (i) the phase velocity is strongly dependent on the value of the vorticity; (ii) the singularities (Wilton singularities) in the Stokes expansion in powers of wave amplitude that correspond to a Bond number of 1/2 and 1/3, which are the consequences of the non-uniformity in the ordering of the Fourier coefficients, are found to be influenced by vorticity; (iii) different surface profiles of capillary-gravity waves are computed and the effect of vorticity on those profiles is shown to be important, in particular that the solutions exhibit type-2-like wave features, characterized by a secondary maximum on the surface profile with a trough between the two maxima.

On asymmetric generalized solitary gravity-capillary waves in finite depth.

Generalized solitary waves propagating at the surface of a fluid of finite depth are considered. The fluid is assumed to be inviscid and incompressible and the flow to be irrotational. Both the effects of gravity and surface tension are included. It is shown that in addition to the classical symmetric waves, there are new asymmetric solutions. These new branches of solutions bifurcate from the branches of symmetric waves. The detailed bifurcation diagrams as well as typical wave profiles are presented.