Waves In Finite Depth Research Articles

Internal gravity–capillary solitary waves in finite depth

We consider a two‐dimensional inviscid irrotational flow in a two layer fluid under the effects of gravity and interfacial tension. The upper fluid is bounded above by a rigid lid, and the lower fluid is bounded below by a rigid bottom. We use a spatial dynamics approach and formulate the steady Euler equations as a Hamiltonian system, where we consider the unbounded horizontal coordinate x as a time‐like coordinate. The linearization of the Hamiltonian system is studied, and bifurcation curves in the (β,α)‐plane are obtained, where α and β are two parameters. The curves depend on two additional parameters ρ and h, where ρ is the ratio of the densities and h is the ratio of the fluid depths. However, the bifurcation diagram is found to be qualitatively the same as for surface waves. In particular, we find that a Hamiltonian‐Hopf bifurcation, Hamiltonian real 1:1 resonance, and a Hamiltonian 02‐resonance occur for certain values of (β,α). Of particular interest are solitary wave solutions of the Euler equations. Such solutions correspond to homoclinic solutions of the Hamiltonian system. We investigate the parameter regimes where the Hamiltonian‐Hopf bifurcation and the Hamiltonian real 1:1 resonance occur. In both these cases, we perform a center manifold reduction of the Hamiltonian system and show that homoclinic solutions of the reduced system exist. In contrast to the case of surface waves, we find parameter values ρ and h for which the leading order nonlinear term in the reduced system vanishes. We make a detailed analysis of this phenomenon in the case of the real 1:1 resonance. We also briefly consider the Hamiltonian 02‐resonance and recover the results found by Kirrmann. Copyright © 2016 John Wiley & Sons, Ltd.

Lagrangian Surface Wave Motion and Stokes Drift Fluctuations

AbstractNonlinear effects in Lagrangian sea surface motions are important to understanding variability in wave-induced mass transport, wave-driven diffusion processes, and the interpretation of measurements obtained with moored or free-drifting buoys. This study evaluates the Lagrangian vertical and horizontal motions of a particle at the surface in a natural, random sea state using second-order, finite-depth wave theory. In deep water, the predicted low-frequency (infragravity) surface height fluctuations are much larger than Eulerian bound wave motions and of the opposite sign. Comparison to surface elevation bispectra observed with a moored buoy in steady, high-wind conditions shows good agreement and confirms that—in contrast to the Eulerian sea surface motion with predominant phase coupling between the spectral peak and double-frequency harmonic components—nonlinearity in Lagrangian wave observations is dominated by phase-coupled infragravity motions. Sea surface skewness estimates obtained from moored buoys in deep and shallow sites, over a wide range of wind–sea and swell conditions, are in good agreement with second-order theory predictions. Theory and field data analysis of surface drift motions in deep water reveal energetic [O(10) cm s−1] infragravity velocity fluctuations that are several orders of magnitude larger and 180° out of phase with Eulerian infragravity motions. These large fluctuations in Stokes drift may be important in upper-ocean diffusion processes.

Stokes coefficients and wave resistance

With the help of the Hamilton variational principle an infinite chain of compactly written quadratic equations with respect to the Stokes coefficients determining the periodic progressive finite-depth waves is constructed. An efficient algorithm of calculation of these coefficients in the form of series in terms of wave-amplitude powers is given. In analytical form, a ten-term expansion in terms of the amplitude for the wave-resistance force arising from motion under the free surface of a two-dimensional body generating the waves is constructed. The obtained expansion is compared with the Kelvin formula, which is single-term in amplitude, and with an accurate numerical solution.

Finite time blow-up and breaking of solitary wind waves.

The evolution of surface water waves in finite depth under wind forcing is reduced to an antidissipative Korteweg-de Vries-Burgers equation. We exhibit its solitary wave solution. Antidissipation accelerates and increases the amplitude of the solitary wave and leads to blow-up and breaking. Blow-up occurs in finite time for infinitely large asymptotic space so it is a nonlinear, dispersive, and antidissipative equivalent of the linear instability which occurs for infinite time. Due to antidissipation two given arbitrary and adjacent planes of constant phases of the solitary wave acquire different velocities and accelerations inducing breaking. Soliton breaking occurs in finite space in a time prior to the blow-up. We show that the theoretical growth in amplitude and the time of breaking are both testable in an existing experimental facility.

Calculating crest statistics of shallow water nonlinear waves based on standard spectra and measured data at the Poseidon platform

This paper introduces a new approach for the calculation of the wave crest distribution of shallow water nonlinear waves by utilizing a Transformed Rayleigh method where the second order nonlinear wave model is incorporated. The proposed Transformed Rayleigh method is based on a deterministic and time instantaneous functional transformation. It is very efficient and accurate and can be used for engineering purposes. Meanwhile, the correction for the effect of bottom on the wave spectrum, a procedure that is frequently overlooked by some researchers, has been integrated into the Transformed Rayleigh method. The proposed new approach has been first applied for calculating the wave crest height exceedance probabilities of sea states with standard JONSWAP spectra corresponding to different water depths, and the calculation results have been favorably validated against Monte Carlo simulation results. Meanwhile, the wave crest height exceedance probabilities of these sea states obtained from using the Jahns and Wheeler finite depth wave crest height distribution model have also been included for comparison purpose. It is found that in all cases the Transformed Rayleigh method can offer better predictions than the Jahns and Wheeler model. The Transformed Rayleigh method is then applied to calculate the wave crest height exceedance probabilities of a combined sea state with a bimodal Torsethaugen spectrum corresponding to a water depth of 25m, and its accuracy and efficiency are again favorably validated by using Monte Carlo simulations. Finally, the Transformed Rayleigh method is applied to predict the wave crest height exceedance probabilities of a sea state with the surface elevation data measured at the Poseidon platform in the Japan Sea, and its accuracy and efficiency have been once again convincingly substantiated.

Noise generation in the solid Earth, oceans and atmosphere, from nonlinear interacting surface gravity waves in finite depth

AbstractOceanic pressure measurements, even in very deep water, and atmospheric pressure or seismic records, from anywhere on Earth, contain noise with dominant periods between 3 and 10 s, which is believed to be excited by ocean surface gravity waves. Most of this noise is explained by a nonlinear wave–wave interaction mechanism, and takes the form of surface gravity waves, acoustic or seismic waves. Previous theoretical work on seismic noise focused on surface (Rayleigh) waves, and did not consider finite-depth effects on the generating wave kinematics. These finite-depth effects are introduced here, which requires the consideration of the direct wave-induced pressure at the ocean bottom, a contribution previously overlooked in the context of seismic noise. That contribution can lead to a considerable reduction of the seismic noise source, which is particularly relevant for noise periods larger than 10 s. The theory is applied to acoustic waves in the atmosphere, extending previous theories that were limited to vertical propagation only. Finally, the noise generation theory is also extended beyond the domain of Rayleigh waves, giving the first quantitative expression for sources of seismic body waves. In the limit of slow phase speeds in the ocean wave forcing, the known and well-verified gravity wave result is obtained, which was previously derived for an incompressible ocean. The noise source of acoustic, acoustic-gravity and seismic modes are given by a mode-specific amplification of the same wave-induced pressure field near zero wavenumber.

A coupled-mode system with application to nonlinear water waves propagating in finite water depth and in variable bathymetry regions

A non-linear coupled-mode system of horizontal equations is presented, modelling the evolution of nonlinear water waves in finite depth over a general bottom topography. The vertical structure of the wave field is represented by means of a local-mode series expansion of the wave potential. This series contains the usual propagating and evanescent modes, plus two additional terms, the free-surface mode and the sloping-bottom mode, enabling to consistently treat the non-vertical end-conditions at the free-surface and the bottom boundaries. The present coupled-mode system fully accounts for the effects of non-linearity and dispersion, and the local-mode series exhibits fast convergence. Thus, a small number of modes (up to 5–6) are usually enough for precise numerical solution. In the present work, the coupled-mode system is applied to the numerical investigation of families of steady travelling wave solutions in constant depth, corresponding to a wide range of water depths, ranging from intermediate depth to shallow-water wave conditions, and its results are compared vs. Stokes and cnoidal wave theories, as well as with fully nonlinear Fourier methods. Furthermore, numerical results are presented for waves propagating over variable bathymetry regions and compared with nonlinear methods based on boundary integral formulation and experimental data, showing good agreement.

Improved model for air pressure due to wind on 2D freak waves in finite depth

This paper presents an improved model for evaluating air pressure acting on 2D freak waves in a finite depth due to the presence of winds. This pressure model is developed by analysing the pressure distribution over freak waves using the QALE-FEM/StarCD approach, which combines the quasi arbitrary Lagrangian-Eulerian finite element method (QALE-FEM) with the commercial software package StarCD and has been proven to be sufficiently accurate for such cases according to our previous publication Yan and Ma (2010) [8]. In this model for air pressure, the pressure is decomposed into the components related to the local wave profiles and others. By coupling with the QALE-FEM, the accuracy of the pressure model is tested using various cases. The results show that the pressure distribution estimated using this model is close to that computed by using the QALE-FEM/StarCD approach when there is no significant vortex shedding and wave breaking. The accuracy investigation in predicting the freak wave heights and elevations demonstrates that this pressure model is much better than others in the literature so far used for modelling wind effects on freak waves in finite depth.

Modeling of depth‐induced wave breaking under finite depth wave growth conditions

Recent studies have shown that the spectral wind wave model SWAN (Simulating Waves Nearshore) underestimates wave heights and periods in situations of finite depth wave growth. In this study, this inaccuracy is addressed through a rescaling of the Battjes and Janssen (1978) bore‐based model for depth‐induced breaking, considering both sloping bed surf zone situations and finite depth wave growth conditions. It is found that the variation of the model error with the breaker index γBJ in this formulation differs significantly between the two types of conditions. For surf zones, clear optimal values are found for the breaker index. By contrast, under finite depth wave growth conditions, model errors asymptotically decrease with increasing values of the breaker index (weaker dissipation). Under both the surf zone and finite depth wave growth conditions, optimal calibration settings of γBJ were found to correlate with the dimensionless depth kpd (where kp is the spectral peak wave number and d is the water depth) and the local mean wave steepness. Subsequently, a new breaker index, based on the local shallow water nonlinearity, expressed in terms of the biphase of the self‐interactions of the spectral peak, is proposed. Implemented in the bore‐based breaker model of Thornton and Guza (1983), this breaker index accurately predicts the large difference in dissipation magnitudes found between surf zone conditions and finite depth growth situations. Hence, the proposed expression yields a significant improvement in model accuracy over the default Battjes and Janssen (1978) model for finite depth growth situations, while retaining good performance for sloping bed surf zones.

The effect of third-order nonlinearity on statistical properties of random directional waves in finite depth

Abstract. It is well established that third-order nonlinearity produces a strong deviation from Gaussian statistics in water of infinite depth, provided the wave field is long crested, narrow banded and sufficiently steep. A reduction of third-order effects is however expected when the wave energy is distributed on a wide range of directions. In water of arbitrary depth, on the other hand, third-order effects tend to be suppressed by finite depth effects if waves are long crested. Numerical simulations of the truncated potential Euler equations are here used to address the combined effect of directionality and finite depth on the statistical properties of surface gravity waves; only relative water depth kh greater than 0.8 are here considered. Results show that random directional wave fields in intermediate water depths, kh=O(1), weakly deviate from Gaussian statistics independently of the degree of directional spreading of the wave energy.

Second-Order Theory and Setup in Surface Gravity Waves: A Comparison with Experimental Data

Abstract The second-order, three-dimensional, finite-depth wave theory is here used to investigate the statistical properties of the surface elevation and wave crests of field data from Lake George, Australia. A direct comparison of experimental and numerical data shows that, as long as the nonlinearity is small, the second-order model describes the statistical properties of field data very accurately. By low-pass filtering the Lake George time series, there is evidence that some energetic wave groups are accompanied by a setup instead of a setdown. A numerical study of the coupling coefficient of the second-order model reveals that such an experimental result is consistent with the second-order theory, provided directional spreading is included in the wave spectrum. In particular, the coupling coefficient of the second-order difference contribution predicts a setup as a result of the interaction of two waves with the same frequency but with different directions. This result is also confirmed by numerical simulations. Bispectral analysis, furthermore, indicates that this setup is a statistically significant feature of the observed wave records.

Secondary criticality of water waves. Part 2. Unsteadiness and the Benjamin–Feir instability from the viewpoint of hydraulics

The theory for criticality presented in Part 1 is extended to the unsteady problem, and a new formulation of the Benjamin–Feir instability for Stokes waves in finite depth coupled to a mean flow, which takes the criticality matrix as an organizing centre, is presented. The generation of unsteady dark solitary waves at points of stability changes and their connection with the steady dark solitary waves of Part 1 are also discussed.

Third-order theory for bichromatic bi-directional water waves

A new third-order solution for bichromatic bi-directional water waves in finite depth is presented. Earlier derivations of steady bichromatic wave theories have been restricted to second-order in finite depth and third-order in infinite depth, while third-order theories in finite depth have been limited to the case of monochromatic short-crested waves. This work generalizes these earlier works. The solution includes explicit expressions for the surface elevation, the amplitude dispersion and the vertical variation of the velocity potential, and it incorporates the effect of an ambient current with the option of specifying zero net volume flux. The nonlinear dispersion relation is generalized to account for many interacting wave components with different frequencies and amplitudes, and it is verified against classical expressions from the literature. Limitations and problems with these classical expressions are identified. Next, third-order resonance curves for finite-amplitude carrier waves and their three-dimensional perturbations are calculated. The influence of nonlinearity on these curves is demonstrated and a comparison is made with the location of dominant class I and class II wave instabilities determined by classical stability analyses. Finally, third-order resonance curves for the interaction of nonlinear waves and an undular sea bottom are calculated. On the basis of these curves, the previously observed downshift/upshift of reflected/transmitted class III Bragg scatter is, for the first time, explained.

Steady dark solitary waves emerging from wave-generated meanflow: The role of modulation equations

Various classes of steady and unsteady dark solitary waves (DSWs) are known to exist in modulation equations for water waves in finite depth. However, there is a class of steady DSWS of the full water-wave problem which are missed by the classical modulation equations such as the Hasimoto-Ono, Benney-Roskes, and Davey-Stewartson. These steady DSWs, recently discovered by Bridges and Donaldson, are pervasive in finite depth, arise through secondary criticality of Stokes gravity waves, and are synchronized with the Stokes wave. In this paper, the role of DSWs in modulation equations for water waves is reappraised. The intrinsic unsteady nature of existing modulation equations filters out some interesting solutions. On the other hand, the geometry of DSWs in modulation equations is very similar to the full water wave problem and these geometrical properties are developed. A model equation is proposed which illustrates the general nature of the emergence of steady DSWs due to wave-generated mean flow coupled to a periodic wave. Although the existing modulation equations are intrinsically unsteady, it is shown that there are also important shortcomings when one wants to use them for stability analysis of DSWs.

Simulation of nonlinear random finite depth waves coupled with an elastic structure

Time accurate computation of random nonlinear shallow-water waves is required for the design of offshore wind turbines, and also in a multitude of other areas. Assuming the fluid to be inviscid and irrotational, nonlinear finite amplitude waves in a liquid body of finite depth are computed based on potential theory, i.e. the velocity potential must satisfy the Laplace equation at every instant in time. The free surface boundary conditions give a partial differential equation for the velocity potential and the free surface elevation on the free boundary. Similar coupling conditions are obtained at the boundary in contact with the deformable body. During the computation, the repeated solution of Laplace’s equation can be obtained by solving a boundary integral equation with the BEM and a fast multipole solver. The incident wave field is given by a stochastic wave process far from the structure, where the scattered (diffracted and radiated) wave field is comparatively small. This is finally coupled to another fluid description at the structure, which has a fully nonlinear description of both incident and scattered wave. The effect of the interacting flexible structure is included via a coupling computation. The chosen software architecture allows a modular coupling concept reflected in the algorithmically strong coupling of the various fluid domains, the structure, and the soil. As a numerical example, an offshore wind turbine is simulated, which has an additional fluid interaction with the stochastic wind field.

On the disappearance of the lowest-order instability for steep gravity waves in finite depth

This paper extends to very large steepness the results of the linear stability analysis of two-dimensional periodic gravity waves over a bottom of uniform depth. Both subharmonic and superharmonic instabilities are investigated. For the case considered herein, it is shown how the lowest-order instability (class I) completely disappears at a critical wave steepness. These results also confirm numerically a theorem connected with the appearance of superharmonic instabilities in finite depth.

Simulation of three-dimensional finite-depth wave phenomenon for moving pressure distributions

The flow field and the bottom pressure signatures due to an air cushion vehicle are calculated by analytical and computational means. The singularities in the integrals from the theoretical analyses are removed by using the Cauchy's residue theorem and the resulting integrals are numerically evaluated by the adaptive quadrature routines of QUADPACK.

Nonlinear energy fluxes and the finite depth equilibrium range in wave spectra

Results using a finite depth representation of Webb's [1978] transformation of Herterich and Hasselmann's [1980] equation for the rate of change of energy density in a random‐phase, spatially homogeneous, finite depth wave spectrum show that the equilibrium range in finite depth preserves a k−2.5 form consistent with Resio and Perrie's [1991] deepwater results and that the relaxation time toward an equilibrium range in shallow water is considerably faster than in deep water. Results from this finite depth nonlinear energy transfer representation compared to previously calculated results of analytical spectral situations show agreement, and the finite depth Zakharov [1968] and Herterich and Hasselmann [1980] forms are shown to be numerically equivalent. Spectral analyses of matching wave spectra sets at sites in 8 and 18 m depths at Duck, North Carolina, show a k−2.5 shape in the equilibrium range and show energy gains above the spectral peak and at high frequencies with energy loss in the midrange of frequencies near the spectral peak, consistent with four‐wave interactions. Spectral energy losses between these two sites correlate with spectral energy fluxes to high frequencies, again consistent with four‐wave interactions. The equilibrium range coefficient shows strong dependence on friction velocity at both gages.

EFFECTS OF A SEA FLOOR STEP ON WAVE DRIFT FORCE

A linear and a fully nonlinear numerical wave tanks (NWTs) were applied to study wave drift force acts on a two-dimensional Lewis form body in finite depth wave flume. These NWTs are based on potential theory and the fluid and floating body motions are directly simulated in time domain. Boundary value problems both on the velocity potential φ and its time derivative ∂φ/∂t are solved at each time step. The coupling condition between fluid and floating body is imposed as the implicit boundary condition of ∂φ/∂t on wetted body surface. The radiation condition at both tank ends are satisfied by artificial damping technique. Using these NWTs, effects of the floor step of the flume on wave drift force were studied. Measurement of wave drift force was also conducted in our two dimensional wave flume. In this report, the results of simulations and measurements are presented and the effect of the floor step on wave drift force is discussed.

On short-crested waves: experimental and analytical investigations

Analytical and experimental investigations were conducted on short-crested wave fields generated by a sea-wall reflection of an incident plane wave. A perturbation method was used to compute analytically the solution of the basic equations up to the sixth order for capillary-gravity waves in finite depth, and up to the ninth order for gravity waves in deep water. For the experiments, we developed a new video-optical tool to measure the full three dimensional wave field η( x, y, t). A good agreement was found between theory and experiments. The spatio-temporal bi-orthogonal decomposition technique was used to exhibit the periodic and progressive properties of the short-crested wave field.