Linear Water Wave Research Articles

ON WELL-POSED BOUNDARY CONDITIONS AND ENERGY STABLE FINITE-VOLUME METHOD FOR THE LINEAR SHALLOW WATER WAVE EQUATION

Abstract We derive and analyse well-posed boundary conditions for the linear shallow water wave equation. The analysis is based on the energy method and it identifies the number, location and form of the boundary conditions so that the initial boundary value problem is well-posed. A finite-volume method is developed based on the summation-by-parts framework with the boundary conditions implemented weakly using penalties. Stability is proven by deriving a discrete energy estimate analogous to the continuous estimate. The continuous and discrete analysis covers all flow regimes. Numerical experiments are presented verifying the analysis.

Effect of shoaling length on rogue wave occurrence

The impact of shoaling on linear water waves is well known, but it has only been recently found to significantly amplify both the intensity and frequency of rogue waves in nonlinear irregular wave trains atop coastal shoals. At least qualitatively, this effect has been partially attributed to the ‘rapid’ nature of the shoaling process, i.e. shoaling occurs over a distance far shorter than that required for waves to modulate themselves and adapt to the reduced water depth. Through a theoretical model and highly accurate nonlinear simulations, we disentangle the respective effects of the length and angle of a shoal's slope. We investigate the effects of the shoaling process rapidness on the evolution of key statistical and spectral sea-state parameters. We let the wave field evolve over a slope with constant angle in all cases while we vary the slope length. Our results indicate that the non-equilibrium dynamics is not affected by the slope length, because further extending the slope length does not influence the magnitude of the statistical and spectral measures as long as the non-equilibrium dynamics dominates the wave evolution. Thus, the shoaling effect on rogue waves is deduced to be mainly driven by the slope magnitude rather than the slope length.

Physical modelling study on wave damping induced by an idealized floating kelp farm

A physical modelling study was carried out to investigate random wave damping promoted by an idealized floating kelp farm. The experimental conditions spanned intermediate water depths and both linear and nonlinear water waves. Unlike previous studies of wave damping promoted by vegetation, the floating kelp farm was placed close to the water surface with a ratio between vegetation height and water depth close to 0.25. The wave transmission coefficient induced by the floating kelp farm ranged between 0.56 and 0.96. This coefficient decreased for longer floating kelp farms and it was a function of the ratio between kelp farm length and incident wavelength and of the relative wave depth. Spectral analysis showed that wave damping was not frequency-dependent for wave frequencies close to the peak frequency. The wave transmission coefficients of a floating kelp farm with about 100 culture lines and with an extension of approximately 200 m were similar to those of submerged detached breakwaters with a relative crest freeboard smaller than -0.4. Furthermore, the bulk drag coefficient of near-surface idealized floating kelp farms can be modelled as a function of the Keulegan-Carpenter number. This study highlights the potential viability of nature-based solutions such as floating kelp farms for coastal protection.

Compressible ocean waves generated by sudden seabed rise near a step-type topography

A typical tsunami generation occurs through submarine earthquakes leading to large volume displacement. The corresponding mathematical problem involves modeling surface water waves generated by an arbitrary temporal motion of the ocean floor. The propagation of tsunami wave and the subsequent scattering from a sudden drop in bathymetry away from the ground motion is studied following linearized water wave theory and a weakly compressible ocean, including static oceanic background compression. The Fourier transformation and eigenfunction expansion techniques are employed to find the surface displacement and pressure profiles by leveraging appropriate matching conditions between regions of different depths. A novel energy balance relationship is derived by considering both the pure-gravity and acoustic-gravity modes. The model is validated in the limit that the depth difference approaches zero, showing a vanishing reflection contribution from the depth change. An efficient numerical code is developed that accurately captures the contribution of the cutoff frequencies of acoustic-gravity modes. Apart from the time-domain propagation of tsunami waves away from the origin, standing wave formations are observed within the shallow region, supported by significantly large pressure fluctuations in time. These standing waves or, equivalently, the pressure fluctuations sustain longer for larger ocean depth. The increase in tsunami speed in the deeper region is readily visible in the time-domain simulations. A three-dimensional axisymmetric solution is also developed, and results show a more gradual sloping tsunami wavefront compared to the equivalent two-dimensional solution for shallower depths. Animation movies corresponding to the two- and three-dimensional surface profiles are provided for better visualization.

An analytical solution to the soil’s dynamic response induced by wave interaction with a marine current characterized by a non-uniform vertical profile

In this work, we obtain an analytical solution to the soil’s dynamic response induced by wave interaction with a marine current with a prescribed non-uniform vertical profile. The control volume used to analyze the present problem is divided into a water region and another composed of a poroelastic soil region. The governing equations for the water hydrodynamics are presented in its dimensionless version and they are simplified for the limit of linear water waves. The velocity field in the water region is determined from the continuity equation and the “inviscid Orr-Sommerfeld linear equation”. We obtain a dispersion equation for wave-current interaction. For modeling the soil’s dynamics response, we adopt the u-p approximation. Considering a characteristic criterion for soil liquefaction, the maximum liquefaction depth is obtained as an eigenvalue problem. The analytical estimations, valid for the limit of linear water waves, provided by the present analysis show that, marine currents traveling in the same direction to the wave propagation induces larger values of the liquefaction depth than those obtained for currents opposite to wave propagation. As a first approximation, the present analysis may help understand the dynamic response of poroelastic soil induced by the wave-non-uniform marine current interaction.

Wave excited motion of a body floating on water confined between a semi-infinite ice sheet and a side wall

In this paper, we investigate the diffraction and radiation coupling problem of a floating body confined between a fixed wall and a semi-infinite ice sheet under the action of incident waves. Based on the linear water wave theory and Kirchhoff Love elastic thin plate assumption, the ice sheet is considered as an elastic beam, and the boundary conditions of fluid boundaries covered by the ice sheet are obtained. Dividing flow domains with different upper surfaces, the eigenfunction expansion method was used to obtain the velocity potential expansion equations for each sub-domain. Then, through the matching conditions at the interface of sub-domains, a system of equations was constructed to solve the unknown coefficients of the expansion equations. The influence of the existence of the ice sheet on the diffraction and radiation motion of the floating body is analyzed. The effects of changes in the draft and open water width of the floating body on the wave excitation force and hydrodynamic coefficient of the floating body are discussed. The influence of the reflection effect of the ice sheet on the added mass and damping coefficient of the floating body has been discovered, especially the changes and oscillation rules of their extreme points. This research achievement provides theoretical support for the safety design of floating structures in frozen harbors.

Atmospheric pressure-induced three-dimensional surface wave propagation in the compressible ocean: Effect of static compression

Under the assumptions of linearized water wave theory, we build a three-dimensional mathematical model that couples atmospheric pressure waves and surface ocean waves, including water compressibility and its static part, to simulate Meteotsunami propagation in the ocean. The solution uses the Laplace–Fourier double transformation technique, emphasizing axisymmetry of the mathematical problem and rigorous treatment of a fairly complicated dispersion relation while using inverse transformations. A novel derivation of the axisymmetric atmospheric pressure front is shown. The impact of water compressibility is shown through a comparative graphical representation against the incompressible case. Faster travel of free-surface waves is observed in the incompressible ocean, followed by the cases with and without static compression of the compressible ocean, respectively. The static compression shifts the phase of the acoustic-gravity modes. The locked wave is hardly influenced by the water compressibility and is entangled with the moving pressure front. The model is validated with the observational pressure data and agrees well with our computed pressure profile. Then, the locked wave profile generated from our model agrees well with the deep-ocean assessment and reporting of tsunami data.

Scattering of Oblique Water Waves by Two Vertical Barriers Over Trench-Type Bottom

The problem of scattering of oblique water waves by two surface-piercing vertical thin rigid barriers over a trench-like bottom is investigated within the framework of linear water wave theory. The eigenfunction expansion technique is used to obtain a system of algebraic equations, which is further solved using the algebraic Least squares method. The reflection and transmission coefficients are computed and contrasted with the prior results. Physical characteristics such as surface elevation and horizontal force on both barriers are also determined and depicted graphically against various parameters to see their behavior. It is possible to strategically construct barriers and trench-like bottoms to regulate how waves reflect and propagate. This might mitigate the wave impact on ships, improving the safety, effectiveness and longevity of ships navigating through the channels. The highlights of the given problem include comparing the impacts of different depth ratios and barrier placements, followed by a study of the modulation parameter to evaluate standing wave patterns in the trench region. The specific circumstances for the formation of the standing wave are identified. Hence, understanding this study is critical for safe navigation because standing waves may produce hazardous conditions for boats, making navigation difficult and causing substantial damage.

Interaction of short-crested waves with cylinder and concentric segmented arc breakwaters

Using the outer breakwater of Palm Jumeirah in Dubai as a physical geometric model, the diffraction problem of short-crested waves interacting with concentric segmented arc structures was theoretically investigated using linear water wave theory. In this study, the watershed was partitioned into two regions, and a unique coordinate system was established to describe the velocity potential function within each region. The interface between the two regions connects the velocity potentials across them using the matching eigenfunction expansion technique. By accounting for the boundary conditions, the integral equation was streamlined into a series of algebraic equations, which were subsequently resolved to determine an unidentified function. The precision of the model was validated through a comparison with findings from the literature. Numerical investigations have elucidated the correlation between changes in wave force, cylinder run-up, and parameters such as the spacing between adjacent arcs, opening angle, and porous coefficient. The numerical results indicated that the impermeable segmented arcs structure provides better protection against short-crested and plane waves, whereas the permeable segmented arc structure provides better protection against standing waves. Interestingly, for a segmented three-arc structure, when the segments are moderately spaced, the protection of the interior region is superior to that of an unsegmented arc structure.

General solution for linear 2D anisotropic surface gravity water waves on uniform current

Measuring and modeling ocean waves is a routine task for oceanographers and marine engineers. Accounting for current effects on waves is less common; however, such effects can be non-trivial, particularly for strong currents. Recent research has also found that currents are increasing with global temperatures. Thus, accounting for currents is of potential importance to a growing subset of real-world environments. One significant impact of currents is their modification of dispersion. It is known that, in contrast to dispersion without current, wave–current dispersion is multivalued and anisotropic, meaning it varies with direction and can yield multiple solutions for the same frequency. Strictly speaking, a complete analysis ought to account for all dispersion solutions. This study presents a general wave field solution for linear steady-state two-dimensional surface gravity waves on current. In contrast to existing formulations, it accounts for the complete set of dispersion solutions. The multiple solutions correspond to linearly independent (same frequency) spatial modes with no analogy in models without current. To fully determine the complete temporal and spatial features of a wave field, one must determine the amplitudes of all spatial modes. This can be achieved through the solution of an inverse problem, provided one has sufficient initial conditions. The significance of accounting for all spatial modes is demonstrated through examples. It is shown that a single time-series measurement is insufficient for determining the spatial features of a monochromatic (single frequency) wave. Rather, additional spatial information must be introduced to render the problem well-posed. The theory and methodology presented can be used to improve wave models and measurements.

The effect of different edge conditions on the motion of a submerged elastic disc

The present study examines the effects of various edge conditions of a submerged flexible disc on wave propagation for infinite-depth water within the framework of linear water waves theory. Three types of edge conditions (namely Free edge, simply supported edge, and clamped edge) are taken into consideration for the analysis. The governing boundary value problem (BVP) has been solved by reducing it to a two-dimensional hypersingular integral equation. The notion of modal analysis has been adopted to examine the structural response of the disc on the propagation of waves. Later, the two-dimensional hypersingular integral has been transformed into a second kind one-dimensional Fredholm integral equation by applying Fourier series expansion. Finally, the Nystrom technique based on Gauss–Legendre quadrature nodes is used to obtain an approximate solution of the one-dimensional integral equation. The computed solution is used to evaluate the numerical estimates of the physical quantities such as added mass, damping coefficient, hydrodynamic force, and surface elevation for different edge conditions mentioned earlier.

Semi-analytical approach to study wave interaction with a pair of obliquely submerged heterogeneous flexible structures

ABSTRACT Analyzing inclined flexible plates of varying thickness is crucial for optimising and attaining precise control over wave reflection and transmission. This is particularly significant in the context of breakwater construction. In contrast to horizontal breakwaters, inclined barriers can penetrate multiple layers of fluid with varying particle velocities, facilitating interactions and causing wave breaking, which results in the dissipation of wave energy. Moreover, compared to vertical structures, inclined ones demonstrate more effective wave attenuation. In the context of the present study, we investigate the interaction of water waves with a pair of obliquely submerged flexible thin plates characterised by non-uniform thickness. In order to model this complex physical scenario, we employ the principles of linear water wave theory in conjunction with Kirchhoff's thin plate theory. By employing repeated integration and Green's integral theorem, we reduce the boundary value problem to a system of coupled integral equations. Through appropriate approximations, we solve this system and obtain numerical values for various hydrodynamic quantities. This model provides comprehensive results for both horizontal and vertical plates, making it highly versatile. We examine the impact of varying thickness and the inclination of the two flexible plates to understand their roles in the wave scattering process and the associated physical parameters. The results obtained from this study shed light on the intricate dynamics involved in wave-structure interactions and can inform the design and optimisation of breakwater systems.

Gravity wave interaction with a heaving membrane above a thick porous bed

The present study analyzes diffraction and radiation phenomena of oblique waves interacting with a heaving floating membrane in the presence of a thick porous bed. Following the linear water wave theory, the physical problem is framed mathematically. The significance of the article resides in the following: (1) progressive wave analysis (water and membrane-covered region), (2) solving the boundary value problem (BVP) using the matched eigenfunction expansion method for diffraction and radiation problems, and (3) numerical illustration of various hydrodynamic coefficients for different membrane and porous bed parameters. Bragg scattering with varying frequency is observed for smaller values of membrane tension. Also, the present study demonstrates that the number of oscillations experienced by the reflection coefficient increases proportionally with the length of the membrane. Furthermore, cut-off membrane properties exist at a given frequency for which the zero minimum of wave force is obtained. Also, the porous bed's thickness impacts wave reflection and membrane deflection significantly. Thus, we found that the maximum reflection is observed for a fully permeable bed; however, it decreases with a decrease in the porosity of the porous medium because of its dissipative nature. Conversely, the added mass and damping coefficient increases with increased membrane length. The collective numerical observations for both diffraction and radiation provide insight into resonance phenomena, the role of membrane properties, and the intricate relationship between wave characteristics and membrane properties. The findings from this study could assist geologists and marine engineers in designing and managing ports and harbor infrastructure.

Effect of static compression on near-field tsunami waves: Three-dimensional solution

An analytical solution of three-dimensional surface wave profiles due to arbitrary spatio-temporal disturbance of a circular ocean bottom in a compressible ocean is obtained by incorporating the influence of the static ocean background compression under the assumptions of linearized water wave theory. Time-domain simulations of the surface profile in three dimensions and the pressure distribution within the water column for a circular uniform rise and tilt are shown. The corresponding animation movies depict the temporal evolution of the surface profile and pressure field inside the water column eloquently, which was not shown in earlier literature. The impact of static compression is also discussed through the simulations. A novel analytical expression of the potential function for a generic tilted motion (rmcos(mθ),m∈Z) of the circular ocean floor is derived. An efficient numerical code is developed to find surface elevation and pressure distribution, implementing the inverse Fourier integral as matrix multiplication. Validation is performed for the specific case of a rising flat ocean floor, showing the oscillations due to acoustic-gravity modes. Initially, a simplified problem of a flat rising ocean bottom is solved using the eigenfunction matching method, which involves finding a particular solution for the nonhomogeneous ocean bottom condition and the solution for its homogeneous counterpart. Solutions are obtained using a newly developed inner product between the depth-dependent functions. Later, a Green function technique is used to incorporate the impact of the arbitrary spatio-temporal motion of the circular portion of the ocean bed. The solution obtained from the eigenfunction matching method is utilized to obtain the analytical form of Green’s function and, eventually, an expression of surface elevation and pressure distribution inside the ocean water column.

Analytical solutions for hydrodynamic responses of arrays of floating truncated cylinders using multi-term Galerkin method and its application to a new wave energy converter device

In the context of linear water wave theory, the analytical solutions for the diffraction and radiation of a truncated cylinder array are developed in the presence of ambient incident waves. Each cylinder in the array can oscillate with five degrees of freedom (DOFs), i.e., surge, sway, heave, roll, and pitch. This paper adopts the multi-term Galerkin method to expand the fluid velocity at the interface of different regions into a set of basis functions containing Gegenbauer polynomials, which accurately and efficiently characterizes the cube root singularity of the fluid velocity near the edges of the truncated cylinders. Using the dynamic equilibrium equations, the amplitudes of each DOF of the cylinders in the array are solved. The analytical solution presented in this paper converges rapidly, and high-precision hydrodynamic response results can be obtained using just a few truncated terms (e.g., the upper bounds of m0 = 5 and p0 = 22 can yield results of five-figure accuracy). For the 4-cylinder array, under the same accuracy conditions (the error less than 1%), the computation time of the conventional method developed by Zeng et al. [“Hydrodynamic interactions between waves and cylinder arrays of relative motions composed of truncated floating cylinders with five degrees of freedom,” J. Fluids Struct. 115, 103785 (2022d)] based on the exact algebraic method [Kagemoto and Yue, “Interactions among multiple three-dimensional bodies in water waves: An exact algebraic method,” J. Fluid Mech. 166, 189–209 (1986)] is 3.9 times longer than that of the present method. As the number of cylinders increases, the advantage of the present method in terms of convergence speed becomes more apparent, e.g., for the 16-cylinder array, the conventional solution takes 6.3 times longer than the present solution. To extract wave energy more efficiently, a new 5DOF wave energy converter (WEC) device that can extract energy in 5DOFs is proposed. The present method is adopted to investigate the hydrodynamic performance of the 5DOF WEC arrays. Compared with the traditional 1DOF (heave) WEC, the 5DOF WEC can significantly improve the energy capture performance of arrays, especially in the high-frequency wave region.

Analysis of Hydrodynamic Loads by a Vertical Hollow Cylindrical Structure in A Two-Layer Fluid of Finite Depth

This paper aims to study the effect of surge radiation by a vertically hollow cylinder floating in a two-layer fluid of finite depth. Since most of the important characteristics of the oscillating water column are preserved by the hollow cylindrical structure, so the proposed device can be considered as an oscillating water column (OWC) which is one of the important wave energy device. In this two-layer fluid system, the upper layer is bounded above by a free surface and the lower layer is bounded below by a solid impermeable bottom, and the submerged hollow cylindrical structure allows it to oscillate in a surge mode of motion in the upper layer. On the assumption that wave amplitudes are small in comparison to wavelengths, hence the linear water wave theory is applied to formulate boundary value problems by dividing whole domain into 2 sub-domains. The formulated boundary value problems for surge radiated potentials in each sub-domain; we execute to obtain solutions in each sub-domain by using variable separation and matched eigenfunction expansion methods. With the help of obtained surge radiated potentials, we derive the radiation forces by the device in terms of added mass and damping coefficients. A set of influences of different parameters of the device on the added mass and damping coefficients are demonstrated in both surface and internal wave modes of motion. Hydrodynamic coefficients oscillate highly near a particular frequency, which leads to the resonance phenomena. In addition to that, the 3D plots of the free surface elevation in surface and internal wave modes are presented. A density ratio γ = 0.97 has a higher oscillation than a density ratio γ = 0.93, 0.95. High oscillations occur around a particular frequency ω = 3.83 due to the resonance phenomenon in which the waves incoming match with the natural motion of the object. HIGHLIGHTS A floating hollow vertical cylindrical structure is considered in a two-layer fluid system having finite depth The separation of variables and matched eigenfunction expansion methods are utilized to obtain the solutions to the boundary-value problems The diffracted velocity potentials under the surge mode of motion are evaluated in the identified regions The added mass, damping coefficients, free surface and interface elevations in surface and internal wave modes are evaluated with different set of parameters of the device A number of results are presented regarding the effect of parameters of the device on the added mass, damping coefficients, free surface and interface elevations It is observed that the density ratio of the two-layer fluid has significant effect on the added mass, damping coefficient and the elevations GRAPHICAL ABSTRACT

A study on the interaction of waves with a submerged vertical elastic plate present in either layer of a two-layer fluid

The problems of interactions between an elastic plate and surface as well as interface waves are considered and analyzed using the theory of linearized water waves. Considering the ocean’s multilayered nature, a two-layered fluid system that serves as the foundation for a more complex multilayered fluid model is examined. It is taken into consideration when an elastic plate is present in either layer of the two-layered water domain. Applying Green’s function technique on the boundary value problem related to the flexible plate and its end conditions, an expression of the normal derivative of the potential function is derived. Employing Green’s identity on the fundamental potential function and the scattered potential, an alternative expression for the normal derivative of the potential function is obtained. Comparison of alternative expressions of the normal derivative of the potential function allows the deduction of a Hypersingular Integral Equation (HSIE) in terms of the unknown potential difference across the plate. Then, using the Expansion-Collocation approach, the HSIE is resolved. The resulting solution provides insights into the reflection coefficient and plate deflection as they vary with different parameters.

Wave trapping by porous breakwater near a wall under the influence of ocean current

In this paper, oblique water wave interaction with a surface piercing box-type porous structure placed in front of a vertical rigid wall is investigated in the presence of current, which flows parallel to vertical wall. The current speed is considered distinct in different subdomains. However, in a particular subdomain current speed is uniform throughout the water depth. In this analysis, Sollitt and Cross model is used for the study of wave motion inside the porous structure. Employing small amplitude linear water wave theory, the present model of our interest is developed into a boundary value problem and solved analytically using the eigenfunction expansion method. Using the developed solution, the reflection coefficient and hydrodynamic wave forces on porous structure are computed for different values of parameters: the dimension of porous structure, the friction coefficient of porous structure, the current speed of following current and opposing current. This research paper suggests that suitable arrangement of rigid walls and surface-piercing partially extended porous structures can be an effective and economical approach for guiding ocean currents and protecting marine facilities against wave attacks.

Design of Infinite-Dimensional Observers for a Linearized Water Wave System With Distributed or Point Measurement

Design of Infinite-Dimensional Observers for a Linearized Water Wave System With Distributed or Point Measurement

Water-Wave Vortices and Skyrmions.

Topological wave structures-phase vortices, skyrmions, merons, etc.-are attracting enormous attention in a variety of quantum and classical wave fields. Surprisingly, these structures have never been properly explored in the most obvious example of classical waves: water-surface (gravity-capillary) waves. Here, we fill this gap and describe (i)water-wave vortices of different orders carrying quantized angular momentum with orbital and spin contributions, (ii)skyrmion lattices formed by the instantaneous displacements of the water-surface particles in wave interference, and (iii)meron (half-skyrmion) lattices formed by the spin-density vectors, as well as (iv)spatiotemporal water-wave vortices and skyrmions. We show that all these topological entities can be readily generated in linear water-wave interference experiments. Our findings can find applications in microfluidics and show that water waves can be employed as an attainable playground for emulating universal topological wave phenomena.