Linear Water Wave Theory Research Articles

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.

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.

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

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.

Hydrodynamic responses and layout optimization of wave energy converter arrays consisting of five-degree-of-freedom truncated cylinders in front of a vertical wall

The hydrodynamic responses and layout optimization of a group of cylindrical wave energy conversion devices (WEC) in front of a fully reflecting vertical wall are investigated. Each truncated floating cylinder can oscillate with five degrees of freedom, i.e., surge, sway, heave, roll, and pitch. Based on the linear water wave theory, an analytical solution is developed for the hydrodynamic problem. The results of specific parameter studies suggest that the wall reflection effect significantly improves the energy extraction performance of the WEC array with the appropriate parameter conditions. A multi-level optimization method based on a genetic algorithm is developed. This paper investigates the optimal layout of the six WEC arrays, composed of 2–7 buoys, respectively. Additionally, the impact of other degrees of freedom (DOFs), besides the heave mode, on the hydrodynamic performance of the array is investigated. For β ≤ π/12, there is no need to consider the impact of other DOFs on the energy extraction in heave mode. The dimensionless amplitudes of other DOFs gradually decrease as the equivalent constraint stiffness increases. For k0a > 1.0, the heave amplitude and energy capture performance of the WEC array are significantly smaller. However, the amplitudes of other DOFs still have considerable magnitudes for k0a > 1.0. Therefore, for the sea area with high-frequency incident waves (k0a > 1.0), setting up a power takeoff system on other DOFs of each buoy to extract energy is a feasible solution to improve the performance of the WEC array.

Linear diffraction and radiation theory of water waves by a truncated vertical circular cylinder with heave plate in deep and shallow drafts

A first-order analytical theory on a truncated and surface-piercing vertical circular cylinder with a circular plate mounted at the bottom of the cylinder was generalized to solve the linear wave diffraction and radiation problems based on potential flow and the hypothesis of small wave and motion amplitudes. The domain was decomposed, and the linearized velocity potentials were derived in each subdomain and matched on each subdomain interface employing pressure and normal velocity continuity. The linear hydrodynamic loads obtained with the analytical method were compared with the results of linear boundary element method (BEM) solvers. Dedicated experimental campaigns were performed on fixed models with regular waves (diffraction) and then on models oscillating in surge, heave, and pitch motions without incident waves (radiation). The analytical method describes well the wave excitation loads from the experiments for small wave steepness (H/λ=0.02). The predictions of added mass and damping are shown to be applicable to the surge motion, only for small Keulegan-Carpenter numbers (KC<1). On the other hand, the analytically predicted radiated waves demonstrate satisfactory agreement with experiments for all motions.

The Moving Element Method for the Dynamic Responses of Floating Double-Plate Structures on the Shallow Water

In this paper, the dynamic behavior of floating double-plate structures under a moving load is investigated in great detail with different factors utilizing the moving element method (MEM). The structures in this study are very large floating structures (VLFSs). The Kirchhoff–Love theory is used to model two floating plates connected by a Winkler-type elastic core, whilst the water surrounding the structure is simulated by the shallow linear water wave theory. In order to improve the computational efficiency of moving load problems, the MEM is adopted. Since the MEM uses fewer discrete elements than those of the traditional finite element method (FEM), the computational cost is reduced. The proposed two-plate model helps to reduce the influence of the surrounding water on the upper plate. Several numerical results are presented to investigate the influence of the ratio of thickness between layers as well as core stiffness on the displacements with different water depths, and load speeds.

A semi-analytical model of an array of moored floating flexible offshore net cages under current loads

An analytical model of an array of moored floating flexible net cage systems is developed to study the dynamics and wave loads on the net cages under the linear water wave theory and small-amplitude wave response assumption. The Fourier–Bessel series expansion of the velocity potential is derived for the regions enclosed under the open-water and cage systems and their proximity. The theoretical solution is obtained by employing the least squares approximation method along with matching the velocity and pressure at the fluid-structure interfaces. The present result of horizontal displacement and horizontal wave load are compared with a published finite element analysis and with experimental data sets, respectively. Further, the results of cage displacements, cage accelerations, and wave loads on the different arrangements for mooring stiffness, cage radius, cage heights, and current speeds are compared and analysed. Moreover, the present analysis will be helpful to understand designing further a multiple floating fish cage system with flexible floaters application to offshore aquaculture.

Radiation study of coaxial porous cylindrical structures for a given oscillation

The paper uses the eigenfunction expansion method to investigate the radiation of coaxial porous cylindrical structures in the ocean for a given oscillation. The study of the coaxial porous cylindrical structures considers two cases: (1) a coaxial porous structure placed on the seabed, and (2) a floating coaxial porous structure. Based on the linear water wave theory and the eigenfunction expansion method, the equations for the velocity potential, the added mass, and the damping coefficient of the coaxial porous cylindrical structure are obtained. Darcy’s law is satisfied at the porous boundary. The continuity condition is satisfied between the fluid regions. The added mass and damping coefficient are calculated based on the velocity potential. The radiation results of the two cases are studied, and the influence of different parameters on the structures can be obtained. In addition, the porous parameters and radius ratio have a significant effect on the hydrodynamic coefficients (the added mass and the damping coefficient) of the structure, so the appropriate porous parameters and radius ratio can be selected to reduce the effect on the structure. The results in this paper provide clear information for the selection of appropriate parameters when engineering in offshore and coastal regions.

Wave power extraction from slanted oscillating water columns in the presence of an array of arbitrary trenches

This paper investigates the impact of arbitrary bottom topography on the efficiency of a slanted oscillating water column device located in near-shore areas. The device incorporates a turbine positioned at its opening, which harnesses the oscillation of the fluid at the free surface, created by recurrent incident waves, to propel an air volume in a uni-direction. To analyze the wave power absorption capability of the system, an eigenfunction expansion and a boundary element model based on linear water wave theory are employed. The notion is to use the anticipated amplification of scattered and reflected wave fields resulting from the existence of a sloping wall to improve the wave power absorption caused by the walls' wave reflections. The accuracy of the current study's numerical results is ratified by conducting a comparative study using the available data given by other researchers. The study demonstrates that multiple efficiencies occur in the case of the long-wave region as compared to the short-wave region. Additionally, both the shape and magnitude of the efficiency curves are influenced by the slanted design of the device's front wall and the number of trenches.

Hydrodynamic response of dual obliquely submerged non-uniform flexible thin plates

Examining inclined flexible plates with variable thickness becomes crucial when it comes to optimizing or achieving controlled reflection and transmission of waves, especially for the construction of breakwaters. Unlike horizontal breakwaters, inclined barriers have the ability to penetrate through numerous layers of the fluid, having different particle velocities and foster their interactions. This causes wave breaking which leads to loss of wave energy. Also, despite exhibiting a similar behavior, vertical structures do not attenuate waves as effectively as inclined ones do. Additionally, the resonant motion of the fluid trapped between two structures proves to enhance the attenuation, thus it is recommended to include dual structures in the model rather than just one. Therefore in the present study, we examine the water wave scattering phenomenon by a pair of symmetric flexible thin plates with non-uniform thickness obliquely submerged in deep water. We use the linear water wave theory and Kirchhoff’s thin plate theory to model the physical problem. The boundary value problem is converted into a system of coupled integral equations using repeated integration and Green’s integral theorem. Using appropriate approximations, this system is solved and its solutions are used to determine numerical values of different hydrodynamic quantities. Results of two horizontal as well as two vertical plates could be obtained from the present model, thus it is a very general model. Also results are illustrated to analyze the contribution of the thickness variation and the inclinations of the two flexible plates towards the wave scattering process and some related physical quantities.

Wave trapping by freely floating obstacles in a discretely stratified fluid

In the context of the linear water wave theory, we model the interaction of time‐harmonic water waves with an infinite array of three‐dimensional freely floating bodies in a multilayer fluid through a spectral boundary value problem, consisting of a differential equation coupled with an algebraic system. We present a variational and operator formulation for the problem and derive sufficient conditions for the existence of trapped waves. We give examples of floating obstacles supporting trapped waves in a discretely stratified fluid in different obstacle configurations with a variable number of layers. Some conclusions regarding the dependence of the sufficient condition on the number of fluid layers are drawn, considering whether the sufficient condition is of the potential or algebraic type.

A mathematical study of a two-layer fluid flow system in the presence of a floating breakwater in front of VLFS

The present study deals with the usefulness of a surface-piercing porous structure of definite width in a two-layer fluid placed at a finite distance from a floating elastic plate to mitigate the hydrodynamic response of the elastic plate. We use linear water wave theory with the eigenfunction expansion approach to develop analytical solutions to the model problem in finite water depth. Here, we identify and explore the complete parameter space associated with the problem by examining the hydrodynamic coefficients for waves in surface and internal modes, plate elevation, shear force, bending moment, wave loads on the elastic plate, flow amplitude in the plate-covered region and wave attenuation. We investigate the effectiveness of structures of various configurations and geometries on wave scattering. This study demonstrates an oscillatory pattern of wave reflection in the confined region between the plate and the breakwater. Moreover, optima in the oscillatory pattern can be observed due to the resonating interaction of waves and the breakwater-plate system. Furthermore, the research shows that an ideal width and distance between the breakwater and floating plate can be calculated while assuming an appropriate configuration of the porous structure to create a breakwater-plate system with a reasonable efficiency that possesses both reflection and dissipation properties. The design of various types of coastal structures employed in the marine environment for the reflection and dissipation of wave energy at continental shelves dominated by stratified fluid, described here as a two-layer fluid, will significantly benefit from the findings of the current study.