Scattering of Linear Water Waves by a Fixed and Infinitely Long Rectangular Structure Parallel to a Vertical Wall in Oblique Seas

With the method of separation of variables and the eigenfunction expansion employed, an analytical solution is presented for the radiation and diffraction of a rectangular structure with an opening near a vertical wall in oblique seas, in which the unknown coefficients are determined by the boundary conditions and matching requirement on the interface. The effects of the width of the opening and the angle of incidence on the hydrodynamic characteristics of a rectangular structure with an opening near a vertical wall are mainly studied. The comparisons of the calculation results with wall-present and with wall-absent are also made. The results indicate that the variation trends of the heave added mass and excitation force with wall-present are almost the same as those with wall-absent, and that the peak values in the former case are obviously larger than those in the latter due to the reflection of the vertical wall.

Wave force analysis of the two vertical cylinders by boundary element method

The diffraction of waves by three bottom fixed vertical circular cylinders is investigated by using the boundary element method. This method has been successfully applied to the isolated vertical circular cylinder and now is used to study the interaction between waves and multiple vertical cylinders. In this paper, a numerical analysis by the boundary element method is developed by the linear potential theory. The numerical analysis by the boundary element method is based on Green's second theorem and introduced to an integral equation for the fluid velocity potential around the vertical circular cylinders. To verify this method, the results obtained in present study are compared with the results computed by the multiple scattering method. The results of the comparisons show strong agreement. Also in this paper, several numerical examples are given to illustrate the effects of various parameters on the wave exciting force such are the separation distance, the wave number and the incident wave angle. This numerical computation method might be used broadly for the design of various offshore structures to be constructed in the future.

In this study, the analytical solution for diffraction near a vertical detached breakwater was suggested by superposing the solutions of diffraction near a semi-infinite breakwater suggested previously using linear wave theory. The solutions of wave forces acting on front, lee and composed wave forces on both side were also derived. Relative wave amplitude changed periodically in space owing to the interactions between diffracting waves and standing waves on front side and the interactions between diffracting waves from both tips of a detached breakwater on lee side. The wave forces on a vertical detached breakwater were investigated with monochromatic, uni-directional random and multi-directional random waves. The maximum composed wave force considering the forces on front and lee side reached maximum 1.6 times of wave forces which doesn’t consider diffraction. This value is larger than the maximum composed wave force of semi-infinite breakwater considering diffraction, 1.34 times, which was suggested by Jung et al. (2021). The maximum composed wave forces were calculated in the order of monochromatic, uni-directional random and multi-directional random waves in terms of intensity. It was also found that the maximum wave force of obliquely incident waves was sometimes larger than that of normally incident waves. It can be known that the considerations of diffraction, the composed wave force on both front and lee side and incident wave angle are important from this study.

The water wave interaction with multiple floating thin porous plates with varying porosity placed over an undulating seabed is investigated using linear potential flow theory. The porous plate is modeled based on [“Wave transmission through permeable breakwaters,” Sollitt and Cross Coastal Eng. 1827–1846 (1972)] theory. The undulated bottom is approximated through stepwise discretization, representing it as a sequence of steps and shelves. The model includes several plate configurations governed by a set P of indices, each corresponding to a region containing a porous plate. Further, the porosity variation is incorporated through a piecewise-constant approximation. The problem is formulated assuming the fluid is incompressible and inviscid. By using the method of step approximation and the eigenfunction expansion method, the problem coins to a system of equations, which is solved numerically to compute the values of reflection, transmission coefficients, energy dissipation, and free surface elevation. The model is validated against existing results and through the energy balance relation. The effects of plate arrangements with or without a gap, wave incidence angle, seabed undulation parameters, and different variable porosity patterns (increasing, sinusoidal, constant, and decreasing) are examined. The influence of varying porosity patterns on wave characteristics indicates that a decreasing porosity pattern provides 5% more energy dissipation as compared to a constant porosity pattern. Further comparison among the three types of decreasing porosity configurations highlights their effectiveness, revealing that the quadratic decreasing pattern results in greater dissipation and reduced reflection. Additionally, the study investigates porous plates with intervening gaps, demonstrating that placing the plates with gaps can result in higher dissipation, but only up to a certain gap size. Also, the effect of the optimal number of plates and the placement of plates is also analyzed.

Trapping of oblique surface gravity waves by dual porous barriers near a wall is studied in the presence of step type varying bottom bed that is connected on both sides by water of uniform depths. The porous barriers are assumed to be fixed at a certain distance in front of a vertical rigid wall. Using linear water wave theory and Darcy's law for flow past porous structure, the physical problem is converted into a boundary value problem. Using eigenfunction expansion in the uniform bottom bed region and modified mild-slope equation in the varying bottom bed region, the mathematical problem is handled for solution. Moreover, certain jump conditions are used to account for mass conservation at slope discontinuities in the bottom bed profile. To understand the effect of dual porous barriers in creating tranquility zone and minimum load on the sea wall, reflection coefficient, wave forces acting on the barrier and the wall, and surface wave elevation are computed and analyzed for different values of depth ratio, porous-effect parameter, incident wave angle, gap between the barriers and wall and slope length of undulated bottom. The study reveals that with moderate porosity and suitable gap between barriers and sea wall, using dual barriers an effective wave trapping system can be developed which will exert less wave force on the barriers and the rigid wall. The proposed wave trapping system is likely to be of immense help for protecting various facilities/ infrastructures in coastal environment.

A study, using potential water wave theory, is conducted on the oblique water wave motion over two fixed submerged rectangular blocks (breakwaters) placed over a finite step bottom. We have considered infinite and semi-infinite fluid domains. In both domains, the Fourier expansion method is employed to obtain the velocity potentials explicitly in terms of the infinite Fourier series. The unknown coefficients appearing in the velocity potentials are determined by the eigenfunction expansion matching method at the interfaces. The derived velocity potentials are used to compute the hydrodynamic horizontal and vertical forces acting on the submerged blocks for different values of block thickness, gap spacing between the two blocks, and submergence depth of the upper block from the mean free surface. In addition, the wave load on the vertical wall is computed in the case of the semi-infinite fluid domain for different values of blocks width and the incident wave angle. It is observed that the amplitudes of hydrodynamic forces are negligible for larger values of the wavenumber. Furthermore, the upper block experiences a higher hydrodynamic force than the lower block, regardless of the gap spacing, submergence depth, and block thickness.

The maximum external force acting on a long continuous harbor structure can be reduced by controlling the phase difference of forces acting longitudinally. This strategy can be used to increase the structural stability of breakwaters consisting of caissons. Breakwaters have been developed using interlocking caissons to effectively respond to the constant increase in wave height due to climate change. In this study, we investigated the wave force characteristics and stability of a detached breakwater consisting of open cell caissons interlocked via crushed stones. We performed wave basin experiments and compared the results with analytical solutions of linear diffraction waves. The results revealed that the maximum wave force acting on the front of the breakwater decreased as the incident angle increased, reducing by as much as 79% for an incident angle of 30°. Although the variability of the maximum wave force for each caisson is large owing to the influence of the diffracted waves, the maximum wave force acting on the entire detached breakwater was not significantly affected by this variability. The analytical solutions based on linear wave theory agreed with the experimental results, indicating that the findings can be applied to actual designs. The structural stability of the breakwater was enhanced, even for low incident wave angles, compared to that of a single integral structure, as the frictional resistance produced by the sliding structure increased due to the shear resistance between the filled crushed stones and the rubble mound.

Results of numerical and experimental study are presented of the wave forces acting on a blunt ship advancing in oblique waves. A 3-D panel method using a new numerical scheme proposed one of the present authors for evaluating the Green function was applied to numerical estimation of wave forces in short waves provided that the line integral and the steady disturbance on the body surface can be neglected, and the numerical results are compared with the experimental ones in full detail.The effect of the number of discretized panels on the body surface is investigated by increasing the number of panels from about 400 to 1600 and it is confirmed that the number of panels affects significantly accuracy of computed second order steady forces but the effect is not so remarkable on the first order wave forces.Two different descriptions of the wave term of the Green function, one is the monopole type and the other is the panel type obtained by integrating analytically the monopole over the panel, are also investigated. We confirm that as large number of panels as the monopole type is necessary to obtain the accurate solutions even if the panel type estimation is adopted.The present study confirmed that 3-D panel method is useful to predict the wave forces comparative-ly in long waves, while in short waves it does not give such accurate solutions as be in good agreement with experiments even if the large number of panels is used.

Wave interaction with multiple bottom-standing rectangular porous structures of different structural parameters is numerically modeled using the multidomain boundary element method and the matched eigenfunction expansion method, while assuming linear water wave theory. The sensitivity of wave reflection and transmission to the wave and structural parameters is analyzed with the objective to maximize wave energy attenuation. Different configurations of multiple structures are tested for maximizing the efficiency of dissipation. Furthermore, wave trapping by multiple porous structures near a sloping rigid wall is studied. Bragg resonance is observed in the case of wave scattering by multiple structures irrespective of wave and structural parameters and is found as proportional to the number of structures deployed. In addition, the study reveals that in the presence of multiple structures, due to more wave energy dissipation, wave transmission in the lee side of the structures is reduced significantly when compared with that observed for a single structure.

Analytical and experimental studies on Bragg scattering of water waves by multiple submerged perforated semi-circular breakwaters

Wave pressure on the wet surface of a V-shaped floating breakwater in random seas is investigated. Considering the diffraction effect, the unit velocity potential caused by the single regular waves around the breakwater is solved using the finite-depth Green function and boundary element method, in which the Green function is solved by integral method. The Response-Amplitude Operator (RAO) of wave pressure is acquired according to the Longuet-Higgins' wave model and the linear Bernoulli equation. Furthermore, the wave pressure's response spectrum is calculated according to the wave spectrum by discretizing the frequency domain. The wave pressure's characteristic value corresponding to certain cumulative probability is determined according to the Rayleigh distribution of wave heights. The numerical results and field test results are compared, which indicates that the wave pressure calculated in random seas agrees with that of field measurements. It is found that the bigger angle between legs will cause the bigger pressure response, while the increase in leg length does not influence the pressure significantly. The pressure at the side of head sea is larger than that of back waves. When the incident wave angle changes from 0A degrees to 90A degrees, the pressure at the side of back waves decreases clearly, while at the side of head sea, the situation is more complicated and there seems no obvious tendency. The concentration of wave energy around low frequency (long wavelength) will induce bigger wave pressure, and more attention should be paid to this situation for the structure safety.

This paper presents the development of a "hybrid" method for computing design wave force and moment loading on offshore structures; the method is illustrated by several-examples. The "hybrid" character of the method is due to the combination of linear and nonlinear wave theories to determine wave loading which incorporates the most significant features of a realistic sea system in nature, 1.e., the non1inearities and the directional spectrum. Wave loading due to nonlinear waves with energy present over a continuum of frequencies and directions is represented as the product of a nonlinear wave force and a linearized force transfer coefficient, the latter representing the effect of the directional spectrum. Loading on a single or multiple-pile platform is first calculated from one of a number of nonlinear theories which are valid-for a wave of single fundamental frequency propagating in a single direction. The force transfer function is based on linear wave theory and, in principle, can represent any distribution of the wave energy over frequency and direction. Examples presented include loading on single and multiple pile structures for idealized directional spectra. For the case of a platform supported by four piling on a square array of 60 ft dimensions in plan and spectra for which the princip1e directions associated with each frequency differ substantially, it is found that the linearized force transfer function can result in significantly lower forces (77% of the forces due to unidirectional waves). This case of highly directional waves may be realistic near the center of hurricanes where considerable variability in wind direction occurs over fairly small areas. INTRODUCTION The problem of computing design wave forces on single or multiple piling 11:1 complicated due to the nonlinearity of the waves and the complexity of the random and directional sea surface. Two essentially different approaches have developed in an attempt to establish realistic design wave loading. One approach is to attempt to represent the nonlinearities of the motion for a single fundamental wave characterized by a period and its higher harmonics. A number of such theories have been developed for waves without currents and Dalrymple has extended the stream function theory to include linear and bilinear shear currents. Although more data for wave theory evaluation are always welcome, it has been demonstrated that nonlinear theories are now available which satisfy fairly accurately the mathematical formulation. This approach accounts for the nonlinearities but avoids the random and directional character of the sea surface. The second approach is to exploit the utility of superposition of linear systems by considering the wave system to be composed of an infinite number of random linear waves which, in the general case, can be characterized by a directional spectrum; that is the energy is distributed over a continuum of frequencies and also at each frequency, spread over a range of directions. It is recognized that in design, elements of randomness, directionality and nonlinearity can all be significant.

Recently, a problem, whose solution was well known in exact form, has been analyzed by Liu (Scattering of Water Waves by a Pair of Semi-Infinite Barriers, ASME Journal of Applied Mechanics, Vol. 42, 1975, p. 777), by the method of matched asymptotic expansions. From the known exact solution a simple expression is obtained for the transmission coefficient. The exact expression for the transmission coefficient when expanded for low frequency incident waves differs from Liu’s result, and therefore casts doubt on Liu’s analysis and physical conclusions.

Study on modulation of near infrared radiation based on plasma photonic crystal